DRAFT CHAPTER 1: Introduction to Calculus Errors will be corrected before printing. Final book will be available August 2008.

Transcription

1 DRAFT CHAPTER 1: Introduction to Calculus Errors will be corrected before printing. Final book will be available August 2008.

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3 08-037_01_VCSB_C01.Intro_pp3.qd 5/23/08 5:30 PM Page 1 Capter 1 INTRODUCTION TO CALCULUS In te Englis language, te rules of grammar are used to speak and write effectively. Asking for a cookie at te age of ten was muc easier tan wen you were first learning to speak. Tese rules developed over time. Calculus developed in a similar way. Sir Isaac Newton and Gottfried Wilelm von Leibniz independently organized an assortment of ideas and metods tat were circulating among te matematicians of teir time. As a tool in te service of science, calculus served its purpose very well. More tan two centuries passed, owever, before matematicians ad identified and agreed on its underlying principles its grammar. In tis capter, you will see some of te ideas tat were brougt togeter to form te underlying principles of calculus. CHAPTER EXPECTATIONS In tis capter, you will simplify radical epressions, Section 1.1 use its to determine te slope and te equation of te tangent to a grap, Section 1.2 pose problems and formulate ypoteses regarding rates of cange, Section 1.3, Career Link calculate and interpret average and instantaneous rates of cange and relate tese values to slopes of secants and tangents, Section 1.3 understand and evaluate its using appropriate properties, Sections 1.4, 1.5 eamine continuous functions and use its to eplain wy a function is discontinuous, Sections 1.5, 1.6

4 Review of Prerequisite Skills Before beginning tis capter, review te following concepts from previous courses: determining te slope of a line: m y determining te equation of a line using function notation for substituting into and evaluating functions simplifying algebraic epressions factoring epressions finding te domain of functions calculating average rate of cange and slopes of secant lines estimating instantaneous rate of cange and slopes of tangent lines Eercise 1. Determine te slope of te line passing troug eac of te following pairs of points: a. 12, 52 and 16, 72 d. 10, 02 and 11, 42 b. 13, 42 and 11, 42 e. 12.1, and 12, 42 c. 10, 02 and 11, 42 f. a 3 and 4, 1 4 b a 7 4, 1 4 b 2 REVIEW OF PREREQUISITE SKILLS 2. Determine te equation of a line for te given information. a. slope 4, y-intercept 2 d. troug 12, 42 and 16, 82 b. slope 2, y-intercept 5 e. vertical, troug 13, 52 c. troug 11, 62 and 14, 122 f. orizontal, troug 13, Evaluate for 2. a. f c. f b. f d. f For f 12 determine eac of te following values: 2 4, a. f 1102 b. f 132 c. f 102 d. f , if Consider te function f given by f 12 e 3, if 0 Calculate eac of te following: a. f 1332 b. f 102 c. f 1782 d. f 13 2

5 1, if 3 6 t 6 0 t 6. A function s is defined for t 7 3 by s1t2 µ 5, if t 0 t 3, if t 7 0 Evaluate eac of te following: a. s122 b. s112 c. s102 d. s112 e. 7. Epand, simplify, and write eac epression in standard form. a d b e. 1a 22 3 c f. 19a Factor eac of te following: a. 3 c e. b. 2 6 d f. 9. Determine te domain of eac of te following: a. y V 5 d. b. y 3 e. c. y 3 f y y Te eigt of a model rocket in fligt can be modelled by te equation 1t2 4.9t 2 25t 2, were is te eigt in metres at t seconds. Determine te average rate of cange in te model rocket s eigt wit respect to time during a. te first second b. te second second 11. Saca drains te water from a ot tub. Te ot tub olds 1600 L of water. It takes 2 for te water to drain completely. Te volume of water in te ot tub is modelled by V1t t22, were V is te volume in litres at 9 t minutes and 0 t 120. a. Determine te average rate of cange in volume during te second our. b. Estimate te instantaneous rate of cange in volume after eactly 60 min. c. Eplain wy all estimates of te instantaneous rate of cange in volume were 0 t 120 result in a negative value. 12. a. Sketc te grap of f b. Draw a tangent line at te point 15, f 1522, and estimate its slope. c. Estimate te instantaneous rate of cange in f 12 wen 5. s

6 CAREER LINK Investigate CHAPTER 1: ASSESSING ATHLETIC PERFORMANCE Time (s) (min) Number of Heartbeats Differential calculus is fundamentally about te idea of instantaneous rate of cange. A familiar rate of cange is eart rate. Elite atletes are keenly interested in te analysis of eart rates. Sporting performance is enanced wen an atlete is able to increase is or er eart rate at a slower pace (tat is, to get tired less quickly). A eart rate is described for an instant in time. Heart rate is te instantaneous rate of cange in te total number of eartbeats wit respect to time. Wen nurses and doctors count eartbeats and ten divide by te time elapsed, tey are not determining te instantaneous rate of cange but are calculating te average eart rate over a period of time (usually 10 s). In tis capter, te idea of te derivative will be developed, progressing from te average rate of cange calculated over smaller and smaller intervals until a iting value is reaced at te instantaneous rate of cange. Case Study Assessing Elite Atlete Performance Te table sows te number of eartbeats of an atlete wo is undergoing a cardiovascular fitness test. Complete te discussion questions to determine if tis atlete is under is or er maimum desired eart rate of 65 beats per minute at precisely 30 s. DISCUSSION QUESTIONS 1. Grap te number of eartbeats versus time (in minutes) on grap paper, joining te points to make a smoot curve. Draw a second relationsip on te same set of aes, sowing te resting eart rate of 50 beats per minute. Use te slopes of te two relationsips graped to eplain wy te test results indicate tat te person must be eercising. 2. Discuss ow te average rate of cange in te number of eartbeats over an interval of time could be calculated using tis grap. Eplain your reasoning. 3. Calculate te atlete s average eart rate over te intervals of 30 s, 60 s4, 310 s, 50 s4, and 320 s, 40 s4. Sow te progression of tese average eart rate calculations on te grap as a series of secants. 4. Use te progression of tese average eart-rate secants to make a grapical prediction of te instantaneous eart rate at t 30 s. Is te atlete s eart rate less tan 65 beats per minute at t 30 s? Estimate te eart rate at t 60 s. 4 CAREER LINK

