AWL/Tomas_cp7-8/9/ :6 AM Page 8 8 Capter : Differentiation f () Secant slope is f (z) f () z Q(z, f (z)) f (z) f () P(, f()) We use te notation ƒ() ra

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1 AWL/Tomas_cp7-8/9/ :6 AM Page 7 Capter DIFFERENTIATION OVERVIEW In Capter, we efine te slope of a curve at a point as te limit of secant slopes. Tis limit, calle a erivative, measures te rate at wic a function canges, an it is one of te most important ieas in calculus. Derivatives are use to calculate velocit an acceleration, to estimate te rate of sprea of a isease, to set levels of prouction so as to maimize efficienc, to fin te best imensions of a clinrical can, to fin te age of a preistoric artifact, an for man oter applications. In tis capter, we evelop tecniques to calculate erivatives easil an learn ow to use erivatives to approimate complicate functions.. Te Derivative as a Function At te en of Capter, we efine te slope of a curve ƒs at te point were to be HISTORICAL ESSAY lim Te Derivative : ƒs + - ƒs. We calle tis limit, wen it eiste, te erivative of ƒ at. We now investigate te erivative as a function erive from ƒ b consiering te limit at eac point of te omain of ƒ. DEFINITION Derivative Function Te erivative of te function ƒ() wit respect to te variable is te function ƒ wose value at is ƒs + - ƒs, : ƒ s lim provie te limit eists. 7 Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

2 AWL/Tomas_cp7-8/9/ :6 AM Page 8 8 Capter : Differentiation f () Secant slope is f (z) f () z Q(z, f (z)) f (z) f () P(, f()) We use te notation ƒ() rater tan simpl ƒ in te efinition to empasize te inepenent variable, wic we are ifferentiating wit respect to. Te omain of ƒ is te set of points in te omain of ƒ for wic te limit eists, an te omain ma be te same or smaller tan te omain of ƒ. If ƒ eists at a particular, we sa tat ƒ is ifferentiable (as a erivative) at. If ƒ eists at ever point in te omain of ƒ, we call ƒ ifferentiable. If we write z +, ten z - an approaces if an onl if z approaces. Terefore, an equivalent efinition of te erivative is as follows (see Figure.). z z Alternative Formula for te Derivative Derivative of f at is f( ) f () f '() lim lim z ƒ s lim f (z) f () z z: FIGURE. Te wa we write te ifference quotient for te erivative of a function ƒ epens on ow we label te points involve. ƒsz - ƒs z -. Calculating Derivatives from te Definition Te process of calculating a erivative is calle ifferentiation. To empasize te iea tat ifferentiation is an operation performe on a function ƒs, we use te notation ƒs as anoter wa to enote te erivative ƒ s. Eamples an of Section.7 illustrate te ifferentiation process for te functions m + b an >. Eample sows tat sm + b m. For instance, a - b. In Eample, we see tat a b -. Here are two more eamples. EXAMPLE Appling te Definition Differentiate ƒs Solution. - Here we ave ƒs - Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

3 AWL/Tomas_cp7-8/9/ :6 AM Page 9. Te Derivative as a Function 9 an s +, so s + - ƒs + - ƒs ƒ s lim : ƒs + lim s + s - - s + - lim # s + - s - : : lim - # s + - s - lim - -. s + - s - s - : : EXAMPLE c a - cb a - b b Derivative of te Square Root Function (a) Fin te erivative of for 7. (b) Fin te tangent line to te curve at. Solution You will often nee to know te erivative of for 7 : (a) We use te equivalent form to calculate ƒ : ƒsz - ƒs z - z:. ƒ s lim lim z: lim z: lim z: (, ) z - A z - B A z + B. z + (b) Te slope of te curve at is ƒ s 兹 z - z -. Te tangent is te line troug te point (, ) wit slope > (Figure.): FIGURE. Te curve an its tangent at (, ). Te tangent s slope is foun b evaluating te erivative at (Eample ). + s - +. We consier te erivative of wen in Eample 6. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

