What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

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1 CHAPTER Limits an Their Properties Section STUDY TIP As ou progress through this course, remember that learning calculus is just one of our goals Your most important goal is to learn how to use calculus to moel an solve real-life problems Here are a few problemsolving strategies that ma help ou Be sure ou unerstan the question What is given? What are ou aske to fin? Outline a plan There are man approaches ou coul use: look for a pattern, solve a simpler problem, work backwars, raw a iagram, use technolog, or an of man other approaches Complete our plan Be sure to answer the question Verbalize our answer For eample, rather than writing the answer as 6, it woul be better to write the answer as The area of the region is 6 square meters Look back at our work Does our answer make sense? Is there a wa ou can check the reasonableness of our answer? A Preview of Calculus Unerstan what calculus is an how it compares with precalculus Unerstan that the tangent line problem is basic to calculus Unerstan that the area problem is also basic to calculus What Is Calculus? Calculus is the mathematics of change velocities an accelerations Calculus is also the mathematics of tangent lines, slopes, areas, volumes, arc lengths, centrois, curvatures, an a variet of other concepts that have enable scientists, engineers, an economists to moel real-life situations Although precalculus mathematics also eals with velocities, accelerations, tangent lines, slopes, an so on, there is a funamental ifference between precalculus mathematics an calculus Precalculus mathematics is more static, whereas calculus is more namic Here are some eamples An object traveling at a constant velocit can be analze with precalculus mathematics To analze the velocit of an accelerating object, ou nee calculus The slope of a line can be analze with precalculus mathematics To analze the slope of a curve, ou nee calculus A tangent line to a circle can be analze with precalculus mathematics To analze a tangent line to a general graph, ou nee calculus The area of a rectangle can be analze with precalculus mathematics To analze the area uner a general curve, ou nee calculus Each of these situations involves the same general strateg the reformulation of precalculus mathematics through the use of a it process So, one wa to answer the question What is calculus? is to sa that calculus is a it machine that involves three stages The first stage is precalculus mathematics, such as the slope of a line or the area of a rectangle The secon stage is the it process, an the thir stage is a new calculus formulation, such as a erivative or integral Precalculus mathematics Limit process Calculus GRACE CHISHOLM YOUNG (868 9) Grace Chisholm Young receive her egree in mathematics from Girton College in Cambrige, Englan Her earl work was publishe uner the name of William Young, her husban Between 9 an 96, Grace Young publishe work on the founations of calculus that won her the Gamble Prize from Girton College Some stuents tr to learn calculus as if it were simpl a collection of new formulas This is unfortunate If ou reuce calculus to the memorization of ifferentiation an integration formulas, ou will miss a great eal of unerstaning, self-confience, an satisfaction On the following two pages some familiar precalculus concepts couple with their calculus counterparts are liste Throughout the tet, our goal shoul be to learn how precalculus formulas an techniques are use as builing blocks to prouce the more general calculus formulas an techniques Don t worr if ou are unfamiliar with some of the ol formulas liste on the following two pages ou will be reviewing all of them As ou procee through this tet, come back to this iscussion repeatel Tr to keep track of where ou are relative to the three stages involve in the stu of calculus For eample, the first three chapters break own as shown Chapter P: Preparation for Calculus Chapter : Limits an Their Properties Chapter : Differentiation Precalculus Limit process Calculus

2 SECTION A Preview of Calculus Without Calculus With Differential Calculus Value of f = f() Limit of f as when c approaches c c c = f() Slope of a line Slope of a curve Secant line to a curve Tangent line to a curve Average rate of change between t a an t b Instantaneous t = a t = b rate of change at t c t = c Curvature of a circle Curvature of a curve Height of a curve when c c Maimum height of a curve on an interval a b Tangent plane to a sphere Tangent plane to a surface Direction of motion along a line Direction of motion along a curve

3 CHAPTER Limits an Their Properties Without Calculus With Integral Calculus Area of a rectangle Area uner a curve Work one b a constant force Work one b a variable force Center of a rectangle Centroi of a region Length of a line segment Length of an arc Surface area of a cliner Surface area of a soli of revolution Mass of a soli of constant ensit Mass of a soli of variable ensit Volume of a rectangular soli Volume of a region uner a surface Sum of a finite number of terms a a a n S Sum of an infinite number of terms a a a S

4 SECTION A Preview of Calculus 5 = f() Tangent line P The tangent line to the graph of f at P Figure Vieo The Tangent Line Problem The notion of a it is funamental to the stu of calculus The following brief escriptions of two classic problems in calculus the tangent line problem an the area problem shoul give ou some iea of the wa its are use in calculus In the tangent line problem, ou are given a function f an a point P on its graph an are aske to fin an equation of the tangent line to the graph at point P, as shown in Figure Ecept for cases involving a vertical tangent line, the problem of fining the tangent line at a point P is equivalent to fining the slope of the tangent line at P You can approimate this slope b using a line through the point of tangenc an a secon point on the curve, as shown in Figure (a) Such a line is calle a secant line If Pc, f c is the point of tangenc an Qc, fc is a secon point on the graph of points is given b f, the slope of the secant line through these two m sec f c f c c c f c f c Q(c +, f(c + )) P(c, f(c)) f(c + ) f(c) Q Secant lines P Tangent line (a) The secant line through c, f c an c, fc Figure Animation (b) As Q approaches P, the secant lines approach the tangent line As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line, as shown in Figure (b) When such a iting position eists, the slope of the tangent line is sai to be the it of the slope of the secant line (Much more will be sai about this important problem in Chapter ) EXPLORATION The following points lie on the graph of f Q 5, f5, Q, f, Q 0, f 0, Q 00, f00, Q 5 000, f000 Each successive point gets closer to the point P, Fin the slope of the secant line through Q an P, Q an P, an so on Graph these secant lines on a graphing utilit Then use our results to estimate the slope of the tangent line to the graph of f at the point P

5 6 CHAPTER Limits an Their Properties a Area uner a curve Figure Vieo = f() b The Area Problem In the tangent line problem, ou saw how the it process can be applie to the slope of a line to fin the slope of a general curve A secon classic problem in calculus is fining the area of a plane region that is boune b the graphs of functions This problem can also be solve with a it process In this case, the it process is applie to the area of a rectangle to fin the area of a general region As a simple eample, consier the region boune b the graph of the function f, the -ais, an the vertical lines a an b, as shown in Figure You can approimate the area of the region with several rectangular regions, as shown in Figure As ou increase the number of rectangles, the approimation tens to become better an better because the amount of area misse b the rectangles ecreases Your goal is to etermine the it of the sum of the areas of the rectangles as the number of rectangles increases without boun = f() = f() HISTORICAL NOTE In one of the most astouning events ever to occur in mathematics, it was iscovere that the tangent line problem an the area problem are closel relate This iscover le to the birth of calculus You will learn about the relationship between these two problems when ou stu the Funamental Theorem of Calculus in Chapter a b a b Approimation using four rectangles Figure Approimation using eight rectangles Animation EXPLORATION Consier the region boune b the graphs of f, 0, an, as shown in part (a) of the figure The area of the region can be approimate b two sets of rectangles one set inscribe within the region an the other set circumscribe over the region, as shown in parts (b) an (c) Fin the sum of the areas of each set of rectangles Then use our results to approimate the area of the region f() = f() = f() = (a) Boune region (b) Inscribe rectangles (c) Circumscribe rectangles

6 SECTION A Preview of Calculus 7 Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 6, ecie whether the problem can be solve using precalculus, or whether calculus is require If the problem can be solve using precalculus, solve it If the problem seems to require calculus, eplain our reasoning an use a graphical or numerical approach to estimate the solution Fin the istance travele in 5 secons b an object traveling at a constant velocit of 0 feet per secon Fin the istance travele in 5 secons b an object moving with a velocit of vt 0 7 cos t feet per secon A bicclist is riing on a path moele b the function f 008, where an f are measure in miles Fin the rate of change of elevation when Figure for Figure for A bicclist is riing on a path moele b the function f 008, where an f are measure in miles Fin the rate of change of elevation when 5 Fin the area of the shae region 5 (, ) (0, 0) ( ) f() = (5, 0) 5 6 Figure for 5 Figure for 6 6 Fin the area of the shae region 7 Secant Lines Consier the function f an the point P, on the graph of f (a) Graph f an the secant lines passing through P, an Q, f for -values of, 5, an 05 (b) Fin the slope of each secant line (c) Use the results of part (b) to estimate the slope of the tangent line of f at P, Describe how to improve our approimation of the slope 8 Secant Lines Consier the function f an the point P, on the graph of f (a) Graph f an the secant lines passing through P, an Q, f for -values of,, an 5 (b) Fin the slope of each secant line f() = (c) Use the results of part (b) to estimate the slope of the tangent line of f at P, Describe how to improve our approimation of the slope 9 (a) Use the rectangles in each graph to approimate the area of the region boune b 5, 0,, an 5 5 (b) Describe how ou coul continue this process to obtain a more accurate approimation of the area 0 (a) Use the rectangles in each graph to approimate the area of the region boune b sin, 0, 0, an 5 (b) Describe how ou coul continue this process to obtain a more accurate approimation of the area Writing About Concepts Consier the length of the graph of f 5 from, 5 to 5, 5 (, 5) (5, ) 5 (a) Approimate the length of the curve b fining the istance between its two enpoints, as shown in the first figure (b) Approimate the length of the curve b fining the sum of the lengths of four line segments, as shown in the secon figure (c) Describe how ou coul continue this process to obtain a more accurate approimation of the length of the curve 5 (, 5) (5, )

7 8 CHAPTER Limits an Their Properties Section Fining Limits Graphicall an Numericall Estimate a it using a numerical or graphical approach Learn ifferent was that a it can fail to eist Stu an use a formal efinition of it Vieo f() = (, ) An Introuction to Limits Suppose ou are aske to sketch the graph of the function f given b f, For all values other than, ou can use stanar curve-sketching techniques However, at, it is not clear what to epect To get an iea of the behavior of the graph of f near, ou can use two sets of -values one set that approaches from the left an one set that approaches from the right, as shown in the table approaches from the left approaches from the right f ? f() = The it of f as approaches is Figure 5 Animation Animation f approaches f approaches The graph of f is a parabola that has a gap at the point,, as shown in Figure 5 Although cannot equal, ou can move arbitraril close to, an as a result f moves arbitraril close to Using it notation, ou can write f This is rea as the it of f as approaches is This iscussion leas to an informal escription of a it If f becomes arbitraril close to a single number L as approaches c from either sie, the it of f, as approaches c, is L This it is written as f L c EXPLORATION The iscussion above gives an eample of how ou can estimate a it numericall b constructing a table an graphicall b rawing a graph Estimate the following it numericall b completing the table f ????????? Then use a graphing utilit to estimate the it graphicall

8 SECTION Fining Limits Graphicall an Numericall 9 f is unefine at = 0 EXAMPLE Estimating a Limit Numericall Evaluate the function at several points near 0 an use the results to estimate the it 0 f Solution The table lists the values of f for several -values near 0 f() = + approaches 0 from the left approaches 0 from the right f ? f approaches f approaches The it of f as approaches 0 is Figure 6 From the results shown in the table, ou can estimate the it to be This it is reinforce b the graph of f (see Figure 6) Eitable Graph Tr It Eploration A Eploration B In Eample, note that the function is unefine at 0 an et f () appears to be approaching a it as approaches 0 This often happens, an it is important to realize that the eistence or noneistence of f at c has no bearing on the eistence of the it of f as approaches c EXAMPLE Fining a Limit Fin the it of f as approaches where f is efine as f, 0, f() =, 0, The it of f as approaches is Figure 7 Solution Because f for all other than, ou can conclue that the it is, as shown in Figure 7 So, ou can write f The fact that f 0 has no bearing on the eistence or value of the it as approaches For instance, if the function were efine as f,, the it woul be the same Eitable Graph Tr It Eploration A So far in this section, ou have been estimating its numericall an graphicall Each of these approaches prouces an estimate of the it In Section, ou will stu analtic techniques for evaluating its Throughout the course, tr to evelop a habit of using this three-pronge approach to problem solving Numerical approach Construct a table of values Graphical approach Draw a graph b han or using technolog Analtic approach Use algebra or calculus

9 50 CHAPTER Limits an Their Properties Limits That Fail to Eist In the net three eamples ou will eamine some its that fail to eist EXAMPLE Behavior That Differs from the Right an Left oes not eist f 0 Figure 8 δ δ f() = Eitable Graph f() = f() = Show that the it oes not eist 0 Solution Consier the graph of the function From Figure 8, ou can see that for positive -values, an for negative -values < 0, This means that no matter how close gets to 0, there will be both positive an negative -values that iel f an f Specificall, if (the lowercase Greek letter elta) is a positive number, then for -values satisfing the inequalit 0 < <, ou can classif the values of as shown, 0 > 0 0, f Negative -values Positive -values iel This implies that the it oes not eist iel Tr It Eploration A Eploration B EXAMPLE Unboune Behavior Discuss the eistence of the it 0 f() = Solution Let f In Figure 9, ou can see that as approaches 0 from either the right or the left, f increases without boun This means that b choosing close enough to 0, ou can force f to be as large as ou want For instance, f ) will be larger than 00 if ou choose that is within of 0 That is, 0 < < 0 f > 00 0 Similarl, ou can force f to be larger than,000,000, as follows f oes not eist 0 Figure 9 0 < < 000 f >,000,000 Because f is not approaching a real number L as approaches 0, ou can conclue that the it oes not eist Eitable Graph Tr It Eploration A Eploration B

10 SECTION Fining Limits Graphicall an Numericall 5 EXAMPLE 5 Oscillating Behavior f() = sin Discuss the eistence of the it Solution Let f sin In Figure 0, ou can see that as approaches 0, f oscillates between an So, the it oes not eist because no matter how small ou choose, it is possible to choose an within units of 0 such that sin an sin, as shown in the table sin / 5 sin Limit oes not eist 0 f oes not eist 0 Figure 0 Eitable Graph Tr It Eploration A Open Eploration Common Tpes of Behavior Associate with Noneistence of a Limit f approaches a ifferent number from the right sie of c than it approaches from the left sie f increases or ecreases without boun as approaches c f oscillates between two fie values as approaches c There are man other interesting functions that have unusual it behavior An often cite one is the Dirichlet function f 0,, if is rational if is irrational Because this function has no it at an real number c, it is not continuous at an real number c You will stu continuit more closel in Section TECHNOLOGY PITFALL When ou use a graphing utilit to investigate the behavior of a function near the -value at which ou are tring to evaluate a it, remember that ou can t alwas trust the pictures that graphing utilities raw If ou use a graphing utilit to graph the function in Eample 5 over an interval containing 0, ou will most likel obtain an incorrect graph such as that shown in Figure The reason that a graphing utilit can t show the correct graph is that the graph has infinitel man oscillations over an interval that contains 0 PETER GUSTAV DIRICHLET ( ) In the earl evelopment of calculus, the efinition of a function was much more restricte than it is toa, an functions such as the Dirichlet function woul not have been consiere The moern efinition of function was given b the German mathematician Peter Gustav Dirichlet MathBio Incorrect graph of Figure f sin

11 5 CHAPTER Limits an Their Properties L + ε L L ε (c, L) c + δ c c δ The - efinition of the it of f as approaches c Figure A Formal Definition of Limit Let s take another look at the informal escription of a it If f becomes arbitraril close to a single number L as approaches c from either sie, then the it of f as approaches c is L, written as At first glance, this escription looks fairl technical Even so, it is informal because eact meanings have not et been given to the two phrases an f L c f becomes arbitraril close to L approaches c The first person to assign mathematicall rigorous meanings to these two phrases was Augustin-Louis Cauch His - efinition of it is the stanar use toa In Figure, let (the lowercase Greek letter epsilon) represent a (small) positive number Then the phrase f becomes arbitraril close to L means that f lies in the interval L, L Using absolute value, ou can write this as f L < Similarl, the phrase approaches c means that there eists a positive number such that lies in either the interval c, c or the interval c, c This fact can be concisel epresse b the ouble inequalit 0 < c < The first inequalit 0 < c The istance between an c is more than 0 epresses the fact that c The secon inequalit c < is within sas that is within a istance of c units of c FOR FURTHER INFORMATION For more on the introuction of rigor to calculus, see Who Gave You the Epsilon? Cauch an the Origins of Rigorous Calculus b Juith V Grabiner in The American Mathematical Monthl MathArticle NOTE Definition of Limit Let f be a function efine on an open interval containing c (ecept possibl at c) an let L be a real number The statement f L c means that for each > 0 there eists a > 0 such that if 0 < c < Throughout this tet, the epression f L c, then f L < implies two statements the it eists an the it is L Some functions o not have its as c, but those that o cannot have two ifferent its as c That is, if the it of a function eists, it is unique (see Eercise 69)

12 SECTION Fining Limits Graphicall an Numericall 5 - The net three eamples shoul help ou evelop a better unerstaning of the efinition of it = 0 = = 099 EXAMPLE 6 Fining a for a Given Given the it = 995 = = 005 f() = 5 The it of f as approaches is Figure 5 fin such that 5 < 00 whenever Solution In this problem, ou are working with a given value of namel, 00 To fin an appropriate, notice that 5 6 Because the inequalit ou can choose 0 < < 0005 implies that 5 as shown in Figure 5 < 00 is equivalent to This choice works because < < < < 00, Tr It Eploration A Eploration B NOTE In Eample 6, note that 0005 is the largest value of that will guarantee 5 < 00 whenever 0 < < An smaller positive value of woul also work In Eample 6, ou foun a -value for a given This oes not prove the eistence of the it To o that, ou must prove that ou can fin a for an, as shown in the net eample = + ε = = ε = + δ = = δ f() = The it of f as approaches is Figure EXAMPLE 7 Using the - Definition of Limit Use the - efinition of it to prove that Solution You must show that for each > 0, there eists a > 0 such that < whenever 0 < < Because our choice of epens on, ou nee to establish a connection between the absolute values an 6 So, for a given > 0 ou can choose This choice works because 0 < < implies that < as shown in Figure Tr It Eploration A

13 5 CHAPTER Limits an Their Properties EXAMPLE 8 Using the - Definition of Limit f() = + ε ( + δ) ( δ) ε + δ δ The it of f as approaches is Figure 5 Use the - efinition of it to prove that Solution You must show that for each > 0, there eists a > 0 such that < whenever To fin an appropriate, begin b writing For all in the interval,, ou know that So, letting be the minimum of 5 an, it follows that, whenever 0 < < 5 <, ou have < 5 5 as shown in Figure 5 0 < < Tr It Eploration A Throughout this chapter ou will use the efinition of it primaril to prove theorems about its an to establish the eistence or noneistence of particular tpes of its For fining its, ou will learn techniques that are easier to use than the efinition of it - -

14 5 CHAPTER Limits an Their Properties Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 8, complete the table an use the result to estimate the it Use a graphing utilit to graph the function to confirm our result 5 f f f f sin 0 0 f f cos 8 0 f f

15 SECTION Fining Limits Graphicall an Numericall 55 In Eercises 9 8, use the graph to fin the it (if it eists) If the it oes not eist, eplain wh 9 0 f sin 6 f, 0, f sec 0 f,, In Eercises 9 an 0, use the graph of the function f to ecie whether the value of the given quantit eists If it oes, fin it If not, eplain wh 9 (a) () 0 (a) (h) In Eercises an, use the graph of f to ientif the values of c for which f eists f (b) f (c) f f f (b) f (c) f 0 () f 0 (e) f (f ) f (g) f f 6 6 c cos 8 0 tan In Eercises an, sketch the graph of f Then ientif the values of c for which f eists f, 8,, c sin, f cos, cos, < < < 0 0 >

16 56 CHAPTER Limits an Their Properties In Eercises 5 an 6, sketch a graph of a function f that satisfies the given values (There are man correct answers) 5 f 0 is unefine 6 f 0 f 0 f 6 f t C 5 6 7? f oes not eist 7 Moeling Data The cost of a telephone call between two cities is $075 for the first minute an $050 for each aitional minute or fraction thereof A formula for the cost is given b Ct t where t is the time in minutes Note: greatest integer n such that n For eample, an 6 (a) Use a graphing utilit to graph the cost function for 0 < t 5 (b) Use the graph to complete the table an observe the behavior of the function as t approaches 5 Use the graph an the table to fin C t t5 f 0 f 0 (c) Use the graph to complete the table an observe the behavior of the function as t approaches 0 The graph of f is shown in the figure Fin such that if f < 00 0 < < The graph of f is shown in the figure Fin such that if f < 0 0 < < = = = 09 The graph of f f then then t C 5 9 5? is shown in the figure Fin such that if f < 0 0 < < then Does the it of Ct as t approaches eist? Eplain 8 Repeat Eercise 7 for Ct 05 0t 9 The graph of f is shown in the figure Fin such that if then 0 < < f < In Eercises 6, fin the it L Then fin > 0 such that f L < 00 whenever 0 < c < f = = =