7 Wat Is Calculus? Two simple geometric problems originally led to te development of wat is now called calculus. Bot problems can be stated in terms of te grap of a function y f 12. Te problem of tangents: Wat is te slope of te tangent to te grap of a function at a given point P? Te problem of areas: Wat is te area under a grap of a function y f 12 between a and b? y slope =? P y = f() 0 area =? a b Interest in te problem of tangents and te problem of areas dates back to scientists suc as Arcimedes of Syracuse ( BCE), wo used is vast ingenuity to solve special cases of tese problems. Furter progress was made in te seventeent century, most notably by Pierre de Fermat ( ) and Isaac Barrow ( ), a professor of Sir Isaac Newton ( ) at te University of Cambridge, England. Professor Barrow recognized tat tere was a close connection between te problem of tangents and te problem of areas. However, it took te genius of bot Newton and Gottfried Wilelm von Leibniz ( ) to sow te way to andle bot problems. Using te analytic geometry of Rene Descartes ( ), Newton and Leibniz sowed independently ow tese two problems could be solved by means of new operations on functions, called differentiation and integration. Teir discovery is considered to be one of te major advances in te istory of matematics. Furter researc by matematicians from many countries using tese operations as created a problem-solving tool of immense power and versatility, wic is known as calculus. It is a powerful branc of matematics, used in applied matematics, science, engineering, and economics. We begin our study of calculus by discussing te meaning of a tangent and te related idea of rate of cange. Tis leads us to te study of its and, at te end of te capter, to te concept of te derivative of a function. CHAPTER 1 5

8 Section 1.1 Radical Epressions: Rationalizing Denominators Now tat we ave reviewed some concepts tat will be needed before beginning te introduction to calculus, we ave to consider simplifying epressions wit radicals in te denominator of radical epressions. Recall tat a rational number is a number tat can be epressed as a fraction (quotient) containing integers. So te process of canging a denominator from a radical (square root) to a rational number (integer) is called rationalizing te denominator. Te reason tat we rationalize denominators is tat dividing by an integer is preferable to dividing by a radical number. In certain situations, it is useful to rationalize te numerator. Practice wit rationalizing te denominator prepares you for rationalizing te numerator. Tere are two situations tat we need to consider: radical epressions wit one-term denominators and tose wit two-term denominators. For bot, te numerator and denominator will be multiplied by te same epression, wic is te same as multiplying by one. EXAMPLE 1 Selecting a strategy to rationalize te denominator 3 Simplify by rationalizing te denominator. 45 Solution 3 4V5 3 V5 4V5 V5 3V V5 20 (Multiply bot te numerator and denominator by 5) (Simplify) Wen te denominator of a radical fraction is a two-term epression, you can rationalize te denominator by multiplying by te conjugate. An epression suc as a b as te conjugate Va Vb. Wy are conjugates important? Recall tat te linear terms are einated wen epanding a difference of squares. For eample, 6 1a b21a b2 a 2 ab ab b 2 a 2 b RADICAL EXPRESSIONS: RATIONALIZING DENOMINATORS

9 If a and b were radicals, squaring tem would rationalize tem. Consider tis product: QVm VnR QVm VnR, m, n rational Notice tat te result is rational! QmR 2 mn mn QnR 2 m n EXAMPLE 2 EXAMPLE 3 Creating an equivalent epression by rationalizing te denominator Simplify Solution by rationalizing te denominator. 2 V6 V3 2 V6 V3 V6 V3 V6 V3 Selecting a strategy to rationalize te denominator 5 Simplify te radical epression by rationalizing te denominator Solution 5 2V V6 3 2V6 3 2V6 3 5 Q2V6 3R 4V Q2V6 3R Q2V6 3R 15 2 Q6 3R Q6 3R 3 2V6 3 3 (Multiply bot te numerator and denominator by 6 3) (Simplify) (Te conjugate 26 3 is 2V6 3) (Simplify) (Divide by te common factor of 5) Te numerator can also be rationalized in te same way as te denominator was in te previous epressions. CHAPTER 1 7

10 EXAMPLE 4 Selecting a strategy to rationalize te numerator 7 3 Rationalize te numerator of te epression 2. Solution V7 V3 2 V7 V3 V7 V3 2 V7 V Q7 3R 4 2 Q7 3R 2 V7 V3 (Multiply te numerator and denominator by 7 3) (Simplify) (Divide by te common factor of 2) IN SUMMARY Key Ideas To rewrite a radical epression wit a one-term radical in te denominator, multiply te numerator and denominator by te one-term denominator. Va Va Vb Vb Vb Vb Vab b Wen te denominator of a radical epression is a two-term epression, rationalize te denominator by multiplying te numerator and denominator by te conjugate, and ten simplify. 1 a b 1 a b a b a b Va Vb a b Need to Know 3 Wen you simplify a radical epression suc as, multiply te numerator 52 and denominator by te radical only. V3 V2 V6 5V2 V V6 10 Va Vb is te conjugate Va Vb, and vice versa RADICAL EXPRESSIONS: RATIONALIZING DENOMINATORS

11 Eercise 1.1 PART A 1. Write te conjugate of eac radical epression. a c. 2V3 V2 e. V2 V5 b. V3 V2 d. 3V3 V2 f. V5 2V2 2. Rationalize te denominator of eac epression. Write your answer in simplest form. a. V3 V5 4V3 3V2 c. V2 2V3 2V3 3V2 b. d. V2 3V5 V2 2V2 K C A PART B 3. Rationalize eac denominator. 3 V3 V2 2V3 V2 a. c. e. V5 V2 V3 V2 5V2 V3 2V5 2V5 8 3V3 2V2 b. d. f. 2V5 3V2 2V5 3 3V3 2V2 4. Rationalize eac numerator. V V2 V5 2 a. b. c V a. Rationalize te denominator of b. Rationalize te denominator of c. Wy are your answers in parts a and b te same? Eplain. 6. Rationalize eac denominator. 2V2 2V2 3V5 a. c. e. 2V3 V8 V16 V12 4V3 5V2 2V6 3V2 2V3 b. d. f. 2V27 V8 V12 V8 V18 V12 V18 V12 7. Rationalize te numerator of eac of te following epressions: Va 2 V 4 2 V a. b. c. a 4 CHAPTER 1 9