4 AWL/Tomas_cp7-8/9/ :6 AM Page 5 5 Capter : Differentiation Notations Tere are man was to enote te erivative of a function ƒs, were te inepenent variable is an te epenent variable is. Some common alternative notations for te erivative are ƒ s ƒ ƒs Dsƒs D ƒs. Te smbols > an D inicate te operation of ifferentiation an are calle ifferentiation operators. We rea > as te erivative of wit respect to, an ƒ> an (>)ƒ() as te erivative of ƒ wit respect to. Te prime notations an ƒ come from notations tat Newton use for erivatives. Te > notations are similar to tose use b Leibniz. Te smbol > soul not be regare as a ratio (until we introuce te iea of ifferentials in Section.8). Be careful not to confuse te notation D(ƒ) as meaning te omain of te function ƒ instea of te erivative function ƒ. Te istinction soul be clear from te contet. To inicate te value of a erivative at a specifie number a, we use te notation ƒ sa f ` ` ƒs `. a a a For instance, in Eample b we coul write ƒ s ` `. To evaluate an epression, we sometimes use te rigt bracket ] in place of te vertical bar ƒ. Graping te Derivative We can often make a reasonable plot of te erivative of ƒs b estimating te slopes on te grap of ƒ. Tat is, we plot te points s, ƒ s in te -plane an connect tem wit a smoot curve, wic represents ƒ s. EXAMPLE Graping a Derivative Grap te erivative of te function ƒs in Figure.a. We sketc te tangents to te grap of ƒ at frequent intervals an use teir slopes to estimate te values of ƒ s at tese points. We plot te corresponing s, ƒ s pairs an connect tem wit a smoot curve as sketce in Figure.b. Solution Wat can we learn from te grap of ƒ s? At a glance we can see... were te rate of cange of ƒ is positive, negative, or zero; te roug size of te growt rate at an an its size in relation to te size of ƒ(); were te rate of cange itself is increasing or ecreasing. Here s anoter eample. EXAMPLE Concentration of Bloo Sugar On April, 988, te uman-powere airplane Daealus flew a recor-breaking 9 km from Crete to te islan of Santorini in te Aegean Sea, souteast of mainlan Greece. Dur- Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

5 AWL/Tomas_cp7-8/9/ :6 AM Page 5 5. Te Derivative as a Function Slope f () A Slope B Slope C E D 5 Slope Slope 8 -units/-unit 8 -units -units 5 (a) 5 Slope f '() E' A' D' 5 C' 5 B' Vertical coorinate (b) FIGURE. We mae te grap of ƒ s in (b) b plotting slopes from te grap of ƒs in (a). Te vertical coorinate of B is te slope at B an so on. Te grap of ƒ is a visual recor of ow te slope of ƒ canges wit. ing te 6-our enurance tests before te fligt, researcers monitore te prospective pilots bloo-sugar concentrations. Te concentration grap for one of te atlete-pilots is sown in Figure.a, were te concentration in milligrams> eciliter is plotte against time in ours. Te grap consists of line segments connecting ata points. Te constant slope of eac segment gives an estimate of te erivative of te concentration between measurements. We calculate te slope of eac segment from te coorinate gri an plotte te erivative as a step function in Figure.b. To make te plot for te first our, for instance, we observe tat te concentration increase from about 79 mg> L to 9 mg> L. Te net increase was 9-79 mg>l. Diviing tis b t our gave te rate of cange as mg>l per our. t Notice tat we can make no estimate of te concentration s rate of cange at times t,, Á, 5, were te grap we ave rawn for te concentration as a corner an no slope. Te erivative step function is not efine at tese times. Coprigt 5 Pearson Eucation, Inc., publising as Pearson AŁ ır Å n «