17 SECTION Fining Limits Graphicall an Numericall 57 In Eercises 7 8, fin the it Then use the - efinition to prove that the it is L Writing In Eercises 9 5, use a graphing utilit to graph the function an estimate the it (if it eists) What is the omain of the function? Can ou etect a possible error in etermining the omain of a function solel b analzing the graph generate b a graphing utilit? Write a short paragraph about the importance of eamining a function analticall as well as graphicall f f ) f f f 9 f 9 f 9 f Writing About Concepts 5 Write a brief escription of the meaning of the notation f If f, can ou conclue anthing about the it of f as approaches? Eplain our reasoning 55 If the it of f as approaches is, can ou conclue anthing about f? Eplain our reasoning L Writing About Concepts (continue) 56 Ientif three tpes of behavior associate with the noneistence of a it Illustrate each tpe with a graph of a function 57 Jewelr A jeweler resizes a ring so that its inner circumference is 6 centimeters (a) What is the raius of the ring? (b) If the ring s inner circumference can var between 55 centimeters an 65 centimeters, how can the raiu var? (c) Use the - efinition of it to escribe this situation Ientif an 58 Sports A sporting goos manufacturer esigns a golf bal having a volume of 8 cubic inches (a) What is the raius of the golf ball? (b) If the ball s volume can var between 5 cubic inches an 5 cubic inches, how can the raius var? (c) Use the - efinition of it to escribe this situation Ientif an 59 Consier the function f Estimate the it 0 b evaluating f at -values near 0 Sketch the graph of f 60 Consier the function f Estimate 0 b evaluating f at -values near 0 Sketch the graph of f 6 Graphical Analsis The statement means that for each > 0 there correspons a > 0 such tha if 0 < <, then < If 000, then < 000 Use a graphing utilit to graph each sie of this inequalit Use the zoom feature to fin an interval, such tha the graph of the left sie is below the graph of the right sie o the inequalit

18 58 CHAPTER Limits an Their Properties 6 Graphical Analsis The statement means that for each > 0 there correspons a > 0 such that if 0 < <, then < If 000, then < 000 Use a graphing utilit to graph each sie of this inequalit Use the zoom feature to fin an interval, such that the graph of the left sie is below the graph of the right sie of the inequalit True or False? In Eercises 6 66, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 6 If f is unefine at c, then the it of f as approaches c oes not eist 6 If the it of f as approaches c is 0, then there must eist a number k such that f k < If f c L, then f L c 66 If f L, then f c L c 67 Consier the function f (a) Is (b) Is a true statement? Eplain 0 a true statement? Eplain 68 Writing The efinition of it on page 5 requires that f is a function efine on an open interval containing c, ecept possibl at c Wh is this requirement necessar? 69 Prove that if the it of f as c eists, then the it must be unique [Hint: Let f L c an an prove that L L ] f L c 70 Consier the line f m b, where m 0 Use the efinition of it to prove that f mc b c 7 Prove that f L is equivalent to f L 0 c c - 7 (a) Given that prove that there eists an open interval a, b containing 0 such that 00 > 0 for all 0 in a, b (b) Given that g L, where L > 0, prove that there c eists an open interval a, b containing c such tha g > 0 for all c in a, b 7 Programming Use the programming capabilities of a graphing utilit to write a program for approimating f c Assume the program will be applie onl to functions whose its eist as approaches c Let f an generate two lists whose entries form the orere pairs c ± 0 n, f c ± 0 n for n 0,,,, an 7 Programming Use the program ou create in Eercise 7 to approimate the it Putnam Eam Challenge 75 Inscribe a rectangle of base b an height h an an isosceles triangle of base b in a circle of raius one as shown For wha value of h o the rectangle an triangle have the same area? h b 76 A right circular cone has base of raius an height A cube is inscribe in the cone so that one face of the cube is containe in the base of the cone What is the sie-length of the cube? These problems were compose b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserve

19 SECTION Evaluating Limits Analticall 59 Vieo Section Evaluating Limits Analticall Evaluate a it using properties of its Develop an use a strateg for fining its Evaluate a it using iviing out an rationalizing techniques Evaluate a it using the Squeeze Theorem Properties of Limits In Section, ou learne that the it of f as approaches c oes not epen on the value of f at c It ma happen, however, that the it is precisel fc In such cases, the it can be evaluate b irect substitution That is, f fc c Substitute c for Such well-behave functions are continuous at c You will eamine this concept more closel in Section f(c) = THEOREM Some Basic Limits c + ε f(c) = c ε = δ Let b an c be real numbers an let n be a positive integer b b c c n c n c c ε = δ c ε c δ Figure 6 c c + δ Proof To prove Propert of Theorem, ou nee to show that for each > 0 there eists a > 0 such that c < whenever 0 < c < To o this, choose The secon inequalit then implies the first, as shown in Figure 6 This completes the proof (Proofs of the other properties of its in this section are liste in Appeni A or are iscusse in the eercises) NOTE When ou encounter new notations or smbols in mathematics, be sure ou know how the notations are rea For instance, the it in Eample (c) is rea as the it of as approaches is EXAMPLE Evaluating Basic Limits a b c Tr It Eploration A The eitable graph feature allows ou to eit the graph of a function to visuall evaluate the it as approaches c a Eitable Graph b Eitable Graph c Eitable Graph THEOREM Properties of Limits Let b an c be real numbers, let n be a positive integer, an let f an g be functions with the following its f L c Scalar multiple: Sum or ifference: Prouct: an bf bl c c c g K c f ± g L ± K fg LK f Quotient: provie K 0 c g L K, 5 Power: c fn L n

20 60 CHAPTER Limits an Their Properties EXAMPLE The Limit of a Polnomial 9 Propert Propert Eample Simplif Tr It Eploration A The eitiable graph feature allows ou to eit the graph of a function to visuall evaluate the it as approaches c Eitable Graph In Eample, note that the it (as ) of the polnomial function p is simpl the value of p at p p 9 This irect substitution propert is vali for all polnomial an rational functions with nonzero enominators THEOREM Limits of Polnomial an Rational Functions If p is a polnomial function an c is a real number, then p pc c If r is a rational function given b r pq an c is a real number such that qc 0, then pc r rc c qc EXAMPLE The Limit of a Rational Function Fin the it: Solution Because the enominator is not 0 when, ou can appl Theorem to obtain Tr It Eploration A The eitiable graph feature allows ou to eit the graph of a function to visuall evaluate the it as approaches c Eitable Graph THE SQUARE ROOT SYMBOL The first use of a smbol to enote the square root can be trace to the siteenth centur Mathematicians first use the smbol, which ha onl two strokes This smbol was chosen because it resemble a lowercase r, to stan for the Latin wor rai, meaning root Vieo Vieo Polnomial functions an rational functions are two of the three basic tpes of algebraic functions The following theorem eals with the it of the thir tpe of algebraic function one that involves a raical See Appeni A for a proof of this theorem THEOREM The Limit of a Function Involving a Raical Let n be a positive integer The following it is vali for all c if n is o, an is vali for c > 0 if n is even n c n c

21 SECTION Evaluating Limits Analticall 6 The following theorem greatl epans our abilit to evaluate its because it shows how to analze the it of a composite function See Appeni A for a proof of this theorem THEOREM 5 The Limit of a Composite Function If f an g are functions such that g L an f fl, then c fg f c g c f L L EXAMPLE The Limit of a Composite Function a Because it follows that b Because it follows that 0 8 an an 8 Tr It Eploration A Open Eploration The eitable graph feature allows ou to eit the graph of a function to visuall evaluate the it as approaches c a Eitable Graph b Eitable Graph You have seen that the its of man algebraic functions can be evaluate b irect substitution The si basic trigonometric functions also ehibit this esirable qualit, as shown in the net theorem (presente without proof) THEOREM 6 Limits of Trigonometric Functions Let c be a real number in the omain of the given trigonometric function sin sin c c tan tan c c cos cos c c cot cot c c 5 sec sec c 6 csc csc c c c EXAMPLE 5 Limits of Trigonometric Functions a b tan tan0 0 0 cos cos cos c 0 sin 0 sin 0 0 Tr It Eploration A

22 6 CHAPTER Limits an Their Properties f() = A Strateg for Fining Limits On the previous three pages, ou stuie several tpes of functions whose its can be evaluate b irect substitution This knowlege, together with the following theorem, can be use to evelop a strateg for fining its A proof of this theorem is given in Appeni A THEOREM 7 Functions That Agree at All But One Point Let c be a real number an let f g for all c in an open interval containing c If the it of g as approaches c eists, then the it of f also eists an f g c c EXAMPLE 6 Fining the Limit of a Function Eitable Graph f an g agree at all but one point Eitable Graph Figure 7 g() = + + Fin the it: Solution Let f B factoring an iviing out like factors, ou can rewrite f as f So, for all -values other than, the functions f an g agree, as shown in Figure 7 Because g eists, ou can appl Theorem 7 to conclue that f an g have the same it at g, Factor Divie out like factors Appl Theorem 7 Use irect substitution Simplif Tr It Eploration A Eploration B Eploration C Eploration D STUDY TIP When appling this strateg for fining a it, remember that some functions o not have a it (as approaches c) For instance, the following it oes not eist A Strateg for Fining Limits Learn to recognize which its can be evaluate b irect substitution (These its are liste in Theorems through 6) If the it of f as approaches c cannot be evaluate b irect substitution, tr to fin a function g that agrees with f for all other than c [Choose g such that the it of g can be evaluate b irect substitution] Appl Theorem 7 to conclue analticall that f g gc c c Use a graph or table to reinforce our conclusion

23 SECTION Evaluating Limits Analticall 6 Diviing Out an Rationalizing Techniques Two techniques for fining its analticall are shown in Eamples 7 an 8 The first technique involves iviing out common factors, an the secon technique involves rationalizing the numerator of a fractional epression EXAMPLE 7 Diviing Out Technique Fin the it: 6 (, 5) 5 f is unefine when Figure 8 Eitable Graph NOTE In the solution of Eample 7, be sure ou see the usefulness of the Factor Theorem of Algebra This theorem states that if c is a zero of a polnomial function, c is a factor of the polnomial So, if ou appl irect substitution to a rational function an obtain rc pc qc 0 0 ou can conclue that c must be a common factor to both p an q f() = Solution Although ou are taking the it of a rational function, ou cannot appl Theorem because the it of the enominator is 0 6 Direct substitution fails Because the it of the numerator is also 0, the numerator an enominator have a common factor of So, for all, ou can ivie out this factor to obtain f 6 Using Theorem 7, it follows that g, Appl Theorem 7 Use irect substitution This result is shown graphicall in Figure 8 Note that the graph of the function f coincies with the graph of the function g, ecept that the graph of f has a gap at the point, 5 Tr It Eploration A Open Eploration In Eample 7, irect substitution prouce the meaningless fractional form 00 An epression such as 00 is calle an ineterminate form because ou cannot (from the form alone) etermine the it When ou tr to evaluate a it an encounter this form, remember that ou must rewrite the fraction so that the new enominator oes not have 0 as its it One wa to o this is to ivie out like factors, as shown in Eample 7 A secon wa is to rationalize the numerator, as shown in Eample 8 δ Incorrect graph of Figure 9 f 5 + ε + δ Glitch near (, 5) 5 ε TECHNOLOGY PITFALL f 6 an Because the graphs of g iffer onl at the point, 5, a stanar graphing utilit setting ma not istinguish clearl between these graphs However, because of the piel configuration an rouning error of a graphing utilit, it ma be possible to fin screen settings that istinguish between the graphs Specificall, b repeatel zooming in near the point, 5 on the graph of f, our graphing utilit ma show glitches or irregularities that o not eist on the actual graph (See Figure 9) B changing the screen settings on our graphing utilit ou ma obtain the correct graph of f

24 6 CHAPTER Limits an Their Properties EXAMPLE 8 Rationalizing Technique Fin the it: 0 Solution B irect substitution, ou obtain the ineterminate form Direct substitution fails 0 0 f() = + In this case, ou can rewrite the fraction b rationalizing the numerator, 0 Now, using Theorem 7, ou can evaluate the it as shown 0 0 The it of f as approaches 0 is Figure 0 A table or a graph can reinforce our conclusion that the it is (See Figure 0) Eitable Graph approaches 0 from the left approaches 0 from the right f ? f approaches 05 f approaches 05 Tr It Eploration A Eploration B Eploration C NOTE The rationalizing technique for evaluating its is base on multiplication b a convenient form of In Eample 8, the convenient form is

25 SECTION Evaluating Limits Analticall 65 f g h h() f() g() The Squeeze Theorem Figure f lies in here c h g f The Squeeze Theorem The net theorem concerns the it of a function that is squeeze between two other functions, each of which has the same it at a given -value, as shown in Figure (The proof of this theorem is given in Appeni A) THEOREM 8 The Squeeze Theorem If h f g for all in an open interval containing c, ecept possibl at c itself, an if h L g c c then f eists an is equal to L c Vieo You can see the usefulness of the Squeeze Theorem in the proof of Theorem 9 THEOREM 9 Two Special Trigonometric Limits sin cos (cos θ, sin θ) (, tan θ) Proof To avoi the confusion of two ifferent uses of, the proof is presente using the variable, where is an acute positive angle measure in raians Figure shows a circular sector that is squeeze between two triangles FOR FURTHER INFORMATION For more information on the function f sin, see the article The Function sin b William B Gearhart an Harris S Shultz in The College Mathematics Journal θ (, 0) A circular sector is use to prove Theorem 9 Figure MathArticle θ Area of triangle Area of sector Area of triangle tan sin Multipling each epression b sin prouces cos sin an taking reciprocals an reversing the inequalities iels cos sin tan θ θ Because cos cos an sin sin, ou can conclue that this inequalit is vali for all nonzero in the open interval, Finall, because cos an, ou can appl the Squeeze Theorem to 0 0 conclue that sin The proof of the secon it is left as an eercise (see 0 Eercise 0) θ sin θ

26 66 CHAPTER Limits an Their Properties EXAMPLE 9 A Limit Involving a Trigonometric Function Fin the it: tan 0 f() = tan The it of f as approaches 0 is Figure Solution Direct substitution iels the ineterminate form 00 To solve this problem, ou can write tan as sin cos an obtain tan 0 0 sin cos Now, because sin 0 ou can obtain (See Figure ) an 0 tan 0 sin 0 0 cos cos Eitable Graph Tr It EXAMPLE 0 Eploration A A Limit Involving a Trigonometric Function Fin the it: 0 sin g() = sin 6 The it of g as approaches 0 is Figure Solution Direct substitution iels the ineterminate form 00 To solve this problem, ou can rewrite the it as sin 0 sin 0 Multipl an ivie b Now, b letting an observing that 0 if an onl if 0, ou can write sin 0 sin 0 (See Figure ) sin 0 Eitable Graph Tr It Eploration A TECHNOLOGY Use a graphing utilit to confirm the its in the eamples an eercise set For instance, Figures an show the graphs of f tan an g sin Note that the first graph appears to contain the point 0, an the secon graph appears to contain the point 0,, which lens support to the conclusions obtaine in Eamples 9 an 0

27 SECTION Evaluating Limits Analticall 67 Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises, use a graphing utilit to graph the function an visuall estimate the its h 5 g (a) (b) f cos (a) (b) In Eercises 5, fin the it h 5 h f 0 f (a) (b) (a) (b) 0 9 g g f t t t ft t ft t tan 6 In Eercises 7 0, use the information to evaluate the its 7 f 8 c c (a) c (b) c (c) () 9 f 0 c (a) (b) (c) () g c c c f c f c f c 5g f g f g f g f (c) () f 7 c (a) f In Eercises, use the graph to etermine the it visuall (if it eists) Write a simpler function that agrees with the given function at all but one point sec 7 6 f c g c (a) c (b) c (b) (c) () c c c c c f c f f g f g f g f 8 f h g In Eercises 6, fin the its f 5, g (a) f (b) g (c) f 7, g (a) f (b) g (c) 5 f, g (a) f (b) g (c) 6 f, g 6 (a) f (b) g (c) In Eercises 7 6, fin the it of the trigonometric function 7 sin 8 tan 9 cos 0 sec cos 0 sin 56 sin 5 cos g f g f g f g f (a) (b) (a) (b) g 0 g g g (a) (b) g f (a) 5 h h 0 f (b) 0 f

28 68 CHAPTER Limits an Their Properties In Eercises 5 8, fin the it of the function (if it eists) Write a simpler function that agrees with the given function at all but one point Use a graphing utilit to confirm our result In Eercises 9 6, fin the it (if it eists) Graphical, Numerical, an Analtic Analsis In Eercises 6 66, use a graphing utilit to graph the function an estimate the it Use a table to reinforce our conclusion Then fin the it b analtic methos In Eercises 67 78, etermine the it of the trigonometric function (if it eists) sin sin cos sin cos h 7 7 h0 h cos cot sin t t0 t sin 0 sin sin Hint: Fin cos 0 cos tan 0 tan 0 sec tan sin cos sin Graphical, Numerical, an Analtic Analsis In Eercises 79 8, use a graphing utilit to graph the function an estimate the it Use a table to reinforce our conclusion Then fin the it b analtic methos sin t t0 t sin In Eercises 8 86, fin 8 f 8 f 85 f 86 f In Eercises 87 an 88, use the Squeeze Theorem to fin f c c 0 f c a b a f b a In Eercises 89 9, use a graphing utilit to graph the given function an the equations an in the same viewing winow Using the graphs to observe the Squeeze Theorem visuall, fin f 89 f cos 90 f sin f sin 9 h cos 99 Writing Use a graphing utilit to graph f, g sin, an h sin in the same viewing winow Compare the magnitues of f an g when is close to 0 Use the comparison to write a short paragraph eplaining wh h Writing About Concepts cos 0 0 sin f f f sin f cos 95 In the contet of fining its, iscuss what is meant b two functions that agree at all but one point 96 Give an eample of two functions that agree at all but one point 97 What is meant b an ineterminate form? 98 In our own wors, eplain the Squeeze Theorem

29 SECTION Evaluating Limits Analticall Writing Use a graphing utilit to graph f, g sin, an h sin in the same viewing winow Compare the magnitues of f an g when is close to 0 Use the comparison to write a short paragraph eplaining wh h 0 0 Free-Falling Object In Eercises 0 an 0, use the position function st 6t 000, which gives the height (in feet) of an object that has fallen for t secons from a height of 000 feet The velocit at time t a secons is given b sa st ta a t 0 If a construction worker rops a wrench from a height of 000 feet, how fast will the wrench be falling after 5 secons? 0 If a construction worker rops a wrench from a height of 000 feet, when will the wrench hit the groun? At what velocit will the wrench impact the groun? Free-Falling Object In Eercises 0 an 0, use the position function st 9t 50, which gives the height (in meters) of an object that has fallen from a height of 50 meters The velocit at time t a secons is given b sa st ta a t 0 Fin the velocit of the object when t 0 At what velocit will the object impact the groun? 05 Fin two functions f an g such that f an g o not eist, but f g oes eist 06 Prove that if f eists an f g oes not eist, then c 0 c g oes not eist 07 Prove Propert of Theorem 08 Prove Propert of Theorem (You ma use Propert of Theorem ) 09 Prove Propert of Theorem 0 Prove that if f 0, then c Prove that if f 0 an for a fie number c (b) Prove that if c Hint: Use the inequalit c 0 f c g M f L, then 0 M an all c, then fg 0 (a) Prove that if c f 0, then f 0 c c (Note: This is the converse of Eercise 0) 0 c f L f L f L True or False? In Eercises 8, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 5 If f g for all real numbers other than 0, an then 6 If f L, then f c L 7 f, where 8 If f < g for all a, then 9 Think About It Fin a function f to show that the converse of Eercise (b) is not true [Hint: Fin a function f such that but f oes not eist] 0 Prove the secon part of Theorem 9 b proving that Let 0 f L, 0 f < g a a cos 0 0 an c sin g 0,, Fin (if possible) f an Graphical Reasoning Consier f (a) Fin the omain of f (b) Use a graphing utilit to graph f Is the omain of f obvious from the graph? If not, eplain (c) Use the graph of f to approimate f 0 () Confirm the answer in part (c) analticall Approimation (a) Fin c f L f 0,, if is rational if is irrational 0 cos 0 g L 0 f, 0, if is rational if is irrational (b) Use the result in part (a) to erive the approimation cos for near 0 (c) Use the result in part (b) to approimate cos0 () Use a calculator to approimate cos0 to four ecima places Compare the result with part (c) Think About It When using a graphing utilit to generate a table to approimate sin, a stuent conclue that 0 c > g 0 sec the it was 0075 rather than Determine the probable cause of the error

30 70 CHAPTER Limits an Their Properties Section Informall, ou might sa that a function is continuous on an open interval if its graph can be rawn with a pencil without lifting the pencil from the paper Use a graphing utilit to graph each function on the given interval From the graphs, which functions woul ou sa are continuous on the interval? Do ou think ou can trust the results ou obtaine graphicall? Eplain our reasoning Function a b c EXPLORATION sin Interval,,,,, 0 e,, > 0 Continuit an One-Sie Limits Determine continuit at a point an continuit on an open interval Determine one-sie its an continuit on a close interval Use properties of continuit Unerstan an use the Intermeiate Value Theorem Continuit at a Point an on an Open Interval In mathematics, the term continuous has much the same meaning as it has in evera usage Informall, to sa that a function f is continuous at c means that there is no interruption in the graph of f at c That is, its graph is unbroken at c an there are no holes, jumps, or gaps Figure 5 ientifies three values of at which the graph of f is not continuous At all other points in the interval a, b, the graph of f is uninterrupte an continuous Animation a f(c) is not efine c Three conitions eist for which the graph of Figure 5 b In Figure 5, it appears that continuit at c can be estroe b an one of the following conitions The function is not efine at c The it of f oes not eist at c The it of f eists at c, but it is not equal to fc a f() c oes not eist c If none of the above three conitions is true, the function f is calle continuous at c, as inicate in the following important efinition b f is not continuous at c a f() f(c) c c b FOR FURTHER INFORMATION For more information on the concept of continuit, see the article Leibniz an the Spell of the Continuous b Har Grant in The College Mathematics Journal MathArticle Definition of Continuit Continuit at a Point: A function f is continuous at c if the following three conitions are met fc is efine eists c f f c c Continuit on an Open Interval: A function is continuous on an open interval a, b if it is continuous at each point in the interval A function that is continuous on the entire real line, is everwhere continuous Vieo