12 Section 1.2 Te Slope of a Tangent You are familiar wit te concept of a tangent to a curve. Wat geometric interpretation can be given to a tangent to te grap of a function at a point P? A tangent is te straigt line tat most resembles te grap near a point. Its slope tells ow steep te grap is at te point of tangency. In te figure below, four tangents ave been drawn. T 2 T 1 T 3 y = f() T 4 Te goal of tis section is to develop a metod for determining te slope of a tangent at a given point on a curve. We begin wit a brief review of lines and slopes. Lines and Slopes y P2 ( 2, y 2 ) l P 1 ( 1, y 1 ) 0 D Dy Te slope m of te line joining points P 1 1 1, y 1 2 and P 2 1 2, y 2 2 is defined as m y y 2 y y Te equation of te line l in point-slope form is y 1 m or y y 1 m Te equation in slope y-intercept form is y m b, were b is te y-intercept of te line THE SLOPE OF A TANGENT To determine te equation of a tangent to a curve at a given point, we first need to know te slope of te tangent. Wat can we do wen we only ave one point? We proceed as follows: y Q Q Q tangent at P 0 P y = f() secant

13 Consider a curve y f 12 and a point P tat lies on te curve. Now consider anoter point Q on te curve. Te line joining P and Q is called a secant. Tink of Q as a moving point tat slides along te curve toward P, so tat te slope of te secant PQ becomes a progressively better estimate of te slope of te tangent at P. Tis suggests te following definition of te slope of te tangent: Slope of a Tangent Te slope of te tangent to a curve at a point P is te iting slope of te secant PQ as te point Q slides along te curve toward P. In oter words, te slope of te tangent is said to be te it of te slope of te secant as Q approaces P along te curve. We will illustrate tis idea by finding te slope of te tangent to te parabola y 2 at P13, 92. INVESTIGATION 1 A. Determine te y-coordinates of te following points tat lie on te grap of te parabola y 2 : i) Q , y2 ii) Q , y2 iii) Q , y2 iv) Q , y2 B. Calculate te slopes of te secants troug P13, 92 and eac of te points Q 1, Q 2, Q 3, and Q 4. C. Determine te y-coordinates of eac point on te parabola, and ten repeat part B using te following points. i) Q , y2 ii) Q , y2 iii) Q , y2 iv) Q , y2 D. Use your results from parts B and C to estimate te slope of te tangent at point P13, 92. E. Grap y 2 and te tangent to te grap at P13, 92. In tis investigation, you found te slope of te tangent by finding te iting value of te slopes of a sequence of secants. Since we are interested in points Q tat are close to P13, 92 on te parabola y 2 it is convenient to write Q as 13, , were is a very small nonzero number. Te variable determines te position of Q on te parabola. As Q slides along te parabola toward P, will take on values successively smaller and closer to zero. We say tat approaces zero and use te notation S 0. CHAPTER 1 11

14 INVESTIGATION 2 A. Using tecnology or grap paper, draw te parabola f B. Let P be te point 11, 12. C. Determine te slope of te secant troug Q 1 and P11, 12, Q 2 and P11, 12 and so on, for points Q , f , Q , f , Q , f , Q , f , and Q , f D. Draw tese secants on te same grap you created in part A. E. Use your results to estimate te slope of te tangent to te grap of f at point P. F. Draw te tangent at point P11, 12. INVESTIGATION 3 A. Determine an epression for te slope of te secant PQ troug points P13, 92 and Q13, B. Eplain ow you could use te epression in a part A to predict te slope of te tangent to te parabola f 12 2 at point P13, 92. Te slope of te tangent to te parabola at point P is te iting slope of te secant line PQ as point Q slides along te parabola; tat is, as S 0, we write as te abbreviation for iting value as approaces 0. S0 Terefore, from te investigation, te slope of te tangent at a point P is 1slope of te secant PQ2. S0 EXAMPLE 1 Reasoning about te slope of a tangent as a iting value Determine te slope of te tangent to te grap of te parabola f 12 2 at P13, 92. Solution Using points P13, 92 and Q13, , 0, te slope of te secant PQ is THE SLOPE OF A TANGENT y y 2 y (Substitute) (Epand) (Simplify and factor) (Divide by te common factor of )

15 As S 0, te value of 16 2 approaces 6, and tus We conclude tat te slope of te tangent at P13, 92 to te parabola y 2 is 6. S0 EXAMPLE 2 Tec Support For elp graping functions using a graping calculator, see Tecnology Appendi X-XX. Selecting a strategy involving a series of secants to estimate te slope of a tangent a. Use your calculator to grap te parabola y Plot te 8 points on te parabola from 1to 6, were is an integer. b. Determine te slope of te secants using eac point from part a and point P15, c. Use te result of part b to estimate te slope of te tangent at P15, Solution a. Using te -intercepts of 1 and 7, te equation of te ais of symmetry is 1 7 3, so te -coordinate of te verte is 3. 2 Substitute 3 into y y Terefore, te verte is 13, Te y-intercept of te parabola is 8. Te points on te parabola are 11, 02, 10, , 11, 1.52, 12, , 13, 22, 14, , 15, 1.52, and 16, Te parabola and te secants troug eac point and point P15, 1.52 in red. Te tangent troug P15, 1.52 is sown in green. are sown y P(5, 1.5) b. Using points 11, 02 and P15, 1.52, te slope is m Using te oter points and P15, 1.52, te slopes are 0.125, 0, 0.125, 0.25, 0.375, and 0.625, respectively. c. Te slope of te tangent at P15, 1.52 is between and It can be determined to be 0.5 using points closer and closer to P15, CHAPTER 1 13