6 AWL/Tomas_cp7-8/9/ :6 AM Page 5 5 Capter : Differentiation Atens a ge Ae GREECE TURKEY ns ea Concentration, mg/l 9 SANTORINI 8 RHODES Time () 5 Sea of Crete t 6 (a) Heraklion Meiterranean Sea 5 CRETE 5 km Daealus's fligt pat on April, Rate of cange of concentration, mg/l ' t FIGURE. (a) Grap of te sugar concentration in te bloo of a Daealus pilot uring a 6-our prefligt enurance test. (b) Te erivative of te pilot s bloo-sugar concentration sows ow rapil te concentration rose an fell uring various portions of te test. 5 Differentiable on an Interval; One-Sie Derivatives Time () A function ƒs is ifferentiable on an open interval (finite or infinite) if it as a erivative at eac point of te interval. It is ifferentiable on a close interval [a, b] if it is ifferentiable on te interior (a, b) an if te limits (b) lim ƒsa + - ƒsa Rigt-an erivative at a lim ƒsb + - ƒsb Left-an erivative at b : + : - Slope f(a ) f(a) lim eist at te enpoints (Figure.5). Rigt-an an left-an erivatives ma be efine at an point of a function s omain. Te usual relation between one-sie an two-sie limits ols for tese erivatives. Because of Teorem 6, Section., a function as a erivative at a point if an onl if it as left-an an rigt-an erivatives tere, an tese one-sie erivatives are equal. Slope f (b ) f (b) lim EXAMPLE 5 Sow tat te function ƒ ƒ is ifferentiable on s - q, an s, q but as no erivative at. f () Solution a a ƒ ƒ Is Not Differentiable at te Origin b b To te rigt of te origin, s s s #. ƒ ƒ sm + b m, ƒ ƒ To te left, FIGURE.5 Derivatives at enpoints are one-sie limits. s s - s - # - ƒ ƒ Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle ƒ ƒ -

7 AWL/Tomas_cp7-8/9/ :6 AM Page 5. Te Derivative as a Function (Figure.6). Tere can be no erivative at te origin because te one-sie erivatives iffer tere: ' ƒ + ƒ - ƒƒ ƒƒ lim+ : : lim+ ƒ ƒ wen 7. : lim+ Rigt-an erivative of ƒ ƒ at zero lim+ ' 5 ' not efine at : rigt-an erivative left-an erivative : ƒ + ƒ - ƒƒ ƒƒ lim : : - limƒ ƒ - wen 6. : lim Left-an erivative of ƒ ƒ at zero lim- FIGURE.6 Te function ƒ ƒ is not ifferentiable at te origin were te grap as a corner. : EXAMPLE 6 Is Not Differentiable at In Eample we foun tat for 7,. We appl te efinition to eamine if te erivative eists at : lim : lim+ q. : Since te (rigt-an) limit is not finite, tere is no erivative at. Since te slopes of te secant lines joining te origin to te points A, B on a grap of approac q, te grap as a vertical tangent at te origin. Wen Does a Function Not Have a Derivative at a Point? A function as a erivative at a point if te slopes of te secant lines troug Ps, ƒs an a nearb point Q on te grap approac a limit as Q approaces P. Wenever te secants fail to take up a limiting position or become vertical as Q approaces P, te erivative oes not eist. Tus ifferentiabilit is a smootness conition on te grap of ƒ. A function wose grap is oterwise smoot will fail to ave a erivative at a point for several reasons, suc as at points were te grap as. a corner, were te one-sie erivatives iffer.. a cusp, were te slope of PQ approaces q from one sie an - q from te oter. P P Q Q Q Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle Q

8 AWL/Tomas_cp7-8/9/ :6 AM Page 5 5 Capter : Differentiation. a vertical tangent, were te slope of PQ approaces q from bot sies or approaces - q from bot sies (ere, - q ). Q P Q. a iscontinuit. P P Q Q Q Q Differentiable Functions Are Continuous A function is continuous at ever point were it as a erivative. THEOREM Differentiabilit Implies Continuit If ƒ as a erivative at c, ten ƒ is continuous at c. Proof Given tat ƒ sc eists, we must sow tat lim:c ƒs ƒsc, or equivalentl, tat lim: ƒsc + ƒsc. If Z, ten ƒsc + ƒsc + sƒsc + - ƒsc ƒsc + - ƒsc #. ƒsc + Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