31 SECTION Continuit an One-Sie Limits 7 Consier an open interval I that contains a real number c If a function f is efine on I (ecept possibl at c), an f is not continuous at c, then f is sai to have a iscontinuit at c Discontinuities fall into two categories: removable an nonremovable A iscontinuit at c is calle removable if f can be mae continuous b appropriatel efining (or reefining) fc For instance, the functions shown in Figure 6(a) an (c) have removable iscontinuities at c an the function shown in Figure 6(b) has a nonremovable iscontinuit at c a c b (a) Removable iscontinuit EXAMPLE Continuit of a Function Discuss the continuit of each function, 0 a b g f c h, > 0 sin a c b (b) Nonremovable iscontinuit Solution a The omain of f is all nonzero real numbers From Theorem, ou can conclue that f is continuous at ever -value in its omain At 0, f has a nonremovable iscontinuit, as shown in Figure 7(a) In other wors, there is no wa to efine f0 so as to make the function continuous at 0 b The omain of g is all real numbers ecept From Theorem, ou can conclue that g is continuous at ever -value in its omain At, the function has a removable iscontinuit, as shown in Figure 7(b) If g is efine as, the newl efine function is continuous for all real numbers c The omain of h is all real numbers The function h is continuous on, 0 an 0,, an, because h, h is continuous on the entire real line, as shown 0 in Figure 7(c) The omain of is all real numbers From Theorem 6, ou can conclue that the function is continuous on its entire omain,,, as shown in Figure 7() a (c) Removable iscontinuit Figure 6 c b f() = (, ) g() = (a) Nonremovable iscontinuit at 0 (b) Removable iscontinuit at Eitable Graph Eitable Graph STUDY TIP Some people ma refer to the function in Eample (a) as iscontinuous We have foun that this terminolog can be confusing Rather than saing the function is iscontinuous, we prefer to sa that it has a iscontinuit at 0 +, 0 h() = +, > 0 (c) Continuous on entire real line Eitable Graph Figure 7 = sin π π () Continuous on entire real line Eitable Graph Tr It Eploration A Eploration B Eploration C

32 7 CHAPTER Limits an Their Properties (a) Limit from right approaches c from the left c > (b) Limit from left Figure 8 approaches c from the right c < One-Sie Limits an Continuit on a Close Interval To unerstan continuit on a close interval, ou first nee to look at a ifferent tpe of it calle a one-sie it For eample, the it from the right means that approaches c from values greater than c [see Figure 8(a)] This it is enote as f L c Limit from the right Similarl, the it from the left means that approaches c from values less than c [see Figure 8(b)] This it is enote as f L c Limit from the left One-sie its are useful in taking its of functions involving raicals For instance, if n is an even integer, n 0 0 EXAMPLE A One-Sie Limit f() = Fin the it of f as approaches from the right Solution As shown in Figure 9, the it as approaches from the right is 0 Tr It Eploration A The it of f as approaches from the right is 0 Figure 9 Eitable Graph One-sie its can be use to investigate the behavior of step functions One common tpe of step function is the greatest integer function, efine b greatest integer n such that n For instance, 5 an 5 Greatest integer function EXAMPLE The Greatest Integer Function f() = [[ ]] Fin the it of the greatest integer function f as approaches 0 from the left an from the right Solution As shown in Figure 0, the it as approaches 0 from the left is given b 0 an the it as approaches 0 from the right is given b Greatest integer function Figure 0 Eitable Graph 0 0 The greatest integer function has a iscontinuit at zero because the left an right its at zero are ifferent B similar reasoning, ou can see that the greatest integer function has a iscontinuit at an integer n Tr It Eploration A Eploration B

33 SECTION Continuit an One-Sie Limits 7 When the it from the left is not equal to the it from the right, the (twosie) it oes not eist The net theorem makes this more eplicit The proof of this theorem follows irectl from the efinition of a one-sie it THEOREM 0 The Eistence of a Limit Let f be a function an let c an L be real numbers The it of f as approaches c is L if an onl if f L c an f L c The concept of a one-sie it allows ou to eten the efinition of continuit to close intervals Basicall, a function is continuous on a close interval if it is continuous in the interior of the interval an ehibits one-sie continuit at the enpoints This is state formall as follows a Continuous function on a close interval Figure b Definition of Continuit on a Close Interval A function f is continuous on the close interval [a, b] if it is continuous on the open interval a, b an f fa a an f fb b The function f is continuous from the right at a an continuous from the left at b (see Figure ) Similar efinitions can be mae to cover continuit on intervals of the form a, b an a, b that are neither open nor close, or on infinite intervals For eample, the function f is continuous on the infinite interval 0,, an the function g is continuous on the infinite interval, EXAMPLE Continuit on a Close Interval Discuss the continuit of f f() = f is continuous on, Figure Solution The omain of f is the close interval, At all points in the open interval,, the continuit of f follows from Theorems an 5 Moreover, because an 0 f 0 f Continuous from the right Continuous from the left ou can conclue that f is continuous on the close interval,, as shown in Figure Eitable Graph Tr It Eploration A

34 7 CHAPTER Limits an Their Properties The net eample shows how a one-sie it can be use to etermine the value of absolute zero on the Kelvin scale EXAMPLE 5 Charles s Law an Absolute Zero V 0 5 V = 008T ( 75, 0) The volume of hrogen gas epens on its temperature Figure Eitable Graph T On the Kelvin scale, absolute zero is the temperature 0 K Although temperatures of approimatel 0000 K have been prouce in laboratories, absolute zero has never been attaine In fact, evience suggests that absolute zero cannot be attaine How i scientists etermine that 0 K is the lower it of the temperature of matter? What is absolute zero on the Celsius scale? Solution The etermination of absolute zero stems from the work of the French phsicist Jacques Charles (76 8) Charles iscovere that the volume of gas at a constant pressure increases linearl with the temperature of the gas The table illustrates this relationship between volume an temperature In the table, one mole of hrogen is hel at a constant pressure of one atmosphere The volume V is measure in liters an the temperature T is measure in egrees Celsius T V The points represente b the table are shown in Figure Moreover, b using the points in the table, ou can etermine that T an V are relate b the linear equation V 008T or B reasoning that the volume of the gas can approach 0 (but never equal or go below 0) ou can etermine that the least possible temperature is given b V T V 0 V Use irect substitution So, absolute zero on the Kelvin scale 0 K is approimatel 75 on the Celsius scale T V 008 Tr It Eploration A The following table shows the temperatures in Eample 5, converte to the Fahrenheit scale Tr repeating the solution shown in Eample 5 using these temperatures an volumes Use the result to fin the value of absolute zero on the Fahrenheit scale In 995, phsicists Carl Wieman an Eric Cornell of the Universit of Colorao at Bouler use lasers an evaporation to prouce a supercol gas in which atoms overlap This gas is calle a Bose-Einstein conensate We get to within a billionth of a egree of absolute zero, reporte Wieman (Source: Time magazine, April 0, 000) T V NOTE Charles s Law for gases (assuming constant pressure) can be state as V RT Charles s Law where V is volume, R is constant, an T is temperature In the statement of this law, what propert must the temperature scale have?

35 SECTION Continuit an One-Sie Limits 75 AUGUSTIN-LOUIS CAUCHY ( ) The concept of a continuous function was first introuce b Augustin-Louis Cauch in 8 The efinition given in his tet Cours Analse state that inefinite small changes in were the result of inefinite small changes in f will be calle a continuous function if the numerical values of the ifference f f ecrease inefinitel with those of MathBio Properties of Continuit In Section, ou stuie several properties of its Each of those properties iels a corresponing propert pertaining to the continuit of a function For instance, Theorem follows irectl from Theorem THEOREM Properties of Continuit If b is a real number an f an g are continuous at c, then the following functions are also continuous at c Scalar multiple: Sum an ifference: Prouct: fg bf f ± g f Quotient: if gc 0 g, The following tpes of functions are continuous at ever point in their omains Polnomial functions: Rational functions: Raical functions: p a n n a n n a a 0 r p q 0 q, f n Trigonometric functions: sin, cos, tan, cot, sec, csc B combining Theorem with this summar, ou can conclue that a wie variet of elementar functions are continuous at ever point in their omains EXAMPLE 6 Appling Properties of Continuit B Theorem, it follows that each of the following functions is continuous at ever point in its omain f sin, f tan, f cos Tr It Eploration A Open Eploration The net theorem, which is a consequence of Theorem 5, allows ou to etermine the continuit of composite functions such as f sin, f, f tan THEOREM Continuit of a Composite Function If g is continuous at c an f is continuous at gc, then the composite function given b f g fg is continuous at c One consequence of Theorem is that if f an g satisf the given conitions, ou can etermine the it of fg as approaches c to be fg fgc c Technolog

36 76 CHAPTER Limits an Their Properties EXAMPLE 7 Testing for Continuit Describe the interval(s) on which each function is continuous a b g sin, 0 f tan c 0, 0 h sin, 0, 0 0 Solution a The tangent function f tan is unefine at n, n is an integer At all other points it is continuous So, f tan is continuous on the open intervals,,,,,,, as shown in Figure (a) b Because is continuous ecept at 0 an the sine function is continuous for all real values of, it follows that sin is continuous at all real values ecept 0 At 0, the it of g oes not eist (see Eample 5, Section ) So, g is continuous on the intervals, 0 an 0,, as shown in Figure (b) c This function is similar to that in part (b) ecept that the oscillations are ampe b the factor Using the Squeeze Theorem, ou obtain sin, 0 an ou can conclue that h 0 0 So, h is continuous on the entire real line, as shown in Figure (c) π π = f() = tan (a) f is continuous on each open interval in its omain Eitable Graph g() = sin, 0 0, = 0 = h() = sin, 0 0, = 0 (b) g is continuous on, 0 an 0, (c) h is continuous on the entire real line Figure Eitable Graph Eitable Graph Tr It Eploration A

37 SECTION Continuit an One-Sie Limits 77 The Intermeiate Value Theorem Theorem is an important theorem concerning the behavior of functions that are continuous on a close interval THEOREM Intermeiate Value Theorem If f is continuous on the close interval a, b an k is an number between fa an fb), then there is at least one number c in a, b such that fc k Vieo NOTE The Intermeiate Value Theorem tells ou that at least one c eists, but it oes not give a metho for fining c Such theorems are calle eistence theorems B referring to a tet on avance calculus, ou will fin that a proof of this theorem is base on a propert of real numbers calle completeness The Intermeiate Value Theorem states that for a continuous function f, if takes on all values between a an b, f must take on all values between fa an fb As a simple eample of this theorem, consier a person s height Suppose that a girl is 5 feet tall on her thirteenth birtha an 5 feet 7 inches tall on her fourteenth birtha Then, for an height h between 5 feet an 5 feet 7 inches, there must have been a time t when her height was eactl h This seems reasonable because human growth is continuous an a person s height oes not abruptl change from one value to another The Intermeiate Value Theorem guarantees the eistence of at least one number c in the close interval a, b There ma, of course, be more than one number c such that fc k, as shown in Figure 5 A function that is not continuous oes not necessaril ehibit the intermeiate value propert For eample, the graph of the function shown in Figure 6 jumps over the horizontal line given b k, an for this function there is no value of c in a, b such that fc k f(a) f(a) k f(b) a c c c b f is continuous on a, b [There eist three c s such that fc k ] Figure 5 k f(b) a b f is not continuous on a, b [There are no c s such that fc k ] Figure 6 The Intermeiate Value Theorem often can be use to locate the zeros of a function that is continuous on a close interval Specificall, if f is continuous on a, b an fa an fb iffer in sign, the Intermeiate Value Theorem guarantees the eistence of at least one zero of f in the close interval a, b

38 78 CHAPTER Limits an Their Properties f() = + EXAMPLE 8 An Application of the Intermeiate Value Theorem (, ) Use the Intermeiate Value Theorem to show that the polnomial function f has a zero in the interval 0, Solution Note that f is continuous on the close interval 0, Because f0 0 0 an f (c, 0) it follows that f0 < 0 an f > 0 You can therefore appl the Intermeiate Value Theorem to conclue that there must be some c in 0, such that fc 0 f has a zero in the close interval 0, as shown in Figure 7 (0, ) Tr It Eploration A f is continuous on 0, with f 0 < 0 an f > 0 Figure 7 Eitable Graph The bisection metho for approimating the real zeros of a continuous function is similar to the metho use in Eample 8 If ou know that a zero eists in the close interval a, b, the zero must lie in the interval a, a b or a b, b From the sign of fa b, ou can etermine which interval contains the zero B repeatel bisecting the interval, ou can close in on the zero of the function TECHNOLOGY You can also use the zoom feature of a graphing utilit to approimate the real zeros of a continuous function B repeatel zooming in on the point where the graph crosses the -ais, an ajusting the -ais scale, ou can approimate the zero of the function to an esire accurac The zero of is approimatel 05, as shown in Figure Figure 8 Zooming in on the zero of f

39 78 CHAPTER Limits an Their Properties Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 6, use the graph to etermine the it, an iscuss the continuit of the function (a) f (b) f (c) c c (, ) c = f c c = (, ) (, ) 6 (, 0) c = 5 6 (, ) c = 5 6 (, ) (, ) (, ) c = c = (, ) (, 0)

40 SECTION Continuit an One-Sie Limits 79 In Eercises 7, fin the it (if it eists) If it oes not eist, eplain wh f, where 6 f, where 7 f, where 8 f, where cot In Eercises 5 8, iscuss the continuit of each function 5 f 6 9 sec 0 5 f,, > f 6, <, f, <, f,, > f 7 f 8 In Eercises 9, iscuss the continuit of the function on the close interval 9 0 In Eercises 5, fin the -values (if an) at which f is not continuous Which of the iscontinuities are removable? Function g 5 f t 9 t f,, 0 > 0 g f f f cos f cos f f f f 9 f f f f f,, 0 > 6 f,, < Interval 5, 5,,, f,,, < >

41 80 CHAPTER Limits an Their Properties f,, f tan,, f csc 6,, f csc f tan f f > f,, < > > In Eercises 65 68, use a graphing utilit to graph the function Use the graph to etermine an -values at which the function is not continuous 65 f 66 h g,, In Eercises 69 7, escribe the interval(s) on which the function is continuous > cos, < 0 f 5, 0 69 f 70 f In Eercises 55 an 56, use a graphing utilit to graph the function From the graph, estimate f an f 0 0 Is the function continuous on the entire real line? Eplain In Eercises 57 60, fin the constant a, or the constants a an b, such that the function is continuous on the entire real line f f f, a, g sin f, a b,, In Eercises 6 6, iscuss the continuit of the composite function h fg 6 f 6 g g 5 >, < 0 a, 0 < < g a a, a 8, a f g 6 f 6 f sin 6 g 7 f sec 7 Writing In Eercises 7 an 7, use a graphing utilit to graph the function on the interval [, ] Does the graph of the func tion appear continuous on this interval? Is the function contin uous on [, ]? Write a short paragraph about the importance of eamining a function analticall as well as graphicall 7 f sin 7 Writing In Eercises 75 78, eplain wh the function has a zero in the given interval Function f 6 f Interval, 0, 77 f cos 0, 78 f tan 8, f (, 0) f 8

42 SECTION Continuit an One-Sie Limits 8 In Eercises 79 8, use the Intermeiate Value Theorem an a graphing utilit to approimate the zero of the function in the interval [0, ] Repeatel zoom in on the graph of the function to approimate the zero accurate to two ecimal places Use the zero or root feature of the graphing utilit to approimate the zero accurate to four ecimal places In Eercises 8 86, verif that the Intermeiate Value Theorem applies to the inicate interval an fin the value of c guarantee b the theorem f f gt cos t t h f, tan f, f 6 8, f, 5,, 0, 5, 0,, 0,, Writing About Concepts f c f c 6 f c 0 f c 87 State how continuit is estroe at c for each of the following graphs (a) (b) Writing About Concepts (continue) 89 Sketch the graph of an function f such that f True or False? In Eercises 9 9, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 9 If f L an f c L, then f is continuous at c c 9 If f g for c an f c gc, then either f or g i not continuous at c 9 A rational function can have infinitel man -values at which it is not continuous 9 The function is continuous on, 95 Swimming Pool Ever a ou issolve 8 ounces o chlorine in a swimming pool The graph shows the amount o chlorine f t in the pool after t as an f f 0 Is the function continuous at? Eplain 90 If the functions f an g are continuous for all real, is f g alwas continuous for all real? Is fg alwas continuous for all real? If either is not continuous, give an eample to verif our conclusion 0 8 (c) c () c Estimate an interpret f t an f t t t 5 t 96 Think About It Describe how the functions f c c an g 88 Describe the ifference between a iscontinuit that is removable an one that is nonremovable In our eplanation, give eamples of the following escriptions (a) A function with a nonremovable iscontinuit at (b) A function with a removable iscontinuit at (c) A function that has both of the characteristics escribe in parts (a) an (b) iffer 97 Telephone Charges A ial-irect long istance call between two cities costs $0 for the first minutes an $06 for each aitional minute or fraction thereof Use the greatest intege function to write the cost C of a call in terms of time t (in minutes) Sketch the graph of this function an iscuss it continuit

43 8 CHAPTER Limits an Their Properties 98 Inventor Management The number of units in inventor in a small compan is given b where t is the time in months Sketch the graph of this function an iscuss its continuit How often must this compan replenish its inventor? 99 Déjà Vu At 8:00 AM on Satura a man begins running up the sie of a mountain to his weeken campsite (see figure) On Suna morning at 8:00 AM he runs back own the mountain It takes him 0 minutes to run up, but onl 0 minutes to run own At some point on the wa own, he realizes that he passe the same place at eactl the same time on Satura Prove that he is correct [Hint: Let st an rt be the position functions for the runs up an own, an appl the Intermeiate Value Theorem to the function f t st rt ] 05 Moeling Data After an object falls for t secons, the spee S (in feet per secon) of the object is recore in the table Nt 5 t t t S (a) Create a line graph of the ata (b) Does there appear to be a iting spee of the object? I there is a iting spee, ientif a possible cause 06 Creating Moels A swimmer crosses a pool of with b b swimming in a straight line from 0, 0 to b, b (See figure (a) Let f be a function efine as the -coorinate of the poin on the long sie of the pool that is nearest the swimmer a an given time uring the swimmer s path across the pool Determine the function f an sketch its graph Is i continuous? Eplain (b) Let g be the minimum istance between the swimmer an the long sies of the pool Determine the function g an sketch its graph Is it continuous? Eplain (b, b) b 00 Volume Use the Intermeiate Value Theorem to show that for all spheres with raii in the interval, 5, there is one with a volume of 75 cubic centimeters 0 Prove that if f is continuous an has no zeros on a, b, then either f > 0 for all in a, b or f < 0 for all in a, b 0 Show that the Dirichlet function f 0,, is not continuous at an real number 0 Show that the function f 0, k, Satura 8:00 AM is continuous onl at 0 (Assume that k is an nonzero real number) 0 The signum function is efine b sgn, 0,, if is rational if is irrational if is rational if is irrational < 0 0 > 0 Sketch a graph of sgn an fin the following (if possible) (a) sgn (b) sgn (c) 0 0 Not rawn to scale Suna 8:00 AM sgn 0 (0, 0) 07 Fin all values of c such that f is continuous on, f,, c > c 08 Prove that for an real number there eists in, such that tan 09 Let f c c, c > 0 What is the omain o f? How can ou efine f at 0 in orer for f to be continuous there? 0 Prove that if f c f c, then f is continuou at c 0 Discuss the continuit of the function h (a) Let f an f be continuous on the close interva a, b If f a < f a an f b > f b, prove that there eists c between a an b such that f c f c (b) Show that there eists c in 0, such that cos Use a graphing utilit to approimate c to three ecimal places Putnam Eam Challenge Prove or isprove: if an are real numbers with 0 an, then Determine all polnomials P such that P P an P0 0 These problems were compose b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserve

44 SECTION 5 Infinite Limits 8 6, as Section 5 6 6, as + f() = 6 f increases an ecreases without boun as approaches Figure 9 Infinite Limits Determine infinite its from the left an from the right Fin an sketch the vertical asmptotes of the graph of a function Infinite Limits Let f be the function given b From Figure 9 an the table, ou can see that f ecreases without boun as approaches from the left, an f increases without boun as approaches from the right This behavior is enote as an f f ecreases without boun as approaches from the left f increases without boun as approaches from the right approaches from the left approaches from the right f ? f ecreases without boun f increases without boun A it in which f increases or ecreases without boun as approaches c is calle an infinite it Definition of Infinite Limits Let f be a function that is efine at ever real number in some open interval containing c (ecept possibl at c itself) The statement M δ δ f() = c f c means that for each M > 0 there eists a > 0 such that f > M whenever (see Figure 0) Similarl, the statement 0 < c < f c means that for each N < 0 there eists a > 0 such that f < N whenever 0 < c To efine the infinite it from the left, replace 0 < c < < b c < < c To efine the infinite it from the right, replace 0 < c < b c < < c Infinite its Figure 0 c Vieo Be sure ou see that the equal sign in the statement f oes not mean that the it eists! On the contrar, it tells ou how the it fails to eist b enoting the unboune behavior of f as approaches c