16 0 y tangent at P P(a, f(a)) Q(a +, f(a + )) y = f() Te Slope of a Tangent at an Arbitrary Point We can now generalize te metod used above to derive a formula for te slope of te tangent to te grap of any function y f 12. Let P1a, f 1a22 be a fied point on te grap of y f 12, and let Q1, y2 Q1, f 122 represent any oter point on te grap. If Q is a orizontal distance of units from P, ten a and y f 1a 2. Point Q ten as coordinates Q1a, f 1a 22. Te slope of te secant PQ is y f 1a 2 f 1a2 a a Tis quotient is fundamental to calculus and is referred to as te difference quotient. Terefore, te slope m of te tangent at P1a, f 1a22 is f 1a 2 f 1a2 1slope of te secant PQ2, wic may be written as m. S0 S0 f 1a 2 f 1a2. Slope of a Tangent as a Limit Te slope of te tangent to te grap y f 12 at point P1a, f 1a22 is y f 1a 2 f 1a2 m, if tis it eists. S0 S0 EXAMPLE 3 Connecting its and te difference quotient to te slope of a tangent a. Using te definition of te slope of a tangent, determine te slope of te tangent to te curve y at te point determined by 3. b. Determine te equation of te tangent. c. Sketc te grap of y and te tangent at 3. Solution a. Te slope of te tangent can be determined using te epression above. In tis eample, f and a 3. Ten f and f THE SLOPE OF A TANGENT

17 Te slope of te tangent at 13, 42 f 13 2 f 132 m S S S S0 2 is (Substitute) (Simplify and factor) (Divide by te common factor) (Evaluate) Te slope of te tangent at 3 is 2. b. Te equation of te tangent at 13, 42 y 4 is or y , c. Using graping software, we obtain y 10 y = (3, 4) y = EXAMPLE 4 Selecting a it strategy to determine te slope of a tangent 3 6 Determine te slope of te tangent to te rational function f 12 at point 12, 62. Solution Using te definition, te slope of te tangent at 12, 62 is f 12 2 f 122 m (Substitute) S (Determine a common denominator) S0 S0 S (Simplify) CHAPTER 1 15

18 S S S0 2 (Multiply by te reciprocal) 3 6 Terefore, te slope of te tangent to f 12 at 12, 62 is 1.5. (Evaluate) EXAMPLE 5 Determining te slope of a line tangent to a root function Find te slope of te tangent to f 12 at 9. Solution f 192 V9 3 f 19 2 V9 Using te it of te difference quotient, te slope of te tangent at 9 is m S0 f 19 2 f 192 S S0 S (Substitute) (Rationalize te numerator) (Simplify) S S (Divide by te common factor of ) (Evalute) 1 6 Terefore, te slope of te tangent to f 12 V at 9 is THE SLOPE OF A TANGENT

19 INVESTIGATION 4 Tec Support For elp graping functions, tracing, and using te table feature on a graping calculator, see Tecnology Appendices X-XX, X-XX, and X-XX A graping calculator can elp us estimate te slope of a tangent at a point. Te eact value can ten be found using te definition of te slope of te tangent using te difference quotient. For eample, suppose tat we wis to find te slope of te tangent to y f 12 3 at 1. A. Grap Y B. Eplain wy te values for te WINDOW were cosen. f 1a 2 f 1a2 Observe tat te function entered in Y 1 is te difference quotient for f 12 3 and Remember tat tis approimates te slope of te tangent and not te grap of f C. Use te TRACE function to find X , Y Tis means tat te slope of te secant passing troug te points were 1 and is about 3.2. Te value 3.2 could be used as an approimation for te slope of te tangent at 1. D. Can you improve tis approimation? Eplain ow you could improve your estimate. Also, if you use different WINDOW values, you can see a different-sized, or differently centred, grap. E. Try once again by setting X min 9, X ma 10, and note te different appearance of te grap. Use te TRACE function to find X , Y , and ten X , Y Wat is your guess for te slope of te tangent at 1 now? Eplain wy only estimation is possible. F. Anoter way of using a graping calculator to approimate te slope of te tangent is to consider as te variable in te difference quotient. For tis f 1a 2 f 1a2 eample, at look at f , 3. G. Trace values of as S 0. You can use te table or grap function of your 11 2 calculator. Grapically, we say tat we are looking at 3 1 in te neigbourood of To do tis, grap y and eamine te value of te function as S 0. CHAPTER 1 17

20 IN SUMMARY Key Ideas Te slope of te tangent to a curve at a point P is te it of te slopes of te secants PQ as Q moves closer to P. m tangent 1slope of secant PQ2 QSP Te slope of te tangent to te grap of y f12 at P1a, f1a22 is given by y f1a 2 f1a2 m tangent. S0 S0 Need to Know To find te slope of te tangent at a point P1a, f 1a22, find te value of f 1a2 find te value of f 1a 2 f1a 2 f1a2 evaluate S0 Eercise 1.2 PART A 1. Calculate te slope of te line troug eac pair of points. a. 12, 72, 13, 82 b. a 1 2, 3 2 b, a 7 2, 7 2 b c. 16.3, 2.62, 11.5, Determine te slope of a line perpendicular to eac of te following: a. y 3 5 b. 13 7y State te equation and sketc te grap of eac line described below. a. passing troug 14, 42 and Q 5 3, 5 3 R b. aving slope 8 and y-intercept 6 c. aving -intercept 5 and y-intercept 3 d. passing troug 15, 62 and 15, THE SLOPE OF A TANGENT

21 4. Simplify eac of te following difference quotients: a. d b. e c. f. 5. Rationalize te numerator of eac epression to obtain an equivalent epression. V16 4 V V5 V5 a. b. c. K PART B 6. Determine an epression, in simplified form, for te slope of te secant PQ. a. P11, 32, Q11, f 11 22, were f b. P11, 32, Q11, c. P19, 32, Q19, Consider te function f a. Copy and complete te following table of values. P and Q are points on te grap of f 12. P Q 12, 2 13, 2 12, , 2 12, , 2 12, , 2 12, 2 11, 2 12, , 2 12, , 2 12, , 2 Slope of Line PQ b. Use your results for part a to approimate te slope of te tangent to te grap of f 12 at point P. c. Calculate te slope of te secant PQ, were te -coordinate of Q is 2. d. Use your result for part c to calculate te slope of te tangent to te grap of f 12 at point P. CHAPTER 1 19