9 AWL/Tomas_cp7-8/9/ :6 AM Page 55. Te Derivative as a Function 55 Now take limits as :. B Teorem of Section., ƒsc + - ƒsc : lim ƒsc + lim ƒsc + lim : : # lim : ƒsc + ƒ sc # ƒsc + ƒsc. Similar arguments wit one-sie limits sow tat if ƒ as a erivative from one sie (rigt or left) at c ten ƒ is continuous from tat sie at c. Teorem on page 5 sas tat if a function as a iscontinuit at a point (for instance, a jump iscontinuit), ten it cannot be ifferentiable tere. Te greatest integer function :; int fails to be ifferentiable at ever integer n (Eample, Section.6). CAUTION Te converse of Teorem is false. A function nee not ave a erivative at a point were it is continuous, as we saw in Eample 5. Te Intermeiate Value Propert of Derivatives U() Not ever function can be some function s erivative, as we see from te following teorem. FIGURE.7 Te unit step function oes not ave te Intermeiate Value Propert an cannot be te erivative of a function on te real line. THEOREM If a an b are an two points in an interval on wic ƒ is ifferentiable, ten ƒ takes on ever value between ƒ sa an ƒ sb. Teorem (wic we will not prove) sas tat a function cannot be a erivative on an interval unless it as te Intermeiate Value Propert tere. For eample, te unit step function in Figure.7 cannot be te erivative of an real-value function on te real line. In Capter 5 we will see tat ever continuous function is a erivative of some function. In Section., we invoke Teorem to analze wat appens at a point on te grap of a twice-ifferentiable function were it canges its bening beavior. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

10 AWL/Tomas_cp7-8/9/ :6 AM Page 55. Te Derivative as a Function EXERCISES. - z ; z Fining Derivative Functions an Values. k sz Using te efinition, calculate te erivatives of te functions in Eercises 6. Ten fin te values of te erivatives as specifie. 5. psu u ;. ƒs - ; ƒ s -, ƒ s, ƒ s. Fs s - + ;. g st ; t F s -, F s, F s g s -, g s, g A B k s -, k s, k A B p s, p s, p s> 6. r ss s + ; r s, r s, r s> In Eercises 7, fin te inicate erivatives. s r if if r s Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle 55

11 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation 9. s t if s. t if t - t. p q if p z. w if z Graps t t + Matc te functions grape in Eercises 7 wit te erivatives grape in te accompaning figures (a) (). ' q + w - 5. s t - t, (b) ' ' t - 6. s +, - In Eercises 7 8, ifferentiate te functions. Ten fin an equation of te tangent line at te inicate point on te grap of te function. 7. ƒs (a) -, + In Eercises 6, ifferentiate te functions an fin te slope of te tangent line at te given value of te inepenent variable. 9. ƒs +, Slopes an Tangent Lines. k s ' 8 - () (c) s, s6,, 8. w g sz + - z, In Eercises 9, fin te values of te erivatives. 9. s ` t t -. `. r ` u u if r. w ` z z if w z + z f () f() sz, w s, s - t if - if 9.. f() f() - u Using te Alternative Formula for Derivatives Use te formula ƒ s lim z: ƒsz - ƒs z -. a. Te grap in te accompaning figure is mae of line segments joine en to en. At wic points of te interval [-, 6] is ƒ not efine? Give reasons for our answer. to fin te erivative of te functions in Eercises 6.. ƒs +. ƒs s - 5. g s - (6, ) (, ) f() (, ) (, ) 6. g s + Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle 6 (, )

12 AWL/Tomas_cp7-8/9/ :6 AM Page 57. Te Derivative as a Function b. Grap te erivative of ƒ. Te grap soul sow a step function. 57 p 5. Recovering a function from its erivative a. Use te following information to grap te function ƒ over te close interval [-, 5]. 5 i) Te grap of ƒ is mae of close line segments joine en to en. ii) Te grap starts at te point s -,. iii) Te erivative of ƒ is te step function in te figure sown ere. 5 5 ' ' f '() 5 t Time (as) b. During wat as oes te population seem to be increasing fastest? Slowest? 5 One-Sie Derivatives Compare te rigt-an an left-an erivatives to sow tat te functions in Eercises 5 8 are not ifferentiable at te point P. b. Repeat part (a) assuming tat te grap starts at s -, instea of s -,.. Growt in te econom Te grap in te accompaning figure sows te average annual percentage cange ƒst in te U.S. gross national prouct (GNP) for te ears Grap >t (were efine). (Source: Statistical Abstracts of te Unite States, t Eition, U.S. Department of Commerce, p. 7.) f () Fruit flies (Continuation of Eample, Section..) Populations starting out in close environments grow slowl at first, wen tere are relativel few members, ten more rapil as te number of reproucing iniviuals increases an resources are still abunant, ten slowl again as te population reaces te carring capacit of te environment. a. Use te grapical tecnique of Eample to grap te erivative of te fruit fl population introuce in Section.. Te grap of te population is reprouce ere. f () P(, ) 98 f () 98 P(, ) P(, ) 6 f () 7% 5 P(, ) 兹 Differentiabilit an Continuit on an Interval Eac figure in Eercises 9 sows te grap of a function over a close interval D. At wat omain points oes te function appear to be a. ifferentiable? b. continuous but not ifferentiable? c. neiter continuous nor ifferentiable? Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