45 8 CHAPTER Limits an Their Properties EXPLORATION Use a graphing utilit to graph each function For each function, analticall fin the single real number c that is not in the omain Then graphicall fin the it of f as approaches c from the left an from the right a f b f c f f EXAMPLE Determining Infinite Limits from a Graph Use Figure to etermine the it of each function as approaches from the left an from the right f() = f() = ( ) (a) f() = (b) f() = ( ) (c) () Eitable Graph Eitable Graph Eitable Graph Eitable Graph Figure Each graph has an asmptote at Solution a an b Limit from each sie is c an Limit from each sie is Tr It Eploration A Vertical Asmptotes If it were possible to eten the graphs in Figure towar positive an negative infinit, ou woul see that each graph becomes arbitraril close to the vertical line This line is a vertical asmptote of the graph of f (You will stu other tpes of asmptotes in Sections 5 an 6) NOTE If the graph of a function f has a vertical asmptote at c, then f is not continuous at c Definition of Vertical Asmptote If f approaches infinit (or negative infinit) as approaches c from the right or the left, then the line c is a vertical asmptote of the graph of f

46 SECTION 5 Infinite Limits 85 In Eample, note that each of the functions is a quotient an that the vertical asmptote occurs at a number where the enominator is 0 (an the numerator is not 0) The net theorem generalizes this observation (A proof of this theorem is given in Appeni A) THEOREM Vertical Asmptotes Let f an g be continuous on an open interval containing c If fc 0, gc 0, an there eists an open interval containing c such that g 0 for all c in the interval, then the graph of the function given b h f g has a vertical asmptote at c f() = ( + ) Vieo (a) EXAMPLE Fining Vertical Asmptotes Determine all vertical asmptotes of the graph of each function a b f f c f cot f() = + (b) 6 π π π (c) Eitable Graph Eitable Graph 6 Eitable Graph f() = cot Functions with vertical asmptotes Figure Solution a When, the enominator of f is 0 an the numerator is not 0 So, b Theorem, ou can conclue that is a vertical asmptote, as shown in Figure (a) b B factoring the enominator as f ou can see that the enominator is 0 at an Moreover, because the numerator is not 0 at these two points, ou can appl Theorem to conclue that the graph of f has two vertical asmptotes, as shown in Figure (b) c B writing the cotangent function in the form f cot cos sin ou can appl Theorem to conclue that vertical asmptotes occur at all values of such that sin 0 an cos 0, as shown in Figure (c) So, the graph of this function has infinitel man vertical asmptotes These asmptotes occur when n, where n is an integer Tr It Eploration A Eploration B Open Eploration Theorem requires that the value of the numerator at c be nonzero If both the numerator an the enominator are 0 at c, ou obtain the ineterminate form 00, an ou cannot etermine the it behavior at c without further investigation, as illustrate in Eample

47 86 CHAPTER Limits an Their Properties EXAMPLE A Rational Function with Common Factors f() = + 8 f increases an ecreases without boun as approaches Figure Eitable Graph Unefine when = Vertical asmptote at = Determine all vertical asmptotes of the graph of Solution f 8 Begin b simplifing the epression, as shown f 8, At all -values other than, the graph of f coincies with the graph of g So, ou can appl Theorem to g to conclue that there is a vertical asmptote at, as shown in Figure From the graph, ou can see that 8 an Note that is not a vertical asmptote 8 Tr It Eploration A Eploration B EXAMPLE Determining Infinite Limits 6 f() = Fin each it an Solution Because the enominator is 0 when (an the numerator is not zero), ou know that the graph of 6 6 f has a vertical asmptote at Figure Eitable Graph has a vertical asmptote at This means that each of the given its is either or A graphing utilit can help etermine the result From the graph of f shown in Figure, ou can see that the graph approaches from the left of an approaches from the right of So, ou can conclue that an f The it from the left is infinit The it from the right is negative infinit Tr It Eploration A TECHNOLOGY PITFALL When using a graphing calculator or graphing software, be careful to interpret correctl the graph of a function with a vertical asmptote graphing utilities often have ifficult rawing this tpe of graph

48 SECTION 5 Infinite Limits 87 THEOREM 5 Properties of Infinite Limits Let c an L be real numbers an let f an g be functions such that f c Sum or ifference: Prouct: Quotient: an g L c f ± g c fg, L > 0 c fg, L < 0 c g c f 0 Similar properties hol for one-sie its an for functions for which the it of f as approaches c is Proof To show that the it of f g is infinite, choose M > 0 You then nee to fin > 0 such that f g > M whenever 0 < c < For simplicit s sake, ou can assume L is positive Let M M Because the it of f is infinite, there eists such that f > M whenever 0 < c Also, because the it of is there eists such that g L < g L, < whenever 0 < c B letting be the smaller of an ou can conclue that 0 < < c implies f > M an, g L < < The secon of these two inequalities implies that g > L, an, aing this to the first inequalit, ou can write f g > M L M L > M So, ou can conclue that f g c The proofs of the remaining properties are left as eercises (see Eercise 7) EXAMPLE 5 Determining Limits a Because an ou can write 0, 0 0 Propert, Theorem 5 b Because an cot, ou can write 0 cot Propert, Theorem 5 c Because an cot, ou can write 0 0 cot 0 Propert, Theorem 5 Tr It Eploration A

49 88 CHAPTER Limits an Their Properties Eercises for Section 5 The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises, etermine whether f approaches or as approaches from the left an from the right f f tan Numerical an Graphical Analsis In Eercises 5 8, etermine whether f approaches or as approaches from the left an from the right b completing the table Use a graphing utilit to graph the function an confirm our answer f f f f f sec f 6 f 9 9 f sec 6 In Eercises 9 8, fin the vertical asmptotes (if an) of the graph of the function 9 f 0 f h g f f s 5 gt t 6 hs t s 5 7 f tan 8 f sec 9 0 g Tt t 6 f f h g 5 f g tan st t sin t In Eercises 9, etermine whether the graph of the function has a vertical asmptote or a removable iscontinuit at Graph the function using a graphing utilit to confirm our answer 9 f 0 f f In Eercises 8, fin the it sin csc 7 sec 8 In Eercises 9 5, use a graphing utilit to graph the function an etermine the one-sie it 9 f 50 f 5 f 5 5 f sec f f ht t t t 6 f cos 0 cot tan f f sin 6

50 SECTION 5 Infinite Limits 89 Writing About Concepts 5 In our own wors, escribe the meaning of an infinite it Is a real number? 5 In our own wors, escribe what is meant b an asmptote of a graph 55 Write a rational function with vertical asmptotes at 6 an, an with a zero at 56 Does the graph of ever rational function have a vertical asmptote? Eplain 57 Use the graph of the function f (see figure) to sketch the graph of g f on the interval, To print an enlarge cop of the graph, select the MathGraph button f 6 Relativit Accoring to the theor of relativit, the mass m of a particle epens on its velocit v That is, m m 0 v c where m 0 is the mass when the particle is at rest an c is the spee of light Fin the it of the mass as v approaches c 6 Rate of Change A 5-foot laer is leaning against a house (see figure) If the base of the laer is pulle awa from the house at a rate of feet per secon, the top will move own the wall at a rate of r ft/sec 65 where is the istance between the base of the laer an the house (a) Fin the rate r when is 7 feet (b) Fin the rate r when is 5 feet (c) Fin the it of r as 5 58 Bole s Law For a quantit of gas at a constant temperature, the pressure P is inversel proportional to the volume V Fin the it of P as V 0 59 Rate of Change A patrol car is parke 50 feet from a long warehouse (see figure) The revolving light on top of the car turns at a rate of revolution per secon The rate at which the light beam moves along the wall is r 50 sec ft/sec (a) Fin the rate r when (b) Fin the rate r when (c) Fin the it of r as is 6 is r 5 ft ft sec 6 Average Spee On a trip of miles to another cit, a truck river s average spee was miles per hour On the return trip the average spee was miles per hour The average spee for the roun trip was 50 miles per hour (a) Verif that 5 What is the omain? 5 (b) Complete the table 50 ft Illegal Drugs The cost in millions of ollars for a governmental agenc to seize % of an illegal rug is C 58 00, 0 < 00 (a) Fin the cost of seizing 5% of the rug (b) Fin the cost of seizing 50% of the rug (c) Fin the cost of seizing 75% of the rug () Fin the it of C as 00 an interpret its meaning Are the values of ifferent than ou epecte? Eplain (c) Fin the it of as 5 an interpret its meaning 6 Numerical an Graphical Analsis Use a graphing utilit to complete the table for each function an graph each function to estimate the it What is the value of the it when the power on in the enominator is greater than? (a) (c) f sin 0 sin 0 (b) sin 0 sin () 0

51 90 CHAPTER Limits an Their Properties 65 Numerical an Graphical Analsis Consier the shae region outsie the sector of a circle of raius 0 meters an insie a right triangle (see figure) (a) Write the area A f of the region as a function of Determine the omain of the function (b) Use a graphing utilit to complete the table an graph the function over the appropriate omain f (c) Fin the it of A as Numerical an Graphical Reasoning A crosse belt connects a 0-centimeter pulle (0-cm raius) on an electric motor with a 0-centimeter pulle (0-cm raius) on a saw arbor (see figure) The electric motor runs at 700 revolutions per minute (a) Determine the number of revolutions per minute of the saw (b) How oes crossing the belt affect the saw in relation to the motor? (c) Let L be the total length of the belt Write L as a function of, where is measure in raians What is the omain of the function? (Hint: A the lengths of the straight sections of the belt an the length of the belt aroun each pulle) 0 m 0 cm 0 cm L () Use a graphing utilit to complete the table (e) Use a graphing utilit to graph the function over the appro priate omain (f) Fin L Use a geometric argument as the basis of a secon metho of fining this it (g) Fin True or False? In Eercises 67 70, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 67 If p is a polnomial, then the graph of the function given b f p has a vertical asmptote at 68 The graph of a rational function has at least one vertica asmptote 69 The graphs of polnomial functions have no vertica asmptotes 70 If f has a vertical asmptote at 0, then f is unefine a 0 7 Fin functions f an g such that f an c g but f g 0 c Prove the remaining properties of Theorem 5 7 Prove that if f, then 7 Prove that if 0, then f oes not eist c f c Infinite Limits In Eercises 75 an 76, use the - efinition o infinite its to prove the statement L 0 c c c 0 f

52 REVIEW EXERCISES 9 Review Eercises for Chapter The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises an, etermine whether the problem can be solve using precalculus or if calculus is require If the problem can be solve using precalculus, solve it If the problem seems to require calculus, eplain our reasoning Use a graphical or numerical approach to estimate the solution Fin the istance between the points, an, 9 along the curve Fin the istance between the points, an, 9 along the line In Eercises an, complete the table an use the result to estimate the it Use a graphing utilit to graph the function to confirm our result f 0 0 In Eercises 5 an 6, use the graph to etermine each it (a) h (b) h (a) g (b) In Eercises 7 0, fin the it L Then use the - efinition to prove that the it is L h 6 g 0 h 9 0 In Eercises, fin the it (if it eists) t t t t t g 8 t 9 t t 0 g 0 s s0 s [Hint: ] [Hint: ] In Eercises 5 an 6, evaluate the it given f an g c c Numerical, Graphical, an Analtic Analsis In Eercises 7 an 8, consier f (a) Complete the table to estimate the it (b) Use a graphing utilit to graph the function an use the graph to estimate the it (c) Rationalize the numerator to fin the eact value of the it analticall cos 0 sin sin6 0 0 f g c f g c f tan f Hint: a b a ba ab b Free-Falling Object In Eercises 9 an 0, use the position function st 9t 00, which gives the height (in meters) of an object that has fallen from a height of 00 meters The velocit at time t a secons is given b sa st ta a t sin sin cos cos sin cos cos cos cos sin sin f 9 Fin the velocit of the object when t 0 At what velocit will the object impact the groun?

53 9 CHAPTER Limits an Their Properties In Eercises 6, fin the it (if it eists) If the it oes not eist, eplain wh f, where g, where 5 ht, where t 6 f s, where s In Eercises 7 6, etermine the intervals on which the function is continuous f f f, 0, f 5,, f f 7 Determine the value of c such that the function is continuous on the entire real line f, c 6, 8 Determine the values of b an c such that the function is continuous on the entire real line f, b c, f,, g,, ht t, t, > > < < > > t < t f s s s, s s 6, s s > f f 5 f csc 6 f tan 9 Use the Intermeiate Value Theorem to show that f has a zero in the interval, 50 Deliver Charges The cost of sening an overnight package from New York to Atlanta is $980 for the first poun an $50 for each aitional poun or fraction thereof Use the greatest integer function to create a moel for the cost C of overnight eliver of a package weighing pouns Use a graphing utilit to graph the function an iscuss its continuit 5 Let f Fin each it (if possible) (a) (b) (c) f f f 5 Let f (a) Fin the omain of f (b) Fin (c) Fin In Eercises 5 56, fin the vertical asmptotes (if an) of the graphs of the function 5 g 5 h 8 55 f 56 f csc 0 In Eercises 57 68, fin the one-sie it sin csc Environment A utilit compan burns coal to generate electricit The cost C in ollars of removing p% of the air pollutants in the stack emissions is C 80,000p 00 p, Fin the cost of removing (a) 5%, (b) 50%, an (c) 90% of the pollutants () Fin the it of C as p The function f is efine as shown f tan, (a) Fin f 0 f 0 tan 0 p < 00 0 (if it eists) sec 0 cos 0 (b) Can the function f be efine at 0 such that it is continuous at 0?

54 PS Problem Solving 9 PS Problem Solving The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph Let P, be a point on the parabola in the first quarant Consier the triangle PAO forme b P, A0,, an the origin O0, 0, an the triangle PBO forme b P, B, 0, an the origin A O (a) Write the perimeter of each triangle in terms of (b) Let r be the ratio of the perimeters of the two triangles, Complete the table (c) Calculate r 0 Let P, be a point on the parabola in the first quarant Consier the triangle PAO forme b P, A0,, an the origin O0, 0, an the triangle PBO forme b P, B, 0, an the origin A Perimeter PAO r Perimeter PBO B P Perimeter PAO Perimeter PBO r P 0 00 (a) Fin the area of a regular heagon inscribe in a circle of raius How close is this area to that of the circle? (b) Fin the area A n of an n-sie regular polgon inscribe in a circle of raius Write our answer as a function of n (c) Complete the table n A n () What number oes approach as n gets larger an larger? Figure for Figure for Let P, be a point on the circle 5 (a) What is the slope of the line joining P an O0, 0? (b) Fin an equation of the tangent line to the circle at P (c) Let Q, be another point on the circle in the first quarant Fin the slope of the line joining P an Q in terms of () Calculate m How oes this number relate to our answer in part (b)? 5 Let P5, be a point on the circle m A n 6 6 O 6 P(, ) Q 6 O B O 5 Q 5 (a) Write the area of each triangle in terms of (b) Let a be the ratio of the areas of the two triangles, a Complete the table (c) Calculate Area PBO Area PAO Area PAO Area PBO a a P(5, ) (a) What is the slope of the line joining P an O0, 0? (b) Fin an equation of the tangent line to the circle at P (c) Let Q, be another point on the circle in the fourth quarant Fin the slope m of the line joining P an Q in terms of () Calculate m How oes this number relate to our 5 answer in part (b)? 6 Fin the values of the constants a an b such that a b 0

55 9 CHAPTER Limits an Their Properties 7 Consier the function (a) Fin the omain of f (b) Use a graphing utilit to graph the function (c) Calculate f () Calculate f 8 Determine all values of the constant a such that the following function is continuous for all real numbers 9 Consier the graphs of the four functions g, g, g, an For each given conition of the function f, which of the graphs coul be the graph of f? (a) f (b) f is continuous at (c) 0 Sketch the graph of the function (a) Evaluate f, f, an f (b) Evaluate the its f 0 7 f a tan, 0 a, < 0 g g f f f, (c) Discuss the continuit of the function Sketch the graph of the function f (a) Evaluate f, f 0, f, an f 7 f, an (b) Evaluate the its f, f, an f f f, (c) Discuss the continuit of the function g g 0 g To escape Earth s gravitational fiel, a rocket must be launche with an initial velocit calle the escape velocit A rocke launche from the surface of Earth has velocit v (in miles per secon) given b v GM r where v 0 is the initial velocit, r is the istance from the rocket to the center of Earth, G is the gravitational constant, M is the mass of Earth, an R is the raius of Earth (approimatel 000 miles) (a) Fin the value of v 0 for which ou obtain an infinite i for r as v tens to zero This value of v 0 is the escape velocit for Earth (b) A rocket launche from the surface of the moon has velocit v (in miles per secon) given b Fin the escape velocit for the moon (c) A rocket launche from the surface of a planet has velocit v (in miles per secon) given b Fin the escape velocit for this planet Is the mass of this planet larger or smaller than that of Earth? (Assume that the mean ensit of this planet is the same as that of Earth) For positive numbers a < b, the pulse function is efine as P a,b H a H b 0,, 0, where v 90 r v 0,600 r 0 is the Heavisie function < 0 (a) Sketch the graph of the pulse function (b) Fin the following its: (i) () Wh is H, 0, a P a,b U b a P a,b v 0 GM R 9,000 v r 0 8 v 0 7 v calle the unit pulse function? Let a be a nonzero constant Prove that if f L, then (ii) a P a,b < a a < b b (iii) P a,b (iv) P a,b b b (c) Discuss the continuit of the pulse function 0 f a L Show b means of an eample that a must be 0 nonzero

56 96 CHAPTER Differentiation Section ISAAC NEWTON (6 77) In aition to his work in calculus, Newton mae revolutionar contributions to phsics, incluing the Law of Universal Gravitation an his three laws of motion MathBio P The Derivative an the Tangent Line Problem Fin the slope of the tangent line to a curve at a point Use the it efinition to fin the erivative of a function Unerstan the relationship between ifferentiabilit an continuit The Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on uring the seventeenth centur The tangent line problem (Section an this section) The velocit an acceleration problem (Sections an ) The minimum an maimum problem (Section ) The area problem (Sections an ) Each problem involves the notion of a it, an calculus can be introuce with an of the four problems A brief introuction to the tangent line problem is given in Section Although partial solutions to this problem were given b Pierre e Fermat (60 665), René Descartes ( ), Christian Hugens (69 695), an Isaac Barrow (60 677), creit for the first general solution is usuall given to Isaac Newton (6 77) an Gottfrie Leibniz (66 76) Newton s work on this problem stemme from his interest in optics an light refraction What oes it mean to sa that a line is tangent to a curve at a point? For a circle, the tangent line at a point P is the line that is perpenicular to the raial line at point P, as shown in Figure For a general curve, however, the problem is more ifficult For eample, how woul ou efine the tangent lines shown in Figure? You might sa that a line is tangent to a curve at a point P if it touches, but oes not cross, the curve at point P This efinition woul work for the first curve shown in Figure, but not for the secon Or ou might sa that a line is tangent to a curve if the line touches or intersects the curve at eactl one point This efinition woul work for a circle but not for more general curves, as the thir curve in Figure shows Tangent line to a circle Figure P = f() P = f() P = f() FOR FURTHER INFORMATION For more information on the creiting of mathematical iscoveries to the first iscoverer, see the article Mathematical Firsts Who Done It? b Richar H Williams an Ro D Mazzagatti in Mathematics Teacher MathArticle Tangent line to a curve at a point Figure EXPLORATION Ientifing a Tangent Line Use a graphing utilit to graph the function f 5 On the same screen, graph 5, 5, an 5 Which of these lines, if an, appears to be tangent to the graph of f at the point 0, 5? Eplain our reasoning

57 SECTION The Derivative an the Tangent Line Problem 97 (c +, f(c + )) Essentiall, the problem of fining the tangent line at a point P boils own to the problem of fining the slope of the tangent line at point P You can approimate this slope using a secant line* through the point of tangenc an a secon point on the curve, as shown in Figure If c, f c is the point of tangenc an c, f c is a secon point on the graph of f, the slope of the secant line through the two points is given b substitution into the slope formula (c, f(c)) f(c + ) f(c) = m fc f c m sec c c Change in Change in The secant line through c, fc an c, fc Figure m sec fc f c Slope of secant line The right-han sie of this equation is a ifference quotient The enominator is the change in, an the numerator f c f c is the change in The beaut of this proceure is that ou can obtain more an more accurate approimations of the slope of the tangent line b choosing points closer an closer to the point of tangenc, as shown in Figure THE TANGENT LINE PROBLEM In 67, mathematician René Descartes state this about the tangent line problem: An I are sa that this is not onl the most useful an general problem in geometr that I know, but even that I ever esire to know (c, f(c)) (c, f(c)) 0 (c, f(c)) (c, f(c)) (c, f(c)) (c, f(c)) 0 (c, f(c)) (c, f(c)) Tangent line approimations Figure Tangent line Tangent line To view a sequence of secant lines approaching a tangent line, select the Animation button Animation Definition of Tangent Line with Slope m If f is efine on an open interval containing c, an if the it f c f c m 0 0 eists, then the line passing through c, f c with slope m is the tangent line to the graph of f at the point c, f c Vieo Vieo The slope of the tangent line to the graph of f at the point c, f c is also calle the slope of the graph of f at c * This use of the wor secant comes from the Latin secare,meaning to cut,an is not a reference to the trigonometric function of the same name

58 98 CHAPTER Differentiation EXAMPLE The Slope of the Graph of a Linear Function = The slope of f at, is m Figure 5 Eitable Graph f() = = m = (, ) Fin the slope of the graph of f at the point, Solution To fin the slope of the graph of f when c, ou can appl the efinition of the slope of a tangent line, as shown f f The slope of f at c, f c, is m, as shown in Figure 5 NOTE In Eample, the it efinition of the slope of f agrees with the efinition of the slope of a line as iscusse in Section P Tr It Eploration A The graph of a linear function has the same slope at an point This is not true of nonlinear functions, as shown in the following eample EXAMPLE Tangent Lines to the Graph of a Nonlinear Function Tangent line at (,) The slope of f at an point c, fc is m c Figure 6 Eitable Graph f() = + Tangent line at (0, ) Fin the slopes of the tangent lines to the graph of at the points 0, an,, as shown in Figure 6 Solution Let c, f c represent an arbitrar point on the graph of f Then the slope of the tangent line at c, f c is given b So, the slope at an point c, f c on the graph of f is m c At the point 0,, the slope is m 0 0, an at,, the slope is m NOTE f 0 f c f c c c 0 c c c 0 c 0 c 0 c In Eample, note that c is hel constant in the it process as 0 Tr It Eploration A