22 e. Compare your answers for parts b and d. f. Sketc te grap of f 12 and te tangent to te grap at point P. 8. Determine te slope of te tangent to eac curve at te given value of. a. y 3 2, 2 b. y 2, 3 c. y 3, 2 9. Determine te slope of te tangent to eac curve at te given value of. a. y 2, 3 b. y 5, 9 c. y 5 1, Determine te slope of te tangent to eac curve at te given value of. a. y 8 b. y 8 c. y 1 3, 2 3, 1 2, 11. Determine te slope of te tangent to eac curve at te given point. a. y 2 3, 12, 22 d. y 7, 116, 32 C b. f , 22 e. y 25 2, 13, 42, c. y f. y 4 33, 11, 32, 18, Sketc te grap of te function in question 11, part e. Sow tat te slope of te tangent can be found using te properties of circles. 13. Eplain ow you would approimate te slope of te tangent at a point witout using te definition of te slope of te tangent. 14. Using tecnology, sketc te grap of y 3 Eplain ow te 4 V16 2. slope of te tangent at P10, 32 can be found witout using te difference quotient. 15. Determine te equation of te tangent to y at (3, 1). 16. Determine te equation of te tangent to y were For f , find a. te coordinates of point A, were 3, b. te coordinates of point B, were 5 c. te equation of te secant AB d. te equation of te tangent at A e. te equation of te tangent at B THE SLOPE OF A TANGENT

23 18. Copy te following figures. Draw an approimate tangent for eac curve at point P and estimate its slope. a. d. P P b. e. P P c. f. P P A T 19. Find te slope of te demand curve D1p2 20, p 7 1, at point 15, 102. Vp It is projected tat, t years from now, te circulation of a local newspaper will be C1t2 100t 2 400t Find ow fast te circulation is increasing after 6 monts. Hint: Find te slope of te tangent wen t Find te coordinates of te point on te curve f were te tangent is parallel to te line y Find te points on te grap of y 1 at wic te tangent is orizontal. PART C 23. Sow tat, at te points of intersection of te quadratic functions y 2 and y 1 te tangents to te functions are perpendicular. 2 2, 24. Determine te equation of te line tat passes troug (2, 2) and is parallel to te line tangent to y at 11, a. Determine te slope of te tangent to te parabola y at te point wose -coordinate is a. b. At wat point on te parabola is te tangent line parallel to te line 10 2y 18 0? c. At wat point on te parabola is te tangent line perpendicular to te line 35y 7 0? CHAPTER 1 21

24 Section 1.3 Rates of Cange Many practical relationsips involve interdependent quantities. For eample, te volume of a balloon varies wit its eigt above te ground, air temperature varies wit elevation, and te surface area of a spere varies wit te lengt of te radius. Tese and oter relationsips can be described by means of a function, often of te form y f 12. Te dependent variable, y, can represent quantities suc as volume, air temperature, and area. Te independent variable,, can represent quantities suc as eigt, elevation, and lengt. We are often interested in ow rapidly te dependent variable canges wen tere is a cange in te independent variable. Recall tat tis concept is called rate of cange. In tis section, we sow tat an instantaneous rate of cange can be calculated by finding te it of a difference quotient in te same way tat we find te slope of a tangent. Velocity as a Rate of Cange An object moving in a straigt line is an eample of a rate-of-cange model. It is customary to use eiter a orizontal or vertical line wit a specified origin to represent te line of motion. On suc a line, movement to te rigt or upward is considered to be in te positive direction, and movement to te left (or down) is considered to be in te negative direction. An eample of an object moving along a line would be a veicle entering a igway and travelling nort 340 km in 4. Te average velocity would be average velocity cange in position cange in time 85 km>, since If s1t2 gives te position of te veicle on a straigt section of te igway at time t, ten te average rate of cange in te position of te veicle over a time interval is average velocity s t. INVESTIGATION You are driving wit a broken speedometer on a igway. At any instant, you do not know ow fast te car is going. Your odometer readings are given t () s(t) (km) RATES OF CHANGE

25 A. Determine te average velocity of te car over eac interval. B. Te speed it is 80 km/. Do any of your results in part A suggest tat you were speeding at any time? If so, wen? C. Eplain wy tere may be oter times wen you were travelling above te posted speed it. D. Compute your average velocity over te interval 4 t 7, if s km and s km. E. After 3 of driving, you decide to continue driving from Goderic to Huntsville, a distance of 345 km. Using te average velocity from part D, ow long would it take you to make tis trip? EXAMPLE 1 Reasoning about average velocity A pebble is dropped from a cliff, 80 m ig. After t seconds, te pebble is s metres above te ground, were s1t2 80 5t 2, 0 t 4. a. Calculate te average velocity of te pebble between te times t 1 s and t 3 s. b. Calculate te average velocity of te pebble between te times t 1 s and t 1.5 s. c. Eplain wy your answers for parts a and b are different. Solution a. average velocity s t s s s132 s112 average velocity m>s Te average velocity in tis 2 s interval is 20 m>s. b. s s11.52 s112 average velocity m>s Te average velocity in tis 0.5 s interval is 12.5 m>s. CHAPTER 1 23

26 c. Since gravity causes te velocity to increase wit time, te smaller interval of 0.5 s gives a lower average velocity, as well as giving a value closer to te actual velocity at time t 1. Te following table sows te results of similar calculations of te average velocity over successively smaller time intervals: Time Interval Average Velocity (m/s) 1 t t t s s (a + ) s(a) P a Q s = s(t) s a + t It appears tat, as we sorten te time interval, te average velocity is approacing te value 10 m>s. Te average velocity over te time interval 1 t 1 is s11 2 s112 average velocity , 0 If te time interval is very sort, ten is small, so 5 is close to 0 and te average velocity is close to 10 m>s. Te instantaneous velocity wen t 1 is defined to be te iting value of tese average values as approaces 0. Terefore, te velocity (te word instantaneous is usually omitted) at time t 1 s is v m>s. S0 In general, suppose tat te position of an object at time t is given by te function s1t2. In te time interval from t a to t a, te cange in position is s s1a 2 s1a2. s s1a 2 s1a2 Te average velocity over tis time interval is, wic is te t same as te slope of te secant PQ were P1a, s1a22 and Q1a, s1a 22. Te velocity at a particular time t a is calculated by finding te iting value of te average velocity as S RATES OF CHANGE