13 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation Give reasons for our answers. 9.. f () D: ⱕ ⱕ. f () D: ⱕ ⱕ. f () D: ⱕ ⱕ f () D: ⱕ ⱕ.. b. Sow tat f () D: ⱕ ⱕ f () D: ⱕ ⱕ 5. Tangent to a parabola Does te parabola ave a tangent wose slope is -? If so, fin an equation for te line an te point of tangenc. If not, w not? 5. Tangent to Does an tangent to te curve cross te -ais at -? If so, fin an equation for te line an te point of tangenc. If not, w not? 5. Greatest integer in Does an function ifferentiable on s - q, q ave int, te greatest integer in (see Figure.55), as its erivative? Give reasons for our answer. 5. Derivative of ƒ ƒ Grap te erivative of ƒs ƒ ƒ. Ten grap s ƒ ƒ - >s - ƒ ƒ >. Wat can ou conclue? 55. Derivative of ƒ Does knowing tat a function ƒ() is ifferentiable at tell ou anting about te ifferentiabilit of te function -ƒ at? Give reasons for our answer. 56. Derivative of multiples Does knowing tat a function g(t) is ifferentiable at t 7 tell ou anting about te ifferentiabilit of te function g at t 7? Give reasons for our answer. 57. Limit of a quotient Suppose tat functions g(t) an (t) are efine for all values of t an g s s. Can limt: sg st>sst eist? If it oes eist, must it equal zero? Give reasons for our answers. 58. a. Let ƒ() be a function satisfing ƒ ƒs ƒ for -. Sow tat ƒ is ifferentiable at an fin ƒ s. ƒs L sin, Z, is ifferentiable at an fin ƒ s. T 59. Grap > A B in a winow tat as. Ten, on te same screen, grap for,.5,.. Ten tr -, -.5, -.. Eplain wat is going on. Teor an Eamples In Eercises 5 8, a. Fin te erivative ƒ s of te given function ƒs. b. Grap ƒs an ƒ s sie b sie using separate sets of coorinate aes, an answer te following questions. c. For wat values of, if an, is ƒ positive? Zero? Negative?. Over wat intervals of -values, if an, oes te function ƒs increase as increases? Decrease as increases? How is tis relate to wat ou foun in part (c)? (We will sa more about tis relationsip in Capter.) > > 8. > 9. Does te curve ever ave a negative slope? If so, were? Give reasons for our answer. 5. Does te curve ave an orizontal tangents? If so, were? Give reasons for our answer. T 6. Grap in a winow tat as -,. Ten, on te same screen, grap s + - for,,.. Ten tr -, -, -.. Eplain wat is going on. T 6. Weierstrass s nowere ifferentiable continuous function Te sum of te first eigt terms of te Weierstrass function ƒ() g nq s>n cos s9np is g s cos sp + s> cos s9p + s> cos s9p + s> cos s9p + Á + s>7 cos s97p. Grap tis sum. Zoom in several times. How wiggl an bump is tis grap? Specif a viewing winow in wic te isplae portion of te grap is smoot. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