59 SECTION The Derivative an the Tangent Line Problem 99 The graph of c, fc Figure 7 Vertical tangent line c (c, f(c)) f has a vertical tangent line at The efinition of a tangent line to a curve oes not cover the possibilit of a vertical tangent line For vertical tangent lines, ou can use the following efinition If f is continuous at c an f c f c 0 or the vertical line c passing through c, f c is a vertical tangent line to the graph of f For eample, the function shown in Figure 7 has a vertical tangent line at c, f c If the omain of f is the close interval a, b, ou can eten the efinition of a vertical tangent line to inclue the enpoints b consiering continuit an its from the right for a an from the left for b The Derivative of a Function f c f c 0 You have now arrive at a crucial point in the stu of calculus The it use to efine the slope of a tangent line is also use to efine one of the two funamental operations of calculus ifferentiation Definition of the Derivative of a Function The erivative of f at is given b f 0 f f provie the it eists For all for which this it eists, is a function of f Vieo Be sure ou see that the erivative of a function of is also a function of This new function gives the slope of the tangent line to the graph of f at the point, f, provie that the graph has a tangent line at this point The process of fining the erivative of a function is calle ifferentiation A function is ifferentiable at if its erivative eists at an is ifferentiable on an open interval a, b if it is ifferentiable at ever point in the interval In aition to f, which is rea as f prime of, other notations are use to enote the erivative of f The most common are f,,, f, D Notation for erivatives The notation is rea as the erivative of with respect to or simpl Using it notation, ou can write 0 0 f f f Histor

60 00 CHAPTER Differentiation EXAMPLE Fining the Derivative b the Limit Process STUDY TIP When using the efinition to fin a erivative of a function, the ke is to rewrite the ifference quotient so that oes not occur as a factor of the enominator Fin the erivative of f Solution f f f Definition of erivative 0 0 Tr It Eploration A Eploration B Eploration C Open Eploration The eitable graph feature below allows ou to eit the graph of a function an its erivative Eitable Graph Remember that the erivative of a function f is itself a function, which can be use to fin the slope of the tangent line at the point, f on the graph of f EXAMPLE Using the Derivative to Fin the Slope at a Point (, ) (, ) m = m = f() = (0, 0) The slope of f at, f, > 0, is m Figure 8 Eitable Graph Fin f for f Then fin the slope of the graph of f at the points, an, Discuss the behavior of f at 0, 0 Solution Use the proceure for rationalizing numerators, as iscusse in Section f f f Definition of erivative , > 0 At the point,, the slope is f At the point the slope is f,, See Figure 8 At the point 0, 0, the slope is unefine Moreover, the graph of f has a vertical tangent line at 0, 0 Tr It Eploration A Eploration B Eploration C

61 SECTION The Derivative an the Tangent Line Problem 0 In man applications, it is convenient to use a variable other than inepenent variable, as shown in Eample 5 as the EXAMPLE 5 Fining the Derivative of a Function Fin the erivative with respect to t for the function t Solution Consiering f t, ou obtain t t 0 t 0 t 0 t 0 t 0 t f t t f t t t t t t t t t tt t t t ttt t tt t Definition of erivative f t t t tan f t t Combine fractions in numerator Divie out common factor of t Simplif Evaluate it as t 0 Tr It Eploration A Open Eploration = t The eitable graph feature below allows ou to eit the graph of a function an its erivative (, ) Eitable Graph 0 0 = t + At the point, the line t is tangent to the graph of t Figure 9 (c, f(c)) c c (, f()) 6 f() f(c) As approaches c, the secant line approaches the tangent line Figure 0 TECHNOLOGY A graphing utilit can be use to reinforce the result given in Eample 5 For instance, using the formula t t, ou know that the slope of the graph of t at the point, is m This implies that an equation of the tangent line to the graph at, is t as shown in Figure 9 Differentiabilit an Continuit The following alternative it form of the erivative is useful in investigating the relationship between ifferentiabilit an continuit The erivative of f at c is provie this it eists (see Figure 0) (A proof of the equivalence of this form is given in Appeni A) Note that the eistence of the it in this alternative form requires that the one-sie its c fc c f f c c f f c c or an t f f c c c Alternative form of erivative eist an are equal These one-sie its are calle the erivatives from the left an from the right, respectivel It follows that f is ifferentiable on the close interval [a, b] if it is ifferentiable on a, b an if the erivative from the right at a an the erivative from the left at b both eist

62 0 CHAPTER Differentiation f() = [[ ]] The greatest integer function is not ifferentiable at 0, because it is not continuous at 0 Figure If a function is not continuous at c, it is also not ifferentiable at c For instance, the greatest integer function is not continuous at 0, an so it is not ifferentiable at 0 (see Figure ) You can verif this b observing that an f f f f f Derivative from the left Derivative from the right Although it is true that ifferentiabilit implies continuit (as shown in Theorem on the net page), the converse is not true That is, it is possible for a function to be continuous at c an not ifferentiable at c Eamples 6 an 7 illustrate this possibilit EXAMPLE 6 A Graph with a Sharp Turn m = f() = m = f is not ifferentiable at, because the erivatives from the left an from the right are not equal Figure The function shown in Figure is continuous at But, the one-sie its an f f f 0 f f 0 Derivative from the left Derivative from the right are not equal So, f is not ifferentiable at an the graph of f oes not have a tangent line at the point, 0 Eitable Graph Tr It Eploration A Open Eploration f() = / f is not ifferentiable at 0, because f has a vertical tangent at 0 Figure EXAMPLE 7 The function f A Graph with a Vertical Tangent Line is continuous at 0, as shown in Figure But, because the it f f is infinite, ou can conclue that the tangent line is vertical at 0 So, f is not ifferentiable at 0 Eitable Graph Tr It Eploration A Eploration B Eploration C From Eamples 6 an 7, ou can see that a function is not ifferentiable at a point at which its graph has a sharp turn or a vertical tangent

63 SECTION The Derivative an the Tangent Line Problem 0 TECHNOLOGY Some graphing utilities, such as Derive, Maple, Mathca, Mathematica,an the TI-89, perform smbolic ifferentiation Others perform numerical ifferentiation b fining values of erivatives using the formula f f f where is a small number such as 000 Can ou see an problems with this efinition? For instance, using this efinition, what is the value of the erivative of f when 0? THEOREM Differentiabilit Implies Continuit If f is ifferentiable at c, then f is continuous at c Proof You can prove that f is continuous at c b showing that f approaches f c as c To o this, use the ifferentiabilit of f at c an consier the following it f f c c c c f f c c c c 0 fc 0 f f c c c Because the ifference f f c approaches zero as c, ou can conclue that f f c So, f is continuous at c c The following statements summarize the relationship between continuit an ifferentiabilit If a function is ifferentiable at c, then it is continuous at c So, ifferentiabilit implies continuit It is possible for a function to be continuous at c an not be ifferentiable at c So, continuit oes not impl ifferentiabilit

64 SECTION The Derivative an the Tangent Line Problem 0 Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises an, estimate the slope of the graph at the points, an, (a) (b) (, ) (, ) (, ) (, ) In Eercises an, use the graph shown in the figure To print an enlarge cop of the graph, select the MathGraph button 6 5 (, ) (, 5) 5 f 6 (a) (b) ( (, ), ) (, ) (, ) Ientif or sketch each of the quantities on the figure (a) f an f (b) f f (c) Insert the proper inequalit smbol < or > between the given quantities (a) (b) f f f f f f f f f f

65 0 CHAPTER Differentiation In Eercises 5 0, fin the slope of the tangent line to the graph of the function at the given point 5 f,, 5 6 g, 7 g,, 8 g 5, In Eercises, fin the erivative b the it process f 5 f 5 hs s 6 f 9 In Eercises 5, (a) fin an equation of the tangent line to the graph of f at the given point, (b) use a graphing utilit to graph the function an its tangent line at the point, an (c) use the erivative feature of a graphing utilit to confirm our results 5 6 f, 5, In Eercises 6, fin an equation of the line that is tangent to the graph of f an parallel to the given line 5 6 f, f, Function f f f f 7 8 f 9 f t t t, 0, 0 0 ht t, f g 5 7 f 8 f 9 f 0 f f f f f, 5, 7 f,, 8 8 f, 9 f,, 0 f, Line In Eercises 7 0, the graph of f of f is given Select the graph 5 f, 0 0 f,,, 7, 5, 0, 9 0 (a) (c) f 5 f f The tangent line to the graph of g at the point 5, passes through the point 9, 0 Fin g5 an g5 The tangent line to the graph of h at the point, passes through the point, 6 Fin h an h (b) () Writing About Concepts 5 In Eercises 6, sketch the graph of f Eplain how ou foun our answer f 5 6 f 7 Sketch a graph of a function whose erivative is alwas negative f f f f f

66 SECTION The Derivative an the Tangent Line Problem 05 Writing About Concepts (continue) 8 Sketch a graph of a function whose erivative is alwas positive In Eercises 9 5, the it represents fc for a function f an a number c Fin f an c In Eercises 5 55, ientif a function f that has the following characteristics Then sketch the function 5 f 0 ; 5 f 55 f 0 0; f 0 0; f > 0 if 0 < 0 for < 0; > 0 for > 0 56 Assume that fc Fin fc if (a) f is an o function an if (b) f is an even function In Eercises 57 an 58, fin equations of the two tangent lines to the graph of f that pass through the inicate point 5, < < 57 f 58 f 59 Graphical Reasoning The figure shows the graph of g (, 5) g (a) g0 (b) g (c) What can ou conclue about the graph of g knowing that g 8? () What can ou conclue about the graph of g knowing that g 7? (e) Is g6 g positive or negative? Eplain 6 (, ) (f) Is it possible to fin g from the graph? Eplain f f f 0 ; f 0 0; 60 Graphical Reasoning Use a graphing utilit to graph each function an its tangent lines at, 0, an Base on the results, etermine whether the slopes of tangen lines to the graph of a function at ifferent values of are alwas istinct (a) (b) Graphical, Numerical, an Analtic Analsis In Eercises 6 an 6, use a graphing utilit to graph f on the interval [, ] Complete the table b graphicall estimating the slopes of the graph at the inicate points Then evaluate the slopes analticall an compare our results with those obtaine graphicall 6 f 6 Graphical Reasoning In Eercises 6 an 6, use a graphing utilit to graph the functions f an g in the same viewing winow where g Label the graphs an escribe the relationship between them 6 f 6 f In Eercises 65 an 66, evaluate f an f an use the results to approimate f 65 f 66 f Graphical Reasoning In Eercises 67 an 68, use a graphing utilit to graph the function an its erivative in the same viewing winow Label the graphs an escribe the relationship between them 67 f 68 Writing In Eercises 69 an 70, consier the functions f an where S S f f 00 f 00 g f f f f f f f (a) Use a graphing utilit to graph f an S in the same viewing winow for, 05, an 0 (b) Give a written escription of the graphs of S for the ifferent values of in part (a) 69 f 70 f

67 06 CHAPTER Differentiation In Eercises 7 80, use the alternative form of the erivative to fin the erivative at c (if it eists) 7 f, c 7 g, c g, c In Eercises 8 86, escribe the -values at which f is ifferentiable 8 f 8 85 f 86 Graphical Analsis In Eercises 87 90, use a graphing utilit to fin the -values at which f is ifferentiable f, c f, c g, c 0 f, c f 6, c 6 h 5, c 5 5 f f 5 f,, 5 6 > f, c f 9 8 f 8 f f,, f > 0 In Eercises 9 9, fin the erivatives from the left an from the right at (if the eist) Is the function ifferentiable at? f f, 9, > In Eercises 95 an 96, etermine whether the function is ifferentiable at 95 f, 96, > 97 Graphical Reasoning A line with slope m passes through the point 0, an has the equation m (a) Write the istance between the line an the point, as a function of m (b) Use a graphing utilit to graph the function in part (a) Base on the graph, is the function ifferentiable at ever value of m? If not, where is it not ifferentiable? 98 Conjecture Consier the functions f an g (a) Graph f an on the same set of aes g (b) Graph g an on the same set of aes (c) Ientif a pattern between f an g an their respective erivatives Use the pattern to make a conjecture abou h if h n, where n is an integer an n () Fin f if f Compare the result with the conjecture in part (c) Is this a proof of our conjecture? Eplain True or False? In Eercises 99 0, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 99 The slope of the tangent line to the ifferentiable function f a f f the point, f is 00 If a function is continuous at a point, then it is ifferentiable at that point 0 If a function has erivatives from both the right an the left a a point, then it is ifferentiable at that point 0 If a function is ifferentiable at a point, then it is continuous at that point 0 Let f sin, 0 an 0, 0 f f f,, f,, g sin, 0, > < 0 0 Show that f is continuous, but not ifferentiable, at 0 Show that g is ifferentiable at 0, an fin g0 0 Writing Use a graphing utilit to graph the two functions f an g in the same viewing winow Use the zoom an trace features to analze the graphs near the point 0, What o ou observe? Which function is ifferentiable at this point? Write a short paragraph escribing the geometric significance of ifferentiabilit at a point

68 SECTION Basic Differentiation Rules an Rates of Change 07 Section Basic Differentiation Rules an Rates of Change Vieo Vieo The slope of a horizontal line is 0 Vieo Vieo Fin the erivative of a function using the Constant Rule Fin the erivative of a function using the Power Rule Fin the erivative of a function using the Constant Multiple Rule Fin the erivative of a function using the Sum an Difference Rules Fin the erivatives of the sine function an of the cosine function Use erivatives to fin rates of change The Constant Rule In Section ou use the it efinition to fin erivatives In this an the net two sections ou will be introuce to several ifferentiation rules that allow ou to fin erivatives without the irect use of the it efinition THEOREM The Constant Rule The erivative of a constant function is 0 f() = c The erivative of a constant function is 0 That is, if c is a real number, then c 0 The Constant Rule Figure NOTE In Figure, note that the Constant Rule is equivalent to saing that the slope of a horizontal line is 0 This emonstrates the relationship between slope an erivative Proof Let f c Then, b the it efinition of the erivative, c f f f c c EXAMPLE Using the Constant Rule Function a 7 b f 0 c st k, k is constant Derivative 0 f 0 st 0 0 Tr It Eploration A The eitable graph feature below allows ou to eit the graph of a function a Eitable Graph b Eitable Graph c Eitable Graph Eitable Graph EXPLORATION Writing a Conjecture Use the efinition of the erivative given in Section to fin the erivative of each function What patterns o ou see? Use our results to write a conjecture about the erivative of f n a f b f c f f e f f f

69 08 CHAPTER Differentiation The Power Rule Before proving the net rule, it is important to review the proceure for epaning a binomial The general binomial epansion for a positive integer n is n n n n nn n n is a factor of these terms This binomial epansion is use in proving a special case of the Power Rule THEOREM The Power Rule If n is a rational number, then the function f n is ifferentiable an n n n n For f to be ifferentiable at 0, n must be a number such that is efine on an interval containing 0 = Proof If n is a positive integer greater than, then the binomial epansion prouces n n n 0 nn n n n n n n 0 0 n nn n n n n n 0 0 n n This proves the case for which n is a positive integer greater than You will prove the case for n Eample 7 in Section proves the case for which n is a negative integer In Eercise 75 in Section 5 ou are aske to prove the case for which n is rational (In Section 55, the Power Rule will be etene to cover irrational values of n ) When using the Power Rule, the case for which n is best thought of as a separate ifferentiation rule That is, The slope of the line is Figure 5 Power Rule when n This rule is consistent with the fact that the slope of the line is, as shown in Figure 5

70 SECTION Basic Differentiation Rules an Rates of Change 09 EXAMPLE Using the Power Rule Function Derivative a b c f g f) g Tr It Eploration A In Eample (c), note that before ifferentiating, Rewriting is the first step in man ifferentiation problems was rewritten as f() = Given: Rewrite: Differentiate: Simplif: EXAMPLE Fining the Slope of a Graph (, ) (, ) Fin the slope of the graph of f when a b 0 c (0, 0) Note that the slope of the graph is negative at the point,, the slope is zero at the point 0, 0, an the slope is positive at the point, Figure 6 Eitable Graph Solution The slope of a graph at a point is the value of the erivative at that point The erivative of f is f a When, the slope is f Slope is negative b When 0, the slope is f0 0 0 Slope is zero c When, the slope is f Slope is positive See Figure 6 Tr It Eploration A Open Eploration EXAMPLE Fining an Equation of a Tangent Line f() = (, ) = The line is tangent to the graph of f at the point, Figure 7 Fin an equation of the tangent line to the graph of f when Solution To fin the point on the graph of f, evaluate the original function at, f, Point on graph To fin the slope of the graph when, evaluate the erivative, f, at m f Slope of graph at, Now, using the point-slope form of the equation of a line, ou can write m See Figure 7 Point-slope form Substitute for, m, an Simplif Eitable Graph Tr It Eploration A Eploration B Open Eploration

71 0 CHAPTER Differentiation The Constant Multiple Rule THEOREM The Constant Multiple Rule If f is a ifferentiable function an c is a real number, then cf is also ifferentiable an cf cf Proof cf cf cf 0 0 c c 0 cf f f f f Definition of erivative Appl Theorem Informall, the Constant Multiple Rule states that constants can be factore out of the ifferentiation process, even if the constants appear in the enominator cf c f cf f c c f c f c f EXAMPLE 5 a b c e Function t ft 5 Tr It Using the Constant Multiple Rule Derivative ft t 5 t 5 t t 5 t 8 5 t 5 5 Eploration A The Constant Multiple Rule an the Power Rule can be combine into one rule The combination rule is D c n cn n

72 SECTION Basic Differentiation Rules an Rates of Change EXAMPLE 6 Using Parentheses When Differentiating Original Function Rewrite Differentiate Simplif a b c Tr It Eploration A The Sum an Difference Rules THEOREM 5 The Sum an Difference Rules The sum (or ifference) of two ifferentiable functions f an g is itself ifferentiable Moreover, the erivative of f g or f g is the sum (or ifference) of the erivatives of f an g f g f g f g f g Proof A proof of the Sum Rule follows from Theorem (The Difference Rule can be prove in a similar wa) f g f g f g 0 0 f f 0 0 f g f g f g f f Sum Rule Difference Rule g g g g 0 The Sum an Difference Rules can be etene to an finite number of functions For instance, if F f g h, then F f g h EXAMPLE 7 Using the Sum an Difference Rules a b Function f 5 g Tr It Eploration A Derivative f g 9 The eitable graph feature below allows ou to eit the graph of a function an its erivative Eitable Graph

73 CHAPTER Differentiation FOR FURTHER INFORMATION For the outline of a geometric proof of the erivatives of the sine an cosine functions, see the article The Spier s Spacewalk Derivation of an b Tim Hesterberg in The College Mathematics Journal sin MathArticle cos Derivatives of Sine an Cosine Functions In Section, ou stuie the following its sin 0 an cos 0 0 These two its can be use to prove ifferentiation rules for the sine an cosine functions (The erivatives of the other four trigonometric functions are iscusse in Section ) THEOREM 6 Derivatives of Sine an Cosine Functions sin cos cos sin = increasing = 0 π = π = sin = π = 0 ecreasing increasing positive negative positive π π = cos π The erivative of the sine function is the cosine function Figure 8 Proof sin sin sin 0 0 cos 0 cos sin 0 cos sin cos cos sin sin sin sin 0 Definition of erivative cos sin sin cos 0 0 cos sin sin cos cos This ifferentiation rule is shown graphicall in Figure 8 Note that for each, the slope of the sine curve is equal to the value of the cosine The proof of the secon rule is left as an eercise (see Eercise 6) Animation EXAMPLE 8 Derivatives Involving Sines an Cosines = sin = sin a b Function sin sin sin Derivative cos cos cos c cos sin = sin = sin a sin a cos Figure 9 TECHNOLOGY A graphing utilit can provie insight into the interpretation of a erivative For instance, Figure 9 shows the graphs of a sin for a,,, an Estimate the slope of each graph at the point 0, 0 Then verif our estimates analticall b evaluating the erivative of each function when 0 Tr It Eploration A Open Eploration

74 SECTION Basic Differentiation Rules an Rates of Change Rates of Change You have seen how the erivative is use to etermine slope The erivative can also be use to etermine the rate of change of one variable with respect to another Applications involving rates of change occur in a wie variet of fiels A few eamples are population growth rates, prouction rates, water flow rates, velocit, an acceleration A common use for rate of change is to escribe the motion of an object moving in a straight line In such problems, it is customar to use either a horizontal or a vertical line with a esignate origin to represent the line of motion On such lines, movement to the right (or upwar) is consiere to be in the positive irection, an movement to the left (or ownwar) is consiere to be in the negative irection The function s that gives the position (relative to the origin) of an object as a function of time t is calle a position function If, over a perio of time t, the object changes its position b the amount s st t st, then, b the familiar formula Rate istance time the average velocit is Change in istance Change in time s t Average velocit EXAMPLE 9 Fining Average Velocit of a Falling Object If a billiar ball is roppe from a height of 00 feet, its height s at time t is given b the position function s 6t 00 Position function where s is measure in feet an t is measure in secons Fin the average velocit over each of the following time intervals a, b, 5 c, Solution a For the interval,, the object falls from a height of s feet to a height of s feet The average velocit is s t feet per secon b For the interval, 5, the object falls from a height of 8 feet to a height of 6 feet The average velocit is s t feet per secon c For the interval,, the object falls from a height of 8 feet to a height of 806 feet The average velocit is s t 0 feet per secon Note that the average velocities are negative, inicating that the object is moving ownwar Tr It Eploration A Eploration B