27 Instantaneous Velocity Te velocity of an object wit position function s1t2, at time t a, is s s1a 2 s1a2 v1a2 ts0 t S0 Note tat te velocity v1a2 is te slope of te tangent to te grap of s1t2 at P1a, s1a22. Te speed of an object is te absolute value of its velocity. It indicates ow fast an object is moving, wereas velocity indicates bot speed and direction (relative to a given coordinate system). EXAMPLE 2 Selecting a strategy to calculate velocity A toy rocket is launced straigt up so tat its eigt s, in metres, at time t, in seconds, is given by s1t2 5t 2 30t 2. Wat is te velocity of te rocket at t 4? Solution Since s1t2 5t 2 30t 2, s s Te velocity at t 4 is s14 2 s142 v142 S S S S0 10 (Substitute) (Factor) (Simplify) (Evaluate) Terefore, te velocity of te rocket is 10 m/s downward at t 4 s. CHAPTER 1 25

28 Comparing Average and Instantaneous Rates of Cange Velocity is only one eample of te concept of rate of cange. In general, suppose tat a quantity y depends on according to te equation y f 12. As te independent variable canges from a to a 1 a a 2, te corresponding cange in te dependent variable y is y f 1a 2 f 1a2. Average Rate of Cange y f 1a 2 f 1a2 Te difference quotient is called te average rate of cange in y wit respect to over te interval from a to a. From te diagram, it follows tat te average rate of cange equals te slope of te secant PQ of te grap of f 12 were P1a, f 1a22 and Q1a, f 1a 22. Te instantaneous rate of cange in y wit respect to wen a is defined to be te iting value of te average rate of cange as S 0. f(a + ) y Q y = f() f(a) P a a + y Instantaneous Rates of Cange Terefore, we conclude tat te instantaneous rate of cange in y f 12 wit respect to wen a y f 1a 2 f 1a2 is, provided tat S0 S0 te it eists. It sould be noted tat, as wit velocity, te instantaneous rate of cange in y wit respect to at a equals te slope of te tangent to te grap of y f 12 at a. EXAMPLE 3 Selecting a strategy to calculate instantaneous rate of cange Te total cost, in dollars, of manufacturing calculators is given by C12 10V a. Wat is te total cost of manufacturing 100 calculators? b. Wat is te rate of cange in te total cost wit respect to te number of calculators,, being produced wen 100? RATES OF CHANGE

29 Solution a. C V Terefore, te total cost of manufacturing 100 calculators is $1100. b. Te rate of cange in te cost at 100 is given by C C11002 S0 10V S0 10V V S0 10V S0 110V S0 S V V V (Substitute) (Rationalize te numerator) (Epand) (Simplify) (Evaluate) Terefore, te rate of cange in te total cost wit respect to te number of calculators being produced, wen 100 calculators are being produced, is $0.50 per calculator. An Alternative Form for Finding Rates of Cange In Eample 1, we determined te velocity of te pebble at t 1 by taking te it of te average velocity over te interval 1 t 1 as approaces 0. We can also determine te velocity at t 1 by considering te average velocity over te interval from 1 to a general time t and letting t approac te value 1. Ten, s1t2 80 5t 2 s CHAPTER 1 27

30 s1t2 s112 v112 ts1 t 1 5 5t 2 ts1 t t211 t2 ts1 t t2 ts1 10 s1t2 s1a2 In general, te velocity of an object at time t a is v1a2. tsa t a Similarly, te instantaneous rate of cange in y f 12 wit respect to wen f 12 f 1a2 a is. Sa a IN SUMMARY Key Ideas Te average velocity can be found in te same way tat we found te slope of te secant. cange in position average velocity cange in time Te instantaneous velocity is te slope of te tangent to te grap of te position function and is found in te same way tat we found te slope of te tangent. Need to Know To find te average velocity (average rate of cange) from t a to t a, we can use te difference quotient and te position function s1t2 s s 1a 2 s 1a2 t Te rate of cange in te position function, s1t2, is te velocity at t a, and we can find it by computing te iting value of te average velocity as S 0: s1a 2 s1a2 v1a2 S RATES OF CHANGE

31 Eercise 1.3 C PART A 1. Te velocity of an object is given by v 1t2 t 1t At wat times, in seconds, is te object at rest? 2. Give a geometrical interpretation of te following epressions, if s is a position function: s 192 s 122 s 16 2 s 162 a. b. 7 S0 3. Give a geometrical interpretation of 4. Use te grap to answer eac question. S y B C y = f() A D E a. Between wic two consecutive points is te average rate of cange in te function te greatest? b. Is te average rate of cange in te function between A and B greater tan or less tan te instantaneous rate of cange at B? c. Sketc a tangent to te grap somewere between points D and E suc tat te slope of te tangent is te same as te average rate of cange in te function between B and C. 5. Wat is wrong wit te statement Te speed of te ceeta was 65 km/ nort? 6. Is tere anyting wrong wit te statement A scool bus ad a velocity of 60 km/ for te morning run, wic is wy it was late arriving? PART B 7. A construction worker drops a bolt wile working on a ig-rise building, 320 m above te ground. After t seconds, te bolt as fallen a distance of s metres, were s1t t 2, 0 t 8. a. Calculate te average velocity during te first, tird, and eigt seconds. b. Calculate te average velocity for te interval 3 t 8. c. Calculate te velocity at t 2. CHAPTER 1 29

32 K A 8. Te function s1t2 8t 1t 22 describes te distance s, in kilometres, tat a car as travelled after a time t, in ours, for 0 t 5. a. Calculate te average velocity of te car during te following intervals: i. from t 3 to t 4 ii. from t 3 to t 3.1 iii. from t 3 to t 3.01 b. Use your results for part a to approimate te instantaneous velocity of te car at t 3. c. Calculate te velocity at t Suppose tat a foreign-language student as learned N1t2 20t t 2 vocabulary terms after t ours of uninterrupted study, were 0 t 10. a. How many terms are learned between time t 2 and t 3? b. Wat is te rate, in terms per our, at wic te student is learning at time t 2? 10. A medicine is administered to a patient. Te amount of medicine M, in milligrams, in 1 ml of te patient s blood, t ours after te injection, is M1t2 1 were 0 t 3. 3 t2 t, a. Find te rate of cange in te amount M, 2 after te injection. b. Wat is te significance of te fact tat your answer is negative? 11. Te time t, in seconds, taken by an object dropped from a eigt of s metres to reac te ground is given by te formula t V s 5. Determine te rate of cange in time wit respect to eigt wen te object is 125 m above te ground. 12. Suppose tat te temperature T, in degrees Celsius, varies wit te eigt, in kilometres, above Eart s surface according to te equation T Find te rate of cange in temperature wit respect to eigt at a eigt of 3 km. 13. A spacesip approacing toucdown on a distant planet as eigt, in metres, at time t, in seconds, given by 25t 2 100t 100. Wen does te spacesip land on te surface? Wit wat speed does it land (assuming it descends vertically)? 14. A manufacturer of soccer balls finds tat te profit from te sale of balls per week is given by P dollars. a. Find te profit on te sale of 40 soccer balls per week. b. Find te rate of cange in profit at te production level of 40 balls per week. c. Using a graping calculator, grap te profit function and, from te grap, determine for wat sales levels of te rate of cange in profit is positive RATES OF CHANGE