14 AWL/Tomas_cp7-8/9/ :6 AM Page 59. Te Derivative as a Function COMPUTER EXPLORATIONS Use a CAS to perform te following steps for te functions in Eercises a. Plot ƒs to see tat function s global beavior. b. Define te ifference quotient q at a general point, wit general step size. c. Take te limit as :. Wat formula oes tis give?. Substitute te value an plot te function ƒs togeter wit its tangent line at tat point. 59 f. Grap te formula obtaine in part (c). Wat oes it mean wen its values are negative? Zero? Positive? Does tis make sense wit our plot from part (a)? Give reasons for our answer. 6. ƒs + -, 6. ƒs > + >,, ƒs sin, p> 6. ƒs -, ƒs cos, p> 65. ƒs e. Substitute various values for larger an smaller tan into te formula obtaine in part (c). Do te numbers make sense wit our picture? Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

15 AWL/Tomas_cp7-8/9/ :6 AM Page 59. Differentiation Rules 59 Differentiation Rules. Tis section introuces a few rules tat allow us to ifferentiate a great variet of functions. B proving tese rules ere, we can ifferentiate functions witout aving to appl te efinition of te erivative eac time. Powers, Multiples, Sums, an Differences Te first rule of ifferentiation is tat te erivative of ever constant function is zero. RULE Derivative of a Constant Function If ƒ as te constant value ƒs c, ten ƒ sc. EXAMPLE If ƒ as te constant value ƒs 8, ten c (, c) (, c) f s8. c Similarl, FIGURE.8 Te rule s>sc is anoter wa to sa tat te values of constant functions never cange an tat te slope of a orizontal line is zero at ever point. p a- b an ab. Proof of Rule We appl te efinition of erivative to ƒs c, te function wose outputs ave te constant value c (Figure.8). At ever value of, we fin tat ƒs + - ƒs c - c lim lim. : : : ƒ s lim Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

16 AWL/Tomas_cp7-8/9/ :6 AM Page 6 6 Capter : Differentiation Te secon rule tells ow to ifferentiate n if n is a positive integer. RULE Power Rule for Positive Integers If n is a positive integer, ten n n n -. To appl te Power Rule, we subtract from te original eponent (n) an multipl te result b n. EXAMPLE HISTORICAL BIOGRAPHY Ricar Courant (888 97) Interpreting Rule ƒ Á ƒ Á First Proof of Rule Te formula z n - n sz - sz n - + z n - + Á + z n - + n - can be verifie b multipling out te rigt-an sie. Ten from te alternative form for te efinition of te erivative, ƒsz - ƒs zn - n lim z - z z: z: ƒ s lim lim sz n - + z n - + Á + z n - + n - z: n n - Secon Proof of Rule If ƒs n, ten ƒs + s + n. Since n is a positive integer, we can epan s + n b te Binomial Teorem to get ƒs + - ƒs s + n - n lim : : ƒ s lim lim c n + n n - + : n n - + lim : lim cn n - + : nsn - n - Á + + n n - + n - n nsn - n - Á + + n n - + n nsn - n - + Á + n n - + n - n n - Te tir rule sas tat wen a ifferentiable function is multiplie b a constant, its erivative is multiplie b te same constant. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

17 AWL/Tomas_cp7-8/9/ :6 AM Page 6. Differentiation Rules 6 RULE Constant Multiple Rule If u is a ifferentiable function of, an c is a constant, ten u scu c. In particular, if n is a positive integer, ten sc n cn n -. Slope () 6 Slope 6() 6 (, ) EXAMPLE (a) Te erivative formula s # 6 Slope Slope () (, ) sas tat if we rescale te grap of b multipling eac -coorinate b, ten we multipl te slope at eac point b (Figure.9). (b) A useful special case Te erivative of te negative of a ifferentiable function u is te negative of te function s erivative. Rule wit c - gives u s - u s - # u - # su -. FIGURE.9 Te graps of an. Tripling te -coorinates triples te slope (Eample ). Proof of Rule cus + - cus cu lim : us + - us : c lim c u Derivative efinition wit ƒs cus Limit propert u is ifferentiable. Te net rule sas tat te erivative of te sum of two ifferentiable functions is te sum of teir erivatives. Denoting Functions b u an Y Te functions we are working wit wen we nee a ifferentiation formula are likel to be enote b letters like ƒ an g. Wen we appl te formula, we o not want to fin it using tese same letters in some oter wa. To guar against tis problem, we enote te functions in ifferentiation rules b letters like u an tat are not likel to be alrea in use. RULE Derivative Sum Rule If u an are ifferentiable functions of, ten teir sum u + is ifferentiable at ever point were u an are bot ifferentiable. At suc points, u +. su + Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