75 CHAPTER Differentiation s P Secant line Tangent line Suppose that in Eample 9 ou wante to fin the instantaneous velocit (or simpl the velocit) of the object when t Just as ou can approimate the slope of the tangent line b calculating the slope of the secant line, ou can approimate the velocit at t b calculating the average velocit over a small interval, t (see Figure 0) B taking the it as t approaches zero, ou obtain the velocit when t Tr oing this ou will fin that the velocit when t is feet per secon In general, if s st is the position function for an object moving along a straight line, the velocit of the object at time t is t = t The average velocit between t an t is the slope of the secant line, an the instantaneous velocit at t is the slope of the tangent line Figure 0 Animation t st t st vt st t 0 t Velocit function In other wors, the velocit function is the erivative of the position function Velocit can be negative, zero, or positive The spee of an object is the absolute value of its velocit Spee cannot be negative The position of a free-falling object (neglecting air resistance) uner the influence of gravit can be represente b the equation st gt v 0 t s 0 Position function where s 0 is the initial height of the object, v 0 is the initial velocit of the object, an g is the acceleration ue to gravit On Earth, the value of g is approimatel feet per secon per secon or 98 meters per secon per secon Histor EXAMPLE 0 Using the Derivative to Fin Velocit ft Velocit is positive when an object is rising, an is negative when an object is falling Figure Animation NOTE In Figure, note that the iver moves upwar for the first halfsecon because the velocit is positive for 0 < t < When the velocit is 0, the iver has reache the maimum height of the ive At time t 0, a iver jumps from a platform iving boar that is feet above the water (see Figure ) The position of the iver is given b st 6t 6t Position function where s is measure in feet an t is measure in secons a When oes the iver hit the water? b What is the iver s velocit at impact? Solution a To fin the time t when the iver hits the water, let s 0 an solve for t 6t 6t 0 Set position function equal to 0 6t t 0 Factor t or Solve for t Because t 0, choose the positive value to conclue that the iver hits the water at t secons b The velocit at time t is given b the erivative st t 6 So, the velocit at time t is s 6 8 feet per secon Tr It Eploration A

76 SECTION Basic Differentiation Rules an Rates of Change 5 Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises an, use the graph to estimate the slope of the tangent line to n at the point, Verif our answer analticall To print an enlarge cop of the graph, select the MathGraph button (a) (b) (, ) (, ) 9 0 In Eercises 8, fin the slope of the graph of the function at the given point Use the erivative feature of a graphing utilit to confirm our results Original Function Function f Rewrite Differentiate Point, Simplif (a) (b) (, ) (, ) ft 5t 5, f 7 5 0, 6 f 5 f sin gt cos t, 8 0, 5, 0 0, 0, In Eercises, fin the erivative of the function 8 f f 5 0 g f g f t t t 6 t t 5 g st t t 8 f 9 0 gt cos t sin cos cos 5 sin sin In Eercises 5 0, complete the table Original Function 5 Rewrite 5 cos Differentiate Simplif In Eercises 9 5, fin the erivative of the function 9 f 5 0 f gt t f t f f 6 8 f 5 9 hs s 5 s 50 f t t t 5 f 6 5 cos 5 In Eercises 5 56, (a) fin an equation of the tangent line to the graph of f at the given point, (b) use a graphing utilit to graph the function an its tangent line at the point, an (c) use the erivative feature of a graphing utilit to confirm our results Function f Point, 0,, 56, 6 h f cos

77 6 CHAPTER Differentiation In Eercises 57 6, etermine the point(s) (if an) at which the graph of the function has a horizontal tangent line In Eercises 6 66, fin k such that the line is tangent to the graph of the function sin, cos, Function f k 9 f k 7 f k f k 0 < 0 < Writing About Concepts Line 67 Use the graph of f to answer each question To print an enlarge cop of the graph, select the MathGraph button (a) Between which two consecutive points is the average rate of change of the function greatest? (b) Is the average rate of change of the function between A an B greater than or less than the instantaneous rate of change at B? (c) Sketch a tangent line to the graph between C an D such that the slope of the tangent line is the same as the average rate of change of the function between C an D 68 Sketch the graph of a function f such that > 0 for all an the rate of change of the function is ecreasing In Eercises 69 an 70, the relationship between f an g is given Eplain the relationship between an g 69 A B C D g f 6 70 g 5 f E f f f Writing About Concepts (continue) In Eercises 7 an 7, the graphs of a function f an its erivative are shown on the same set of coorinate aes Label the graphs as f or an write a short paragraph stating the criteria use in making the selection To print an enlarge cop of the graph, select the MathGraph button f Sketch the graphs of an 6 5, an sketch the two lines that are tangent to both graphs Fin equations of these lines 7 Show that the graphs of the two equations an have tangent lines that are perpenicular to each other at their point of intersection 75 Show that the graph of the function f sin oes not have a horizontal tangent line 76 Show that the graph of the function f 5 5 oes not have a tangent line with a slope of In Eercises 77 an 78, fin an equation of the tangent line to the graph of the function f through the point 0, 0 not on the graph To fin the point of tangenc, on the graph of f solve the equation f f 78 f 0, 0, 0 79 Linear Approimation Use a graphing utilit, with a square winow setting, to zoom in on the graph of f to approimate f Use the erivative to fin f 80 Linear Approimation Use a graphing utilit, with a square winow setting, to zoom in on the graph of f f 0, 0 5, 0 to approimate f Use the erivative to fin f

78 SECTION Basic Differentiation Rules an Rates of Change 7 8 Linear Approimation Consier the function with the solution point, 8 (a) Use a graphing utilit to graph f Use the zoom feature to obtain successive magnifications of the graph in the neighborhoo of the point, 8 After zooming in a few times, the graph shoul appear nearl linear Use the trace feature to etermine the coorinates of a point near, 8 Fin an equation of the secant line S through the two points (b) Fin the equation of the line tangent to the graph of f passing through the given point Wh are the linear functions S an T nearl the same? (c) Use a graphing utilit to graph f an T on the same set of coorinate aes Note that T is a goo approimation of f when is close to What happens to the accurac of the approimation as ou move farther awa from the point of tangenc? () Demonstrate the conclusion in part (c) b completing the table T f f f T f T 8 Linear Approimation Repeat Eercise 8 for the function f where T is the line tangent to the graph at the point, Eplain wh the accurac of the linear approimation ecreases more rapil than in Eercise 8 True or False? In Eercises 8 88, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 8 If f g, then f g 8 If f g c, then f g 85 If, then 86 If, then 87 If g f, then g f 88 If f n, then f n n 05 In Eercises 89 9, fin the average rate of change of the function over the given interval Compare this average rate of change with the instantaneous rates of change at the enpoints of the interval 89 f t t 7,, 90 f t t, f , 0 9 f, 9 f sin,, Vertical Motion In Eercises 9 an 9, use the position function st 6t v 0 t s 0 for free-falling objects 9 A silver ollar is roppe from the top of a builing that is 6 feet tall (a) Determine the position an velocit functions for the coin (b) Determine the average velocit on the interval, (c) Fin the instantaneous velocities when t an t () Fin the time require for the coin to reach groun level (e) Fin the velocit of the coin at impact 9 A ball is thrown straight own from the top of a 0-foo builing with an initial velocit of feet per secon Wha is its velocit after secons? What is its velocit after falling 08 feet? Vertical Motion In Eercises 95 an 96, use the position function st 9t v 0 t s 0 for free-falling objects 95 A projectile is shot upwar from the surface of Earth with an initial velocit of 0 meters per secon What is its velocit after 5 secons? After 0 secons? 96 To estimate the height of a builing, a stone is roppe from the top of the builing into a pool of water at groun level How high is the builing if the splash is seen 68 secons after the stone is roppe? Think About It In Eercises 97 an 98, the graph of a position function is shown It represents the istance in miles that a person rives uring a 0-minute trip to work Make a sketch of the corresponing velocit function 97 s 98 Distance (in miles) Think About It In Eercises 99 an 00, the graph of a velocit function is shown It represents the velocit in miles per hour uring a 0-minute rive to work Make a sketch of the corresponing position function 99 v 00 Velocit (in mph) (0, 0) (, ) (0, 6) (6, ) Time (in minutes) Time (in minutes) t t Distance (in miles) Velocit (in mph) (0, 0) s Time (in minutes) v 0, 6 (0, 6) (6, 5) (8, 5) Time (in minutes) t t

79 8 CHAPTER Differentiation 0 Moeling Data The stopping istance of an automobile, on r, level pavement, traveling at a spee v (kilometers per hour) is the istance R (meters) the car travels uring the reaction time of the river plus the istance B (meters) the car travels after the brakes are applie (see figure) The table shows the results of an eperiment Driver sees obstacle Reaction time R Driver applies brakes Braking istance Car stops Spee, v Reaction Time Distance, R Braking Time Distance, B (a) Use the regression capabilities of a graphing utilit to fin a linear moel for reaction time istance (b) Use the regression capabilities of a graphing utilit to fin a quaratic moel for braking istance (c) Determine the polnomial giving the total stopping istance T () Use a graphing utilit to graph the functions R, B, an T in the same viewing winow (e) Fin the erivative of T an the rates of change of the total stopping istance for v 0, v 80, an v 00 (f) Use the results of this eercise to raw conclusions about the total stopping istance as spee increases 0 Fuel Cost A car is riven 5,000 miles a ear an gets miles per gallon Assume that the average fuel cost is $55 per gallon Fin the annual cost of fuel C as a function of an use this function to complete the table Who woul benefit more from a one-mile-per-gallon increase in fuel efficienc the river of a car that gets 5 miles per gallon or the river of a car that gets 5 miles per gallon? Eplain 0 Volume The volume of a cube with sies of length s is given b V s Fin the rate of change of the volume with respect to s when s centimeters 0 Area The area of a square with sies of length s is given b A s Fin the rate of change of the area with respect to s when s meters B C C/ 05 Velocit Verif that the average velocit over the time interval t 0 t, t 0 t is the same as the instantaneous velocit at t t 0 for the position function 06 Inventor Management The annual inventor cost C for a manufacturer is where Q is the orer size when the inventor is replenishe Fin the change in annual cost when Q is increase from 50 to 5, an compare this with the instantaneous rate of change when Q Writing The number of gallons N of regular unleae gasoline sol b a gasoline station at a price of p ollars pe gallon is given b N f p (a) Describe the meaning of f79 (b) Is f79 usuall positive or negative? Eplain 08 Newton s Law of Cooling This law states that the rate o change of the temperature of an object is proportional to the ifference between the object s temperature T an the temperature T a of the surrouning meium Write an equation for this law 09 Fin an equation of the parabola a b c that passes through 0, an is tangent to the line at, 0 0 Let a, b be an arbitrar point on the graph of > 0 Prove that the area of the triangle forme b the tangent line through a, b an the coorinate aes is Fin the tangent line(s) to the curve 9 through the point, 9 Fin the equation(s) of the tangent line(s) to the parabola through the given point (a) 0, a (b) a, 0 Are there an restrictions on the constant a? In Eercises an, fin a an b such that f is ifferen tiable everwhere st at c C,008,000 Q f a, b, f cos, a b, 5 Where are the functions an f sin ifferentiable? 6 Prove that 6Q > < 0 0 cos sin f sin FOR FURTHER INFORMATION For a geometric interpretation of the erivatives of trigonometric functions, see the article Sines an Cosines of the Times b Victor J Katz in Math Horizons MathArticle

80 SECTION Prouct an Quotient Rules an Higher-Orer Derivatives 9 Section Prouct an Quotient Rules an Higher-Orer Derivatives Fin the erivative of a function using the Prouct Rule Fin the erivative of a function using the Quotient Rule Fin the erivative of a trigonometric function Fin a higher-orer erivative of a function The Prouct Rule In Section ou learne that the erivative of the sum of two functions is simpl the sum of their erivatives The rules for the erivatives of the prouct an quotient of two functions are not as simple NOTE A version of the Prouct Rule that some people prefer is fg fg fg The avantage of this form is that it generalizes easil to proucts involving three or more factors THEOREM 7 The Prouct Rule The prouct of two ifferentiable functions f an g is itself ifferentiable Moreover, the erivative of fg is the first function times the erivative of the secon, plus the secon function times the erivative of the first fg fg gf Vieo fg 0 Proof Some mathematical proofs, such as the proof of the Sum Rule, are straightforwar Others involve clever steps that ma appear unmotivate to a reaer This proof involves such a step subtracting an aing the same quantit which is shown in color 0 g f g g 0 g 0 f g 0 f g fg f g f g f g fg g g f f f g fg gf f f f f g 0 THE PRODUCT RULE When Leibniz originall wrote a formula for the Prouct Rule, he was motivate b the epression from which he subtracte (as being negligible) an obtaine the ifferential form This erivation resulte in the traitional form of the Prouct Rule (Source:The Histor of Mathematics b Davi M Burton) Note that f f because f is given to be ifferentiable an therefore 0 is continuous The Prouct Rule can be etene to cover proucts involving more than two factors For eample, if f, g, an h are ifferentiable functions of, then fgh fgh fgh fgh For instance, the erivative of sin cos is sin cos cos cos sin sin sin cos cos sin

81 0 CHAPTER Differentiation The erivative of a prouct of two functions is not (in general) given b the prouct of the erivatives of the two functions To see this, tr comparing the prouct of the erivatives of f an g 5 with the erivative in Eample EXAMPLE Using the Prouct Rule Fin the erivative of h 5 Solution Derivative Derivative First of secon Secon of first h 5 5 Appl Prouct Rule In Eample, ou have the option of fining the erivative with or without the Prouct Rule To fin the erivative without the Prouct Rule, ou can write D 5 D Tr It Eploration A In the net eample, ou must use the Prouct Rule EXAMPLE Using the Prouct Rule Fin the erivative of sin Solution sin sin sin cos sin 6 cos 6 sin cos sin Tr It Eploration A Technolog Appl Prouct Rule The eitable graph feature below allows ou to eit the graph of a function an its erivative Eitable Graph EXAMPLE Using the Prouct Rule Fin the erivative of cos sin NOTE In Eample, notice that ou use the Prouct Rule when both factors of the prouct are variable, an ou use the Constant Multiple Rule when one of the factors is a constant Solution sin Prouct Rule sin cos cos Constant Multiple Rule cos cos sin Tr It Eploration A Technolog

82 SECTION Prouct an Quotient Rules an Higher-Orer Derivatives The Quotient Rule THEOREM 8 The Quotient Rule The quotient fg of two ifferentiable functions f an g is itself ifferentiable at all values of for which g 0 Moreover, the erivative of fg is given b the enominator times the erivative of the numerator minus the numerator times the erivative of the enominator, all ivie b the square of the enominator g f gf fg, g g 0 TECHNOLOGY A graphing utilit can be use to compare the graph of a function with the graph of its erivative For instance, in Figure, the graph of the function in Eample appears to have two points that have horizontal tangent lines What are the values of at these two points? = ( + ) 7 = Graphical comparison of a function an its erivative Figure 8 Vieo Proof As with the proof of Theorem 7, the ke to this proof is subtracting an aing the same quantit g f 0 Note that is continuous 0 EXAMPLE gf fg 0 gg g 0 f g gf fg g Definition of erivative gf fg fg fg 0 gg g f f fg g 0 0 g g because g is given to be ifferentiable an therefore Using the Quotient Rule 5 Fin the erivative of Solution f g 0 f 0 f f gg 0 gg Tr It Eploration A Eploration B g g Appl Quotient Rule The eitable graph feature below allows ou to eit the graph of a function an its erivative Eitable Graph

83 CHAPTER Differentiation Note the use of parentheses in Eample A liberal use of parentheses is recommene for all tpes of ifferentiation problems For instance, with the Quotient Rule, it is a goo iea to enclose all factors an erivatives in parentheses, an to pa special attention to the subtraction require in the numerator When ifferentiation rules were introuce in the preceing section, the nee for rewriting before ifferentiating was emphasize The net eample illustrates this point with the Quotient Rule EXAMPLE 5 Rewriting Before Differentiating f() = = (, ) The line is tangent to the graph of f at the point, Figure Fin an equation of the tangent line to the graph of f Solution f Begin b rewriting the function 5 f To fin the slope at,, evaluate f f Write original function at, Multipl numerator an enominator b Rewrite Quotient Rule Simplif 5 Slope of graph at, Then, using the point-slope form of the equation of a line, ou can etermine that the equation of the tangent line at, is See Figure Tr It Eploration A The eitable graph feature below allows ou to eit the graph of a function Eitable Graph Not ever quotient nees to be ifferentiate b the Quotient Rule For eample, each quotient in the net eample can be consiere as the prouct of a constant times a function of In such cases it is more convenient to use the Constant Multiple Rule EXAMPLE 6 Using the Constant Multiple Rule NOTE To see the benefit of using the Constant Multiple Rule for some quotients, tr using the Quotient Rule to ifferentiate the functions in Eample 6 ou shoul obtain the same results, but with more work a b c Original Function Rewrite Differentiate Simplif Tr It Eploration A

84 SECTION Prouct an Quotient Rules an Higher-Orer Derivatives In Section, the Power Rule was prove onl for the case where the eponent n is a positive integer greater than The net eample etens the proof to inclue negative integer eponents EXAMPLE 7 Proof of the Power Rule (Negative Integer Eponents) If n is a negative integer, there eists a positive integer k such that n k So, b the Quotient Rule, ou can write n k So, the Power Rule k 0 k k k 0 kk k k k n n D n n n Quotient Rule an Power Rule n k Power Rule is vali for an integer In Eercise 75 in Section 5, ou are aske to prove the case for which n is an rational number Tr It Eploration A Derivatives of Trigonometric Functions Knowing the erivatives of the sine an cosine functions, ou can use the Quotient Rule to fin the erivatives of the four remaining trigonometric functions THEOREM 9 tan sec sec sec tan Derivatives of Trigonometric Functions cot csc csc csc cot Vieo Proof Consiering tan sin cos an appling the Quotient Rule, ou obtain cos cos sin sin tan cos cos sin cos cos sec Appl Quotient Rule The proofs of the other three parts of the theorem are left as an eercise (see Eercise 89)

85 CHAPTER Differentiation EXAMPLE 8 Differentiating Trigonometric Functions NOTE Because of trigonometric ientities, the erivative of a trigonometric function can take man forms This presents a challenge when ou are tring to match our answers to those given in the back of the tet a b Function tan sec Derivative sec sec tan sec sec tan Tr It Eploration A Open Eploration EXAMPLE 9 Different Forms of a Derivative Differentiate both forms of Solution First form: Secon form: sin cos cos sin cos sin sin sin cos cos sin cos sin csc cot cos sin csc cot csc csc cot To show that the two erivatives are equal, ou can write cos sin sin sin cos sin csc csc cot Tr It Eploration A Technolog The summar below shows that much of the work in obtaining a simplifie form of a erivative occurs after ifferentiating Note that two characteristics of a simplifie form are the absence of negative eponents an the combining of like terms Eample Eample Eample Eample 5 Eample 9 f After Differentiating 5 sin cos cos sin sin cos cos sin f After Simplifing 5 sin cos sin

86 SECTION Prouct an Quotient Rules an Higher-Orer Derivatives 5 NOTE: The secon erivative of f is the erivative of the first erivative of f Higher-Orer Derivatives Just as ou can obtain a velocit function b ifferentiating a position function, ou can obtain an acceleration function b ifferentiating a velocit function Another wa of looking at this is that ou can obtain an acceleration function b ifferentiating a position function twice st vt st at vt st Position function Velocit function Acceleration function The function given b at is the secon erivative of st an is enote b st The secon erivative is an eample of a higher-orer erivative You can efine erivatives of an positive integer orer For instance, the thir erivative is the erivative of the secon erivative Higher-orer erivatives are enote as follows First erivative: Secon erivative: Thir erivative: Fourth erivative: nth erivative:,,,, n, f, f, f, f, f n,,,,, n n, f, f, f, f, n n f, D D D D D n EXAMPLE 0 Fining the Acceleration Due to Gravit THE MOON The moon s mass is 79 0 kilograms, an Earth s mass is kilograms The moon s raius is 77 kilometers, an Earth s raius is 678 kilometers Because the gravitational force on the surface of a planet is irectl proportional to its mass an inversel proportional to the square of its raius, the ratio of the gravitational force on Earth to the gravitational force on the moon is Because the moon has no atmosphere, a falling object on the moon encounters no air resistance In 97, astronaut Davi Scott emonstrate that a feather an a hammer fall at the same rate on the moon The position function for each of these falling objects is given b where st is the height in meters an t is the time in secons What is the ratio of Earth s gravitational force to the moon s? Solution st 08t To fin the acceleration, ifferentiate the position function twice st 08t st 6t st 6 Position function Velocit function Acceleration function So, the acceleration ue to gravit on the moon is 6 meters per secon per secon Because the acceleration ue to gravit on Earth is 98 meters per secon per secon, the ratio of Earth s gravitational force to the moon s is Earth s gravitational force 98 Moon s gravitational force Vieo Tr It Eploration A