33 f 12 f 1a2 15. Use te alternate definition to calculate te instantaneous rate Sa a of cange of f 12 at te given point or value of. a. f , 12,52 b. f , c. f 12 V 1, Te average annual salary of a professional baseball player can be modelled by te function S , were S represents te average annual salary, in tousands of dollars, and is te number of years since Determine te rate at wic te average salary was canging in Te motion of an avalance is described by s1t2 3t 2, were s is te distance, in metres, travelled by te leading edge of te snow at t seconds. a. Find te distance travelled from 0 s to 5 s. b. Find te rate at wic te avalance is moving from 0 s to 10 s. c. Find te rate at wic te avalance is moving at 10 s. d. How long, to te nearest second, does te leading edge of te snow take to move 600 m? T PART C 18. Let (a, b) be any point on te grap of y 1 0. Prove tat te area of te, triangle formed by te tangent troug (a, b) and te coordinate aes is MegaCorp s total weekly cost to produce pencils can be written as C12 F V12, were F, a constant, represents fied costs suc as rent and utilities and V12 represents variable costs, wic depend on te production level. Sow tat te rate of cange in te weekly cost is independent of fied costs. 20. A circular oil spill on te surface of te ocean spreads outward. Find te approimate rate of cange in te area of te oil slick wit respect to its radius wen te radius is 100 m. 21. Sow tat te rate of cange in te volume of a cube wit respect to its edge lengt is equal to alf te surface area of te cube. 22. Determine te instantaneous rate of cange in a. te surface area of a sperical balloon (as it is inflated) at te point in time wen te radius reaces 10 cm b. te volume of a sperical balloon (as it is deflated) at te point in time wen te radius reaces 5 cm CHAPTER 1 31

34 Mid-Capter Review 1. Calculate te product of eac radical epression and its corresponding conjugate. a. V5 V2 b. 3V5 2V2 c. 9 2V5 d. 3V5 2V10 2. Rationalize eac denominator. a. 6 V2 5 5V3 c. e. V3 V7 4 2V3 4 b. 2V3 4 2V3 3V2 d. f. V3 V3 2 2V Rationalize eac numerator. a. V2 V7 4 V3 V7 c. e b. V3 2V3 5 2V3 V7 d. f. 6 V2 3V Determine te equation of te line described by te given information. a. slope 2 passing troug point 10, 62 3, b. passing troug points 12, 72 and 16, 112 c. parallel to y 4 6, passing troug point 12, 62 d. perpendicular to y 5 3, passing troug point 11, Find te slope of PQ, in simplified form, given P11, 12 and Q11, f 11 22, were f Consider te function y a. Copy and complete te following tables of values. P and Q are points on te grap of f 12. P Q 11, 12 12, 62 Slope of Line PQ P Q 11, 12 10, 2 Slope of Line PQ 11, , , , 2 11, , 2 11, , 2 11, , 2 11, , 2 11, , 2 11, , 2 32 MID-CHAPTER REVIEW

35 b. Use your results for part a to approimate te slope of te tangent to te grap of f 12 at point P. c. Calculate te slope of te secant were te -coordinate of Q is 1. d. Use your results for part c to calculate te slope of te tangent to te grap of f 12 at point P. e. Compare your answers for parts b and d. 7. Calculate te slope of te tangent to eac curve at te given point or value of. a. f , 13,52 c. y 4, 16, 12 b. y 1, 1 2 d. f 12 V 4, Te function s1t2 6t1t 12 describes te distance (in kilometres) tat a car as travelled after a time t (in ours), for 0 t 6. a. Calculate te average velocity of te car during te following intervals. i. from t 2 to t 3 ii. from t 2 to t 2.1 iii. from t 2 to t 2.01 b. Use your results for part a to approimate te instantaneous velocity of te car wen t 2. c. Find te average velocity of te car from t 2 to t 2. d. Use your results for part c to find te velocity wen t Calculate te instantaneous rate of cange of f 12 wit respect to at te given value of. a. f b. f ,, An oil tank is being drained for cleaning. After t minutes, tere are V litres of oil left in te tank, were V1t t2 2, 0 t 30. a. Calculate te average rate of cange in volume during te first 20 min. b. Calculate te rate of cange in volume at time t Find te equation of te tangent at te given value of. a. y 2 3, 4 b. y 2 2 7, 2 c. f , 1 d. f , Find te equation of te tangent to te grap of te function at te given value of. a. f 12 3, 5 b. f , 1 CHAPTER 1 33

36 Section 1.4 Te Limit of a Function Te notation f 12 L is read te it of f 12 as approaces a equals L Sa and means tat te value of f 12 can be made arbitrarily close to L by coosing sufficiently close to a (but not equal to a). But f 12 eists if and only if te Sa iting value from te left equals te iting value from te rigt. We sall use tis definition to evaluate some its. Note: Tis is an intuitive eplanation of te it of a function. A more precise definition using inequalities is important for advanced work but is not necessary for our purposes. INVESTIGATION 1 Determine te it of y 2 1, as approaces 2. A. Copy and complete te table of values y 2 1 B. As approaces 2 from te left, starting at 1, wat is te approimate value of y? C. As approaces 2 from te rigt, starting at 3, wat is te approimate value of y? D. Grap y 2 1 using graping software or grap paper. E. Using arrows, illustrate tat, as we coose a value of tat is closer and closer to 2, te value of y gets closer and closer to a value of 3. F. Eplain wy te it of y 2 1 eists as approaces 2, and give its approimate value. EXAMPLE 1 Determine 2 1 by graping. 1 Solution S1 On a graping calculator, display te grap of f , THE LIMIT OF A FUNCTION