18 AWL/Tomas_cp7-8/9/ :6 AM Page 6 6 Capter : Differentiation EXAMPLE Derivative of a Sum + s + s + Proof of Rule We appl te efinition of erivative to ƒs us + s: [us + + s + ] - [us + s] [us + s] lim : lim c : s + - s us + - us + us + - us s + - s u + lim +. : : lim Combining te Sum Rule wit te Constant Multiple Rule gives te Difference Rule, wic sas tat te erivative of a ifference of ifferentiable functions is te ifference of teir erivatives. u u + s - su - [u + s - ] Te Sum Rule also etens to sums of more tan two functions, as long as tere are onl finitel man functions in te sum. If u, u, Á, un are ifferentiable at, ten so is u + u + Á + un, an un u u + + Á +. su + u + Á + un EXAMPLE 5 Derivative of a Polnomial a b s5 + s + # Notice tat we can ifferentiate an polnomial term b term, te wa we ifferentiate te polnomial in Eample 5. All polnomials are ifferentiable everwere. Proof of te Sum Rule for Sums of More Tan Two Functions We prove te statement un u u + + Á + su + u + Á + un b matematical inuction (see Appeni ). Te statement is true for n, as was just prove. Tis is Step of te inuction proof. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

19 AWL/Tomas_cp7-8// 9: AM Page 6. Differentiation Rules 6 Step is to sow tat if te statement is true for an positive integer n k, were k Ú n, ten it is also true for n k +. So suppose tat uk u u + + Á +. su + u + Á + uk () Ten (u + u + Á + uk + uk + ) (++++)++++* ()* Call te function efine b tis sum u. Call tis function. uk + su + u + Á + uk + Rule for uk uk + u u + + Á + +. Eq. () su + Wit tese steps verifie, te matematical inuction principle now guarantees te Sum Rule for ever integer n Ú. EXAMPLE 6 Fining Horizontal Tangents Does te curve - + ave an orizontal tangents? If so, were? Solution Te orizontal tangents, if an, occur were te slope > is zero. We ave, s (, ) (, ) Now solve te equation (, ) for : - s -,, -. FIGURE. Te curve - + an its orizontal tangents (Eample 6). Te curve - + as orizontal tangents at,, an -. Te corresponing points on te curve are (, ), (, ) an s -,. See Figure.. Proucts an Quotients Wile te erivative of te sum of two functions is te sum of teir erivatives, te erivative of te prouct of two functions is not te prouct of teir erivatives. For instance, s # s, wile s # s #. Te erivative of a prouct of two functions is te sum of two proucts, as we now eplain. RULE 5 Derivative Prouct Rule If u an are ifferentiable at, ten so is teir prouct u, an u +. su u Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

20 AWL/Tomas_cp7-8/9/ :6 AM Page 6 6 Capter : Differentiation Picturing te Prouct Rule If u() an () are positive an increase wen increases, an if 7, Te erivative of te prouct u is u times te erivative of plus times te erivative of u. In prime notation, su u + u. In function notation, [ƒsg s] ƒsg s + gsƒ s. ( ) u() u u()() () u () EXAMPLE 7 Using te Prouct Rule Fin te erivative of u() u u( ) Solution ten te total sae area in te picture is us + s + - uss us + + s + u - u. Diviing bot sies of tis equation b gives us + s + - uss u + s + us + - u. As : +, u # :#, leaving a + b. We appl te Prouct Rule wit u > an + s>: u +, an su u a b - b Eample, Section.7. c a + b a - b + a + b a- b Proof of Rule 5 us + s + - uss su lim : To cange tis fraction into an equivalent one tat contains ifference quotients for te erivatives of u an, we subtract an a us + s in te numerator: us + s + - us + s + us + s - uss su lim : lim cus + u