87 6 CHAPTER Differentiation Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 6, use the Prouct Rule to ifferentiate the function g f 6 5 ht tt gs s s 5 f cos 6 g sin 7 f f 0 hs s f f h In Eercises 7, use the Quotient Rule to ifferentiate the function 7 f 8 g sin In Eercises 8, fin f an fc In Eercises 9, complete the table without using the Quotient Rule 9 0 Function Function 7 5 f 5 f f f f cos f sin Rewrite In Eercises 5 8, fin the erivative of the algebraic function 5 f 6 gt t t 7 s 9 h 0 hs s f t cos t t Differentiate Value of c c 0 c c c c c 6 Simplif f f f c c is a constant c, 8 f c c is a constant c, In Eercises 9 5, fin the erivative of the trigonometric function 9 ft t sin t 0 f cos ft cos t t f tan cot 5 gt t 8 sec t 6 sin 7 8 cos 9 csc sin 50 sin cos 5 f tan 5 f sin cos 5 sin cos 5 h 5 sec In Eercises 55 58, use a computer algebra sstem to ifferentiate the function g 58 sin In Eercises 59 6, evaluate the erivative of the function at the given point Use a graphing utilit to verif our result f 5 f g 5 f Function csc csc f tan cot ht sec t t Point,, 6 f sin sin cos, 6, g f sin hs 0 csc s s sec f sin cos tan

88 SECTION Prouct an Quotient Rules an Higher-Orer Derivatives 7 In Eercises 6 68, (a) fin an equation of the tangent line to the graph of f at the given point, (b) use a graphing utilit to graph the function an its tangent line at the point, an (c) use the erivative feature of a graphing utilit to confirm our results 6 6 f, f, 65 f, 66 f,, 67 f tan, 68 f sec,,, Famous Curves In Eercises 69 7, fin an equation of the tangent line to the graph at the given point (The graphs in Eercises 69 an 70 are calle witches of Agnesi The graphs in Eercises 7 an 7 are calle serpentines) f() = (, 8 5) 8 6 In Eercises 7 76, etermine the point(s) at which the graph of the function has a horizontal tangent line 75 f 76 f() = (, ) 7 f 7 f 0,, (, ) f 7 77 Tangent Lines Fin equations of the tangent lines to the graph of f that are parallel to the line 6 Then graph the function an the tangent lines 78 Tangent Lines Fin equations of the tangent lines to the graph of f that pass through the point, 5 Then graph the function an the tangent lines f() = f() = (, 5), In Eercises 79 an 80, verif that the relationship between f an g f f g, an eplain In Eercises 8 an 8, use the graphs of f an g Let p f g an q f g 8 (a) Fin p 8 (a) Fin p (b) Fin q (b) Fin q7 8 Area The length of a rectangle is given b t an its height is t, where t is time in secons an the imensions are in centimeters Fin the rate of change of the area with respec to time 8 Volume The raius of a right circular cliner is given b t an its height is t, where t is time in secons an the imensions are in inches Fin the rate of change of the volume with respect to time, sin, 85 Inventor Replenishment The orering an transportation cost C for the components use in manufacturing a prouct is C , where C is measure in thousans of ollars an is the orer size in hunres Fin the rate of change of C with respect to when (a) 0, (b) 5, an (c) 0 What o these rates of change impl about increasing orer size? 86 Bole s Law This law states that if the temperature of a gas remains constant, its pressure is inversel proportional to its volume Use the erivative to show that the rate of change of the pressure is inversel proportional to the square of the volume 87 Population Growth A population of 500 bacteria is introuce into a culture an grows in number accoring to the equation Pt 500 g g f g t 50 t 5 sin where t is measure in hours Fin the rate at which the population is growing when t f f g

89 8 CHAPTER Differentiation 88 Gravitational Force Newton s Law of Universal Gravitation states that the force F between two masses, an m, is F Gm m where G is a constant an is the istance between the masses Fin an equation that gives an instantaneous rate of change of F with respect to (Assume m an m represent moving points) 89 Prove the following ifferentiation rules (a) (c) sec sec tan cot csc 90 Rate of Change Determine whether there eist an values of in the interval 0, such that the rate of change of f sec an the rate of change of g csc are equal 9 Moeling Data The table shows the numbers n (in thousans) of motor homes sol in the Unite States an the retail values v (in billions of ollars) of these motor homes for the ears 996 through 00 The ear is represente b t, with t 6 corresponing to 996 (Source: Recreation Vehicle Inustr Association) (a) Use a graphing utilit to fin cubic moels for the number of motor homes sol nt an the total retail value vt of the motor homes (b) (b) Graph each moel foun in part (a) (c) Fin A vtnt, then graph A What oes this function represent? () Interpret At in the contet of these ata 9 Satellites When satellites observe Earth, the can scan onl part of Earth s surface Some satellites have sensors that can measure the angle shown in the figure Let h represent the satellite s istance from Earth s surface an let r represent Earth s raius csc csc cot Year, t n v m In Eercises 9 98, fin the secon erivative of the function 9 f 9 f 95 f 96 f 97 f sin 98 f sec In Eercises 99 0, fin the given higher-orer erivative 99 f f 00 f,, 0 f, f 0 f, Writing About Concepts f f 6 0 Sketch the graph of a ifferentiable function f such that f 0, < 0 for < <, an > 0 for < < 0 Sketch the graph of a ifferentiable function f such that f > 0 an < 0 for all real numbers In Eercises 05 08, use the given information to fin f g h an an In Eercises 09 an 0, the graphs of f, f, an are shown on the same set of coorinate aes Which is which? Eplain our reasoning To print an enlarge cop of the graph, select the MathGraph button 09 0 f f g h 05 f g h 06 f h 07 f g 08 f gh h f f r r h In Eercises, the graph of f is shown Sketch the graphs of an f To print an enlarge cop of the graph select the MathGraph button f (a) Show that h rcsc (b) Fin the rate at which h is changing with respect to when (Assume r 960 miles) 0 f 8 f 8

90 SECTION Prouct an Quotient Rules an Higher-Orer Derivatives 9 5 Acceleration The velocit of an object in meters per secon is vt 6 t, 0 t 6 Fin the velocit an acceleration of the object when t What can be sai about the spee of the object when the velocit an acceleration have opposite signs? 6 Acceleration An automobile s velocit starting from rest is vt where v is measure in feet per secon Fin the acceleration at (a) 5 secons, (b) 0 secons, an (c) 0 secons 7 Stopping Distance A car is traveling at a rate of 66 feet per secon (5 miles per hour) when the brakes are applie The position function for the car is st 85t 66t, where s is measure in feet an t is measure in secons Use this function to complete the table, an fin the average velocit uring each time interval t st vt at 8 Particle Motion The figure shows the graphs of the position, velocit, an acceleration functions of a particle 6 8 f 00t t (a) Cop the graphs of the functions shown Ientif each graph Eplain our reasoning To print an enlarge cop of the graph, select the MathGraph button (b) On our sketch, ientif when the particle spees up an when it slows own Eplain our reasoning Fining a Pattern In Eercises 9 an 0, evelop a general rule for f n given f 9 f 0 f n t f Fining a Pattern Consier the function f gh (a) Use the Prouct Rule to generate rules for fining f f, an f (b) Use the results in part (a) to write a general rule for f n Fining a Pattern Develop a general rule for f n where f is a ifferentiable function of In Eercises an, fin the erivatives of the function f for n,,, an Use the results to write a general rule for f in terms of n f n sin Differential Equations In Eercises 5 8, verif that the function satisfies the ifferential equation Function, > sin cos sin True or False? In Eercises 9, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 9 If fg, then fg 0 If, then If fc an gc are zero an h fg, then hc 0 If f is an nth-egree polnomial, then f n 0 The secon erivative represents the rate of change of the firs erivative If the velocit of an object is constant, then its acceleration is zero 5 Fin a secon-egree polnomial f a b c such that its graph has a tangent line with slope 0 at the poin, 7 an an -intercept at, 0 6 Consier the thir-egree polnomial f a b c, Determine conitions for a, b, c, an if the graph of f has (a) no horizontal tangents, (b) eactl one horizontal tangent an (c) eactl two horizontal tangents Give an eample for each case 7 Fin the erivative of f Does f0 eist? 8 Think About It Let f an g be functions whose first an secon erivatives eist on an interval I Which of the following formulas is (are) true? (a) fg f g fg fg (b) fg fg fg Differential Equation 0 0 a 0 f cos n

91 0 CHAPTER Differentiation Section The Chain Rule Fin the erivative of a composite function using the Chain Rule Fin the erivative of a function using the General Power Rule Simplif the erivative of a function using algebra Fin the erivative of a trigonometric function using the Chain Rule The Chain Rule This tet has et to iscuss one of the most powerful ifferentiation rules the Chain Rule This rule eals with composite functions an as a surprising versatilit to the rules iscusse in the two previous sections For eample, compare the functions shown below Those on the left can be ifferentiate without the Chain Rule, an those on the right are best one with the Chain Rule Without the Chain Rule sin tan With the Chain Rule sin 6 5 tan Basicall, the Chain Rule states that if changes u times as fast as u, an u changes u times as fast as, then changes uu times as fast as Vieo Gear Gear Ale Gear Ale Gear Ale Ale : revolutions per minute Ale : u revolutions per minute Ale : revolutions per minute Figure Animation EXAMPLE The Derivative of a Composite Function A set of gears is constructe, as shown in Figure, such that the secon an thir gears are on the same ale As the first ale revolves, it rives the secon ale, which in turn rives the thir ale Let, u, an represent the numbers of revolutions per minute of the first, secon, an thir ales Fin u, u, an, an show that u u Solution Because the circumference of the secon gear is three times that of the first, the first ale must make three revolutions to turn the secon ale once Similarl, the secon ale must make two revolutions to turn the thir ale once, an ou can write u an Combining these two results, ou know that the first ale must make si revolutions to turn the thir ale once So, ou can write u Rate of change of first ale with respect to secon ale u 6 u Rate of change of secon ale with respect to thir ale Rate of change of first ale with respect to thir ale In other wors, the rate of change of with respect to is the prouct of the rate of change of with respect to u an the rate of change of u with respect to Tr It Eploration A

92 SECTION The Chain Rule EXPLORATION Using the Chain Rule Each of the following functions can be ifferentiate using rules that ou stuie in Sections an For each function, fin the erivative using those rules Then fin the erivative using the Chain Rule Compare our results Which metho is simpler? a b c sin Eample illustrates a simple case of the Chain Rule The general rule is state below THEOREM 0 The Chain Rule If f u is a ifferentiable function of u an u g is a ifferentiable function of, then f g is a ifferentiable function of an u u or, equivalentl, f g fgg Proof Let h f g Then, using the alternative form of the erivative, ou nee to show that, for c, hc fgcgc An important consieration in this proof is the behavior of g as approaches c A problem occurs if there are values of, other than c, such that g gc Appeni A shows how to use the ifferentiabilit of f an g to overcome this problem For now, assume that g gc for values of other than c In the proofs of the Prouct Rule an the Quotient Rule, the same quantit was ae an subtracte to obtain the esire form This proof uses a similar technique multipling an iviing b the same (nonzero) quantit Note that because g is ifferentiable, it is also continuous, an it follows that g gc as c hc c f g f gc c f g f gc g gc c g gc c, f g f gc c g gc c fgcgc g gc c g gc When appling the Chain Rule, it is helpful to think of the composite function f g as having two parts an inner part an an outer part Outer function f g f u Inner function The erivative of f u is the erivative of the outer function (at the inner function u) times the erivative of the inner function fu u

93 CHAPTER Differentiation EXAMPLE Decomposition of a Composite Function fg u g fu a b c sin tan u u u u tan u sin u u u Tr It Eploration A EXAMPLE Using the Chain Rule STUDY TIP You coul also solve the problem in Eample without using the Chain Rule b observing that an Verif that this is the same as the erivative in Eample Which metho woul ou use to fin 50? Fin for Solution For this function, ou can consier the insie function to be u B the Chain Rule, ou obtain 6 u u Tr It Eploration A Eploration B The eitable graph feature below allows ou to eit the graph of a function an its erivative Eitable Graph The General Power Rule The function in Eample is an eample of one of the most common tpes of composite functions, u n The rule for ifferentiating such functions is calle the General Power Rule, an it is a special case of the Chain Rule THEOREM The General Power Rule If u n, where u is a ifferentiable function of an n is a rational number, then u nun or, equivalentl, un nu n u Vieo Proof Because u n, ou appl the Chain Rule to obtain u u u un u B the (Simple) Power Rule in Section, ou have D u u n nu n, an it follows that u nun

94 SECTION The Chain Rule Vieo EXAMPLE Appling the General Power Rule Fin the erivative of f Solution Let u Then f u an, b the General Power Rule, the erivative is n u n u f() = ( ) f () = The erivative of f is 0 at 0 an is unefine at ± Figure 5 Eitable Graph Appl General Power Rule Differentiate The eitable graph feature below allows ou to eit the graph of a function EXAMPLE 5 Differentiating Functions Involving Raicals Fin all points on the graph of f for which f 0 an those for which f oes not eist Solution f Tr It Eitable Graph Begin b rewriting the function as f Then, appling the General Power Rule (with u prouces f n u n u Appl General Power Rule Write in raical form So, f 0 when 0 an f oes not eist when ±, as shown in Figure 5 Tr It Eploration A Eploration A EXAMPLE 6 Differentiating Quotients with Constant Numerators NOTE Tr ifferentiating the function in Eample 6 using the Quotient Rule You shoul obtain the same result, but using the Quotient Rule is less efficient than using the General Power Rule Differentiate gt 7 t Solution Begin b rewriting the function as gt 7t Then, appling the General Power Rule prouces n u n gt 7t u Appl General Power Rule Constant Multiple Rule 8t 8 t Simplif Write with positive eponent Tr It Eploration A Eploration B

95 CHAPTER Differentiation Simplifing Derivatives The net three eamples illustrate some techniques for simplifing the raw erivatives of functions involving proucts, quotients, an composites EXAMPLE 7 Simplifing b Factoring Out the Least Powers f f Original function Rewrite Prouct Rule General Power Rule Simplif Factor Simplif Tr It EXAMPLE 8 Eploration A Simplifing the Derivative of a Quotient TECHNOLOGY Smbolic ifferentiation utilities are capable of ifferentiating ver complicate functions Often, however, the result is given in unsimplifie form If ou have access to such a utilit, use it to fin the erivatives of the functions given in Eamples 7, 8, an 9 Then compare the results with those given on this page f f Original function Rewrite Quotient Rule Factor Simplif Tr It EXAMPLE 9 Eploration A Simplifing the Derivative of a Power n u n u Original function General Power Rule Quotient Rule Multipl Simplif Tr It Eploration A Open Eploration

96 SECTION The Chain Rule 5 Trigonometric Functions an the Chain Rule The Chain Rule versions of the erivatives of the si trigonometric functions are as shown sin u cos u u tan u sec u u sec u sec u tan u u cos u sin u u cot u csc u u csc u csc u cot u u Technolog EXAMPLE 0 Appling the Chain Rule to Trigonometric Functions u cos u u a sin b c cos tan cos cos cos sin sec Tr It Eploration A Be sure that ou unerstan the mathematical conventions regaring parentheses an trigonometric functions For instance, in Eample 0(a), sin is written to mean sin EXAMPLE Parentheses an Trigonometric Functions a b c cos cos9 sin sin 9 cos cos cos sin e cos cos sin 6 6 sin cos cos cos cos cos cos sin cos sin sin cos Tr It Eploration A To fin the erivative of a function of the form k fgh, ou nee to appl the Chain Rule twice, as shown in Eample EXAMPLE Repeate Application of the Chain Rule ft sin t sin t ft sin t Tr It sin t t sin t cos t t t sin t cos t sin t cos t Eploration A Original function Rewrite Appl Chain Rule once Appl Chain Rule a secon time Simplif

97 6 CHAPTER Differentiation EXAMPLE Tangent Line of a Trigonometric Function π Figure 6 f() = sin + cos π ( π, ) π π STUDY TIP To become skille at ifferentiation, ou shoul memorize each rule As an ai to memorization, note that the cofunctions (cosine, cotangent, an cosecant) require a negative sign as part of their erivatives Fin an equation of the tangent line to the graph of at the point,, as shown in Figure 6 Then etermine all values of in the interval 0, at which the graph of f has a horizontal tangent Solution f sin cos Begin b fining f f sin cos f cos sin cos sin Write original function Appl Chain Rule to cos Simplif To fin the equation of the tangent line at,, evaluate f f cos sin Substitute Slope of graph at, Now, using the point-slope form of the equation of a line, ou can write m Point-slope form Substitute for, m, an Equation of tangent line at, You can then etermine that f 0 when an So, f has a 6,, 6, horizontal tangent at 6,, Tr It 5 an 6, Eploration A This section conclues with a summar of the ifferentiation rules stuie so far 5 Summar of Differentiation Rules General Differentiation Rules Let f, g, an u be ifferentiable functions of Derivatives of Algebraic Functions Derivatives of Trigonometric Functions Chain Rule Constant Multiple Rule: cf cf Prouct Rule: fg fg gf Constant Rule: c 0 sin cos cos sin Chain Rule: fu fu u Sum or Difference Rule: f ± g f ± g Quotient Rule: g f gf fg g Simple Power Rule: n n n, tan sec cot csc General Power Rule: un nu n u sec sec tan csc csc cot

98 SECTION The Chain Rule 7 Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 6, complete the table 5 6 In Eercises 7, fin the erivative of the function g 9 0 f t 9t 9 g 5 6 f 9 9 f t 0 gt t f f fg 6 5 tan csc cos t g 5 ht t t f v v v g u g In Eercises 8, use a computer algebra sstem to fin the erivative of the function Then use the utilit to graph the function an its erivative on the same set of coorinate aes Describe the behavior of the function that correspons to an zeros of the graph of the erivative fu f t t g st t t 5 t g 7 cos 8 tan In Eercises 9 an 0, fin the slope of the tangent line to the sine function at the origin Compare this value with the number of complete ccles in the interval [0, ] What can ou conclue about the slope of the sine function sin a at the origin? 9 (a) (b) 0 (a) (b) In Eercises 58, fin the erivative of the function cos sin g tan h sec 5 sin 6 cos 7 h sin cos 8 9 f cot 50 sin 5 sec 5 gt 5 cos t 5 f sin 5 ht cot t 55 f t sec t 56 5 cos 57 sin 58 sin sin In Eercises 59 66, evaluate the erivative of the function at the given point Use a graphing utilit to verif our result Function st t t 8 5 = sin f f t f t t = sin Point,,,, 6 0, 6 f, 65 7 sec 0, 6 66 cos, g sec tan gv cos v csc v 5 = sin = sin

99 8 CHAPTER Differentiation In Eercises 67 7, (a) fin an equation of the tangent line to the graph of f at the given point, (b) use a graphing utilit to graph the function an its tangent line at the point, an (c) use the erivative feature of the graphing utilit to confirm our results In Eercises 75 78, (a) use a graphing utilit to fin the erivative of the function at the given point, (b) fin an equation of the tangent line to the graph of the function at the given point, an (c) use the utilit to graph the function an its tangent line in the same viewing winow Function f f 5 f 9 f sin cos f tan tan gt s t t t t, f, t t, t 9t, Famous Curves In Eercises 79 an 80, fin an equation of the tangent line to the graph at the given point Then use a graphing utilit to graph the function an its tangent line in the same viewing winow 79 Top half of circle 80 Bullet-nose curve 8 6 f() = 5 (, ) 6 6, 8, 0, 0, Point, 5,,,, 0 8 Horizontal Tangent Line Determine the point(s) in the interval 0, at which the graph of f cos sin has a horizontal tangent 8 Horizontal Tangent Line Determine the point(s) at which the graph of f has a horizontal tangent,,, f() = (, ) In Eercises 8 86, fin the secon erivative of the function 8 f 8 85 f sin 86 f sec In Eercises 87 90, evaluate the secon erivative of the function at the given point Use a computer algebra sstem to verif our result h 9, f, 89 f cos, 0, 90 gt tan t, 6, 6, 9 0, Writing About Concepts f In Eercises 9 9, the graphs of a function f an its erivative are shown Label the graphs as f or an write a short paragraph stating the criteria use in making the selection To print an enlarge cop of the graph, select the MathGraph button f In Eercises 95 an 96, the relationship between f an g is given Eplain the relationship between an g 95 g f 96 g f 97 Given that g5, g5 6, h5, an h5, fin f5 (if possible) for each of the following If it is not possible, state what aitional information is require (a) f gh (b) f gh (c) f g () f g h f f

100 SECTION The Chain Rule 9 98 Think About It The table shows some values of the erivative of an unknown function f Complete the table b fining (if possible) the erivative of each transformation of f (a) g f (c) r f f g h r s In Eercises 99 an 00, the graphs of f an g are shown Let h f g an s g f Fin each erivative, if it eists If the erivative oes not eist, eplain wh (b) () 99 (a) Fin h 00 (a) Fin h (b) Fin s (b) Fin s9 0 Doppler Effect The frequenc F of a fire truck siren hear b a stationar observer is F,00 ± v f g h f s f 0 where ±v represents the velocit of the accelerating fire truck in meters per secon (see figure) Fin the rate of change of F with respect to v when (a) the fire truck is approaching at a velocit of 0 meters per secon (use v) (b) the fire truck is moving awa at a velocit of 0 meters per secon (use v),00,00 F = F = + v v 0 8 f g Harmonic Motion The isplacement from equilibrium of an object in harmonic motion on the en of a spring is cos t sin t where is measure in feet an t is the time in secons Determine the position an velocit of the object when t 8 0 Penulum A 5-centimeter penulum moves accoring to the equation where is the angular isplacement from the vertical in raians an t is the time in secons Determine the maimum angular isplacement an the rate of change of when t secons 0 Wave Motion A buo oscillates in simple harmonic motion A cos t as waves move past it The buo moves a tota of 5 feet (verticall) from its low point to its high point I returns to its high point ever 0 secons (a) Write an equation escribing the motion of the buo if i is at its high point at t 0 (b) Determine the velocit of the buo as a function of t 05 Circulator Sstem The spee S of bloo that is r centimeters from the center of an arter is S CR r where C is a constant, R is the raius of the arter, an S is measure in centimeters per secon Suppose a rug is aministere an the arter begins to ilate at a rate of Rt At a constant istance r, fin the rate at which S changes with respect to t for C , R 0, an Rt Moeling Data The normal ail maimum temperatures T (in egrees Fahrenheit) for Denver, Colorao, are shown in the table (Source: National Oceanic an Atmospheric Aministration) (a) Use a graphing utilit to plot the ata an fin a moel for the ata of the form Tt a b sint6 c where T is the temperature an t is the time in months with t corresponing to Januar (b) Use a graphing utilit to graph the moel How well oes the moel fit the ata? (c) Fin T 0 cos 8t, Month Jan Feb Mar Apr Ma Jun Temperature Month Jul Aug Sep Oct Nov Dec Temperature an use a graphing utilit to graph the erivative () Base on the graph of the erivative, uring what times oes the temperature change most rapil? Most slowl? Do our answers agree with our observations of the temperature changes? Eplain

101 0 CHAPTER Differentiation 07 Moeling Data The cost of proucing units of a prouct is C For one week management etermine the number of units prouce at the en of t hours uring an eight-hour shift The average values of for the week are shown in the table t (a) Use a graphing utilit to fit a cubic moel to the ata (b) Use the Chain Rule to fin Ct (c) Eplain wh the cost function is not increasing at a constant rate uring the 8-hour shift 08 Fining a Pattern Consier the function where is a constant (a) Fin the first-, secon-, thir-, an fourth-orer erivatives of the function (b) Verif that the function an its secon erivative satisf the equation f f 0 (c) Use the results in part (a) to write general rules for the even- an o-orer erivatives f k an f k [Hint: k is positive if k is even an negative if k is o] 09 Conjecture Let f be a ifferentiable function of perio p (a) Is the function perioic? Verif our answer (b) Consier the function g f Is the function g perioic? Verif our answer 0 Think About It Let r f g an s g f where f an g are shown in the figure Fin (a) r an (b) s (, ) g (a) Fin the erivative of the function in two was (b) For f sec an g tan, show that f g f f (6, 6) (6, 5) f sin, g sin cos (a) Show that the erivative of an o function is even That is, if f f, then f f (b) Show that the erivative of an even function is o That is, if f f, then f f Let u be a ifferentiable function of Use the fact tha to prove that In Eercises 7, use the result of Eercise to fin the erivative of the function u u u u u u, g f h cos f sin Linear an Quaratic Approimations The linear an quaratic approimations of a function f at a are P fa a f a an P f a a fa a f a) In Eercises 8 an 9, (a) fin the specifie linear an quaratic approimations of f, (b) use a graphing utilit to graph f an the approimations, (c) etermine whether P or P is the better approimation, an () state how the accurac changes as ou move farther from a 8 f tan 9 f sec a True or False? In Eercises 0, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 0 If, then If f sin, then f sin cos If is a ifferentiable function of u, u is a ifferentiable function of v, an v is a ifferentiable function of, then u v u v u 0 a 6 Putnam Eam Challenge Let f a sin a sin a n sin n, where a, a,, a n are real numbers an where n is a positive integer Given that for all real prove tha a a f sin, na n Let k be a fie positive integer The nth erivative of k has the form P n k n where P n is a polnomial Fin P n These problems were compose b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserve

102 SECTION 5 Implicit Differentiation Section 5 Implicit Differentiation Distinguish between functions written in implicit form an eplicit form Use implicit ifferentiation to fin the erivative of a function EXPLORATION Graphing an Implicit Equation How coul ou use a graphing utilit to sketch the graph of the equation? Here are two possible approaches a Solve the equation for Switch the roles of an an graph the two resulting equations The combine graphs will show a 90 rotation of the graph of the original equation b Set the graphing utilit to parametric moe an graph the equations an t t t t t t From either of these two approaches, can ou ecie whether the graph has a tangent line at the point 0,? Eplain our reasoning Implicit an Eplicit Functions Up to this point in the tet, most functions have been epresse in eplicit form For eample, in the equation 5 Eplicit form the variable is eplicitl written as a function of Some functions, however, are onl implie b an equation For instance, the function is efine implicitl b the equation Suppose ou were aske to fin for this equation You coul begin b writing eplicitl as a function of an then ifferentiating Implicit Form This strateg works whenever ou can solve for the function eplicitl You cannot, however, use this proceure when ou are unable to solve for as a function of For instance, how woul ou fin for the equation Eplicit Form Derivative where it is ver ifficult to epress as a function of eplicitl? To o this, ou can use implicit ifferentiation To unerstan how to fin implicitl, ou must realize that the ifferentiation is taking place with respect to This means that when ou ifferentiate terms involving alone, ou can ifferentiate as usual However, when ou ifferentiate terms involving, ou must appl the Chain Rule, because ou are assuming that is efine implicitl as a ifferentiable function of Vieo EXAMPLE Differentiating with Respect to a Variables agree: use Simple Power Rule Variables agree u n nu n u b Variables isagree: use Chain Rule Variables isagree c Chain Rule: Prouct Rule Chain Rule Simplif Tr It Eploration A

103 CHAPTER Differentiation Implicit Differentiation Guielines for Implicit Differentiation Differentiate both sies of the equation with respect to Collect all terms involving on the left sie of the equation an move all other terms to the right sie of the equation Factor out of the left sie of the equation Solve for EXAMPLE Implicit Differentiation Fin given that 5 NOTE In Eample, note that implicit ifferentiation can prouce an epression for that contains both an, 0, 0, Point on Graph (, ) The implicit equation has the erivative Figure 7 (, ) (, 0) + 5 = Unefine 5 5 Slope of Graph Solution Differentiate both sies of the equation with respect to Collect the terms on the left sie of the equation an move all other terms to the right sie of the equation 5 Factor out of the left sie of the equation 5 Solve for b iviing b Tr It Eploration A Vieo Vieo To see how ou can use an implicit erivative,consier the graph shown in Figure 7 From the graph, ou can see that is not a function of Even so, the erivative foun in Eample gives a formula for the slope of the tangent line at a point on this graph The slopes at several points on the graph are shown below the graph TECHNOLOGY With most graphing utilities, it is eas to graph an equation that eplicitl represents as a function of Graphing other equations, however, can require some ingenuit For instance, to graph the equation given in Eample, use a graphing utilit, set in parametric moe, to graph the parametric representations t t 5t, t, an t t 5t, t, for 5 t 5 How oes the result compare with the graph shown in Figure 7?

104 SECTION 5 Implicit Differentiation + = 0 (0, 0) It is meaningless to solve for in an equation that has no solution points (For eample, has no solution points) If, however, a segment of a graph can be represente b a ifferentiable function, will have meaning as the slope at each point on the segment Recall that a function is not ifferentiable at (a) points with vertical tangents an (b) points at which the function is not continuous (a) Eitable Graph = (, 0) (, 0) = (b) Eitable Graph = (, 0) = (c) EXAMPLE Representing a Graph b Differentiable Functions If possible, represent as a ifferentiable function of a 0 b c Solution a The graph of this equation is a single point So, it oes not efine as a ifferentiable function of See Figure 8(a) b The graph of this equation is the unit circle, centere at 0, 0 The upper semicircle is given b the ifferentiable function, < < an the lower semicircle is given b the ifferentiable function, < < At the points, 0 an, 0, the slope of the graph is unefine See Figure 8(b) c The upper half of this parabola is given b the ifferentiable function, < an the lower half of this parabola is given b the ifferentiable function, < At the point, 0, the slope of the graph is unefine See Figure 8(c) Tr It Eploration A Eploration B Eitable Graph EXAMPLE Fining the Slope of a Graph Implicitl Some graph segments can be represente b ifferentiable functions Figure 8 Determine the slope of the tangent line to the graph of at the point, See Figure 9 + = Figure 9 Eitable Graph (, ) Solution So, at,, the slope is Write original equation Differentiate with respect to Solve for Evaluate when an Tr It Eploration A Eploration B Open Eploration NOTE To see the benefit of implicit ifferentiation, tr oing Eample using the eplicit function

105 CHAPTER Differentiation EXAMPLE 5 Fining the Slope of a Graph Implicitl Determine the slope of the graph of 00 at the point, Solution (, ) ( + ) = 00 Lemniscate Figure 0 At the point,, the slope of the graph is as shown in Figure 0 This graph is calle a lemniscate Tr It Eploration A Eploration B π, ( ) The erivative is Figure π π π Eitable Graph sin = (, π ) EXAMPLE 6 Determining a Differentiable Function Fin implicitl for the equation sin Then fin the largest interval of the form a < < a on which is a ifferentiable function of (see Figure ) Solution sin cos cos The largest interval about the origin for which is a ifferentiable function of is < < To see this, note that cos is positive for all in this interval an is 0 at the enpoints If ou restrict to the interval < <, ou shoul be able to write eplicitl as a function of To o this, ou can use cos sin an conclue that, < < Tr It Eploration A

106 SECTION 5 Implicit Differentiation 5 ISAAC BARROW (60 677) The graph in Figure is calle the kappa curve because it resembles the Greek letter kappa, The general solution for the tangent line to this curve was iscovere b the English mathematician Isaac Barrow Newton was Barrow s stuent, an the correspone frequentl regaring their work in the earl evelopment of calculus MathBio With implicit ifferentiation, the form of the erivative often can be simplifie (as in Eample 6) b an appropriate use of the original equation A similar technique can be use to fin an simplif higher-orer erivatives obtaine implicitl EXAMPLE 7 Given fin 5, Fining the Secon Derivative Implicitl Solution Differentiating each term with respect to prouces 0 Differentiating a secon time with respect to iels 5 Quotient Rule Substitute for Simplif Substitute 5 for Tr It Eploration A EXAMPLE 8 Fining a Tangent Line to a Graph Fin the tangent line to the graph given b,, as shown in Figure at the point The kappa curve Figure (, ) ( + ) = Solution B rewriting an ifferentiating implicitl, ou obtain 0 At the point,, the slope is an the equation of the tangent line at this point is 0 Tr It Eploration A

107 6 CHAPTER Differentiation Eercises for Section 5 The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 6, fin / b implicit ifferentiation sin cos sin cos sin cos sin tan cot 5 sin 6 sec Bifolium: Folium of Descartes: Point:, 6 0 Point:, 8 In Eercises 7 0, (a) fin two eplicit functions b solving the equation for in terms of, (b) sketch the graph of the equation an label the parts given b the corresponing eplicit functions, (c) ifferentiate the eplicit functions, an () fin / an show that the result is equivalent to that of part (c) In Eercises 8, fin / b implicit ifferentiation an evaluate the erivative at the given point , 0,, Famous Curves In Eercises 9, fin the slope of the tangent line to the graph at the given point 9 Witch of Agnesi: 0 Cissoi: 8 Point:,, 5,,, 0,, tan, 0, 0 cos,, 8,,, Point:, Famous Curves In Eercises 0, fin an equation of the tangent line to the graph at the given point To print an enlarge cop of the graph, select the MathGraph button Parabola Circle Rotate hperbola 6 Rotate ellipse 7 Cruciform 8 Astroi 9 = 0 6 (, ) ( ) = ( ) (, 0) 6 6 = (, ) ( + ) + ( ) = = 0 8 / + / = 5 (8, ) (, ) (, )

108 SECTION 5 Implicit Differentiation 7 9 Lemniscate 0 Kappa curve ( + ) = 00( ) 6 (a) Use implicit ifferentiation to fin an equation of the tangent line to the ellipse at, 8 (b) Show that the equation of the tangent line to the ellipse at is 0 a 0 0, 0 a b b (a) Use implicit ifferentiation to fin an equation of the tangent line to the hperbola at, 6 8 (b) Show that the equation of the tangent line to the hperbola at is 0 a 0 0, 0 b a b In Eercises an, fin / implicitl an fin the largest interval of the form a < < a or 0 < < a such that is a ifferentiable function of Write / as a function of In Eercises 5 50, fin / in terms of an In Eercises 5 an 5, use a graphing utilit to graph the equation Fin an equation of the tangent line to the graph at the given point an graph the tangent line in the same viewing winow 5, 9, 5 (, ) tan cos ( + ) =, 5, 5 In Eercises 5 an 5, fin equations for the tangent line an normal line to the circle at the given points (The normal line at a point is perpenicular to the tangent line at the point) Use a graphing utilit to graph the equation, tangent line, an normal line (, ) In Eercises 57 an 58, fin the points at which the graph of the equation has a vertical or horizontal tangent line Orthogonal Trajectories In Eercises 59 6, use a graphing utilit to sketch the intersecting graphs of the equations an show that the are orthogonal [Two graphs are orthogonal if at their point(s) of intersection their tangent lines are perpenicular to each other] sin 5 9 Orthogonal Trajectories In Eercises 6 an 6, verif that the two families of curves are orthogonal where C an K are rea numbers Use a graphing utilit to graph the two families for two values of C an two values of K 6 C, K 6 C, In Eercises 65 68, ifferentiate (a) with respect to ( is a function of ) an (b) with respect to t ( an are functions of t) cos sin 68 sin cos Writing About Concepts K 69 Describe the ifference between the eplicit form of a function an an implicit equation Give an eample of each 70 In our own wors, state the guielines for implicit ifferentiation 7 Orthogonal Trajectories The figure below shows the topographic map carrie b a group of hikers The hikers are in a wooe area on top of the hill shown on the map an the ecie to follow a path of steepest escent (orthogona trajectories to the contours on the map) Draw their routes if the start from point A an if the start from point B If their goal is to reach the roa along the top of the map, which starting point shoul the use? To print an enlarge cop of the graph, select the MathGraph button 5 5 5,,, 9 0,,, Show that the normal line at an point on the circle r passes through the origin 56 Two circles of raius are tangent to the graph of at the point, Fin equations of these two circles A B

109 8 CHAPTER Differentiation 7 Weather Map The weather map shows several isobars curves that represent areas of constant air pressure Three high pressures H an one low pressure L are shown on the map Given that win spee is greatest along the orthogonal trajectories of the isobars, use the map to etermine the areas having high win spee H L H H 76 Slope Fin all points on the circle 5 where the slope is 77 Horizontal Tangent Determine the point(s) at which the graph of has a horizontal tangent 78 Tangent Lines Fin equations of both tangent lines to the ellipse that passes through the point, Normals to a Parabola The graph shows the normal lines from the point, 0 to the graph of the parabola How man normal lines are there from the point 0, 0 to the graph of the parabola if (a) (b) 0 0,, an (c) 0? For what value of 0 are two of the normal lines perpenicular to each other? 7 Consier the equation (a) Use a graphing utilit to graph the equation (b) Fin an graph the four tangent lines to the curve for (c) Fin the eact coorinates of the point of intersection of the two tangent lines in the first quarant 7 Let L be an tangent line to the curve c Show that the sum of the - an -intercepts of L is c 75 Prove (Theorem ) that n n n for the case in which n is a rational number (Hint: Write pq in the form q p an ifferentiate implicitl Assume that p an q are integers, where q > 0 ) 80 Normal Lines (a) Fin an equation of the normal line to the ellipse 8 (, 0) = at the point, (b) Use a graphing utilit to graph the ellipse an the normal line (c) At what other point oes the normal line intersect the ellipse?

110 SECTION 6 Relate Rates 9 Section 6 r r h h Relate Rates Fin a relate rate Use relate rates to solve real-life problems Fining Relate Rates You have seen how the Chain Rule can be use to fin implicitl Another important use of the Chain Rule is to fin the rates of change of two or more relate variables that are changing with respect to time For eample, when water is raine out of a conical tank (see Figure ), the volume V, the raius r, an the height h of the water level are all functions of time t Knowing that these variables are relate b the equation V r h Original equation ou can ifferentiate implicitl with respect to t to obtain the relate-rate equation t V t r h V Differentiate with respect to t t r h t h r r t r h t rh r t From this equation ou can see that the rate of change of V is relate to the rates of change of both h an r r h EXPLORATION Fining a Relate Rate In the conical tank shown in Figure, suppose that the height is changing at a rate of 0 foot per minute an the raius is changing at a rate of 0 foot per minute What is the rate of change in the volume when the raius is r foot an the height is h feet? Does the rate of change in the volume epen on the values of r an h? Eplain EXAMPLE Two Rates That Are Relate Volume is relate to raius an height Figure Animation FOR FURTHER INFORMATION To learn more about the histor of relaterate problems, see the article The Lengthening Shaow: The Stor of Relate Rates b Bill Austin, Don Barr, an Davi Berman in Mathematics Magazine Suppose an are both ifferentiable functions of t an are relate b the equation Fin t when, given that t when Solution Using the Chain Rule, ou can ifferentiate both sies of the equation with respect to t t t t t When an t, ou have t Write original equation Differentiate with respect to t Chain Rule MathArticle Tr It Eploration A

111 50 CHAPTER Differentiation Problem Solving with Relate Rates In Eample, ou were given an equation that relate the variables an an were aske to fin the rate of change of when Equation: Given rate: Fin: t t when when In each of the remaining eamples in this section, ou must create a mathematical moel from a verbal escription EXAMPLE Ripples in a Pon A pebble is roppe into a calm pon, causing ripples in the form of concentric circles, as shown in Figure The raius r of the outer ripple is increasing at a constant rate of foot per secon When the raius is feet, at what rate is the total area A of the isturbe water changing? Total area increases as the outer raius increases Figure Solution The variables r an A are relate b A r The rate of change of the raius r is rt Equation: Given rate: Fin: when With this information, ou can procee as in Eample t A t r A t A r r t A t r r t r A t 8 Differentiate with respect to t Chain Rule Substitute for r an for rt When the raius is feet, the area is changing at a rate of 8 square feet per secon Tr It Eploration A Vieo Vieo NOTE When using these guielines, be sure ou perform Step before Step Substituting the known values of the variables before ifferentiating will prouce an inappropriate erivative Guielines For Solving Relate-Rate Problems Ientif all given quantities an quantities to be etermine Make a sketch an label the quantities Write an equation involving the variables whose rates of change either are given or are to be etermine Using the Chain Rule, implicitl ifferentiate both sies of the equation with respect to time t After completing Step, substitute into the resulting equation all known values for the variables an their rates of change Then solve for the require rate of change

112 SECTION 6 Relate Rates 5 The table below lists eamples of mathematical moels involving rates of change For instance, the rate of change in the first eample is the velocit of a car Verbal Statement The velocit of a car after traveling for hour is 50 miles per hour Water is being pumpe into a swimming pool at a rate of 0 cubic meters per hour A gear is revolving at a rate of 5 revolutions per minute ( revolution ra) Mathematical Moel istance travele 50 when t t V volume of water in pool V t 0 m hr angle of revolution t 5 ramin EXAMPLE An Inflating Balloon Air is being pumpe into a spherical balloon (see Figure 5) at a rate of 5 cubic feet per minute Fin the rate of change of the raius when the raius is feet Solution Let V be the volume of the balloon an let r be its raius Because the volume is increasing at a rate of 5 cubic feet per minute, ou know that at time t the rate of change of the volume is Vt 9 So, the problem can be state as shown Given rate: Fin: V t 9 r t when (constant rate) r To fin the rate of change of the raius, ou must fin an equation that relates the raius r to the volume V Equation: V r Volume of a sphere Inflating a balloon Figure 5 Animation Differentiating both sies of the equation with respect to t prouces V t r r t r t r V t Differentiate with respect to t Solve for rt Finall, when r, the rate of change of the raius is r t foot per minute Tr It Eploration A Vieo In Eample, note that the volume is increasing at a constant rate but the raius is increasing at a variable rate Just because two rates are relate oes not mean that the are proportional In this particular case, the raius is growing more an more slowl as t increases Do ou see wh?

113 5 CHAPTER Differentiation EXAMPLE The Spee of an Airplane Tracke b Raar s 6 mi An airplane is fling on a flight path that will take it irectl over a raar tracking station, as shown in Figure 6 If s is ecreasing at a rate of 00 miles per hour when s 0 miles, what is the spee of the plane? Solution Let be the horizontal istance from the station, as shown in Figure 6 Notice that when s 0, Not rawn to scale An airplane is fling at an altitue of 6 miles, s miles from the station Figure 6 Given rate: when Fin: t when s 0 an You can fin the velocit of the plane as shown Equation: st 00 6 s s s t t t s s t t s miles per hour 8 Pthagorean Theorem Differentiate with respect to t Solve for t Substitute for s,, an st Simplif Because the velocit is 500 miles per hour, the spee is 500 miles per hour Tr It Eploration A Open Eploration EXAMPLE 5 A Changing Angle of Elevation Fin the rate of change in the angle of elevation of the camera shown in Figure 7 at 0 secons after lift-off tan θ = 000 s θ 000 ft Not rawn to scale A television camera at groun level is filming the lift-off of a space shuttle that is rising verticall accoring to the position equation s 50t, where s is measure in feet an t is measure in secons The camera is 000 feet from the launch pa Figure 7 s Solution Let be the angle of elevation, as shown in Figure 7 When t 0, the height s of the rocket is s 50t feet Given rate: st 00t velocit of rocket Fin: t when t 0 an Using Figure 7, ou can relate s an b the equation tan s000 Equation: tan s See Figure When t 0 an s 5000, ou have t sec t 000 s t t cos 00t s t 000 raian per secon So, when t 0, is changing at a rate of raian per secon s Differentiate with respect to t Substitute 00t for st cos 000s 000 Animation Tr It Eploration A