37 Te grap sown on your calculator is a line 1 f 12 12, wereas it sould be a line wit point (1, 2) deleted 1 f 12 1, 12. Te WINDOW used is X min 10, X ma 10, X scl 1, and similarly for Y. Use te TRACE function to find X , Y and X , Y Click ZOOM ; select 4:ZDecimal, ENTER. Now, te grap of f is displayed as a straigt line wit point (1, 2) deleted. Te WINDOW 1 as new values, too. Use te TRACE function to find X 0.9, Y 1.9; X 1, Y as no value given; and X 1.1, Y 2.1. We can estimate f 12. As approaces 1 from te left, written as S 1, we observe tat f 12 approaces te value 2 from below. As approaces 1 from te rigt, written as S 1, f 12 approaces te value 2 from above. We say tat te it at 1 eists only if te value approaced from te left is te same as te value approaced from te rigt. From tis investigation, we conclude tat S1 S1 EXAMPLE 2 Selecting a table of values strategy to evaluate a it Determine 2 1 by using a table. 1 S1 Solution We select sequences of numbers for S 1 and S 1. CHAPTER 1 35

38 approaces 1 from te left S d approaces 1 from te rigt undefined f approaces 2 from below S d f approaces 2 from above Tis pattern of numbers suggests tat in Eample , S1 1 as we found wen graping EXAMPLE 3 Tec Support For elp graping piecewise functions on a graping calculator, see Tecnology Appendi X-XX. Selecting a graping strategy to evaluate a it Sketc te grap of te piecewise function: 1, if 6 1 f 12 1, if 1 2 V 1, if 7 1 Determine f 12. S1 Solution Te grap of te function f consists of te line y 1 for 6 1, te point (1, 1) and te square root function y 2 1 for 7 1. From te grap of f 12, observe tat te it of f 12 as S 1 depends on weter 6 1 or 7 1. As S 1, f 12 approaces te value of 0 from below. We write tis as f S1 S y y = f() Similarly, as S 1, f 12 approaces te value 2 from above. We write tis as f 12 (Tis is te same wen 1 is substituted S1 S1Q2 1R 2. into te epression 2 1. ) Tese two its are referred to as one-sided THE LIMIT OF A FUNCTION

39 its because, in eac case, only values of on one side of 1 are considered. How-ever, te one-sided its are unequal f f 12 or more S1 S1 briefly, f 12 f 12. Tis implies tat f 12 does not approac a single value S1 S1 as S 1. We say te it of f 12 as S 1 does not eist and write f 12 does S1 not eist. Tis may be surprising, since te function f 12 was defined at 1 tat is, f We can now summarize te ideas introduced in tese eamples. Limits and Teir Eistence We say tat te number L is te it of a function y f 12 as approaces te value a, written as f 12 L, if f 12 L f 12. Oterwise, Sa Sa Sa f 12 does not eist. Sa IN SUMMARY Key Idea Te it of a function y f12 at a is written as f12 L, wic means tat f12 approaces te value L as approaces te value a from bot te left and rigt side. Sa Need to Know f12 may eist even if f1a2 is not defined. Sa f12 can be equal to f1a2. In tis case, te grap of f12 passes troug Sa te point 1a, f1a22. If and ten L is te it of f12 as approaces Saf12 L, Saf12 L a, tat is f12 L. Sa Eercise 1.4 C PART A 1. Wat do you tink is te appropriate it of eac sequence? a. 0.7, 0.72, 0.727, ,... b. 3, 3.1, 3.14, 3.141, , , , Eplain a process for finding a it. 3. Write a concise description of te meaning of te following: a. a rigt-sided it b. a left-sided it c. a (two-sided) it CHAPTER 1 37

40 4. Calculate eac it. a. c. e. S 5 b d f. S3 2 S10 S 2 5. Determine 1, if 4 f 12, were f 12 e 1, if 4. S4 4 S1 S3 2 PART B 6. For te function f 12 in te grap below, determine te following: a. f 12 b. f 12 c. f 12 d. S2 S2 y 4 2 (2, 2) ( 2, 0) (2, 1) S2 f 122 K 7. Use te grap to find te it, if it eists. a. f 12 b. f 12 c. S2 S2 y y 4 4 f 12 S3 y Evaluate eac it. 20 a b. c. 1 S 1 S0 2 5 S5 9. Find , and illustrate your result wit a grap indicating te S2 iting value. 10. Evaluate eac it. If te it does not eist, eplain wy. 1 a. c. e. S S0 4 S b. d. f. S S1 3 S THE LIMIT OF A FUNCTION

41 11. For eac function, sketc te grap of te function. Determine te indicated it if it eists. 2, if 6 1 a. f 12 e 2, if 1 ; f 12 S1 b. 4, if 2 f 12 e 2 6, if 7 2 ; f 12 S2 A T c. d. 1, if f 12 e , if 0.5 ; f 12 S Sketc te grap of any function tat satisfies te given conditions. a. f 112 1, f 12 3, f 12 2 S1 S1 b. f 122 1, f 12 0 S2 c. if and f 12 2 S1 d. f 132 0, f 12 0 S3 13. Let f 12 m b, were m and b are constants. If f 12 2and f 12 4, find m and b. S1 S 1 PART C 14. Determine te real values of a, b, and c for te quadratic function f 12 a 2 b c, a 0, tat satisfy te conditions f 102 0, f 12 5, and f S1 4, if 1 2 f 12 1, if S 2 ; f 12 S Te fis population, in tousands, in a lake at time t, in years, is modelled by te following function: t2, if 0 t 6 p1t2 µ t2, if 6 6 t 12 Tis function describes a sudden cange in te population at time t 6, due to a cemical spill. a. Sketc te grap of p1t2. b. Evaluate p1t2 and p1t2. ts6 ts6 c. Determine ow many fis were killed by te spill. d. At wat time did te population recover to te level before te spill? CHAPTER 1 39