The Derivative and the Tangent Line Problem. The Tangent Line Problem

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1 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 limit 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 limit, 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

2 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 limit f c f c lim lim 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

3 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 lim lim 0 0 lim 0 lim 0 lim 0 The slope of f at c, f c, is m, as shown in Figure 5 NOTE In Eample, the limit 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 lim 0 f c f c c lim c 0 c lim c c 0 c lim 0 lim c 0 c In Eample, note that c is hel constant in the limit process as 0 Tr It Eploration A

4 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 lim 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 limits from the right for a an from the left for b The Derivative of a Function f c f c lim 0 You have now arrive at a crucial point in the stu of calculus The limit 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 lim 0 f f provie the limit eists For all for which this limit 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 limit notation, ou can write lim 0 lim 0 f f f Histor

5 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 lim 0 lim 0 lim 0 lim 0 f f Definition of erivative lim 0 lim 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 lim Definition of erivative 0 lim 0 lim 0 lim 0 lim 0 lim 0, > 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

6 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 lim t t 0 lim t 0 lim t 0 lim t 0 lim 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 limit 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 limit form of the erivative is useful in investigating the relationship between ifferentiabilit an continuit The erivative of f at c is provie this limit eists (see Figure 0) (A proof of the equivalence of this form is given in Appeni A) Note that the eistence of the limit in this alternative form requires that the one-sie limits lim c fc lim c f f c c f f c c or an t f f c lim c c Alternative form of erivative eist an are equal These one-sie limits 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

7 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 0 lim 0 0 lim 0 f f lim 0 0 lim 0 0 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 limits an f f f lim lim 0 f f lim lim 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 limit f f 0 lim lim lim 0 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

8 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 limit lim f f c lim c c c f f c c lim c c lim 0 fc 0 f f c c c Because the ifference f f c approaches zero as c, ou can conclue that lim 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

9 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

10 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 limit 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

11 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 limit represents fc for a function f an a number c Fin f an c 5 9 lim 50 0 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 6 lim lim 6 5 lim 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

12 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

13 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 limit 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 limit 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 limit efinition of the erivative, c f lim 0 lim 0 lim 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

14 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 lim n n 0 nn n n n n n n lim 0 lim 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

15 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

16 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 lim 0 lim 0 c c lim 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

17 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 lim 0 lim 0 f f lim 0 lim 0 f g f g f g f f Sum Rule Difference Rule g g g g lim 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

18 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 limits sin lim 0 an cos lim 0 0 These two limits 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 lim 0 lim 0 cos lim 0 cos sin 0 cos sin cos cos sin sin sin sin lim 0 Definition of erivative cos sin sin cos lim 0 lim 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

19 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

20 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 limit 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 lim 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

21 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

22 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

23 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

24 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

25 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 lim 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 lim 0 g lim f g g 0 g lim 0 f g lim 0 f g fg f g f g f g fg g g f f f lim lim g lim fg gf f f f f lim 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 lim 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

26 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

27 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 lim 0 Note that lim is continuous 0 EXAMPLE gf fg lim 0 gg g lim 0 f g gf fg g Definition of erivative gf fg fg fg lim 0 gg g f f fg g lim lim 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 lim 0 f lim 0 f f gg lim 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

28 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

29 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)

30 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

31 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

32 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

33 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

34 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

35 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

36 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

37 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 lim c f g f gc c f g f gc g gc lim c g gc c, f g f gc lim c g gc lim 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

38 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

39 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

40 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

41 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

42 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

43 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

44 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

45 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

46 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

47 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

48 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?

49 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

50 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

51 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

52 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, ) (, ) (, )

53 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

54 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?

55 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

56 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

57 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?

58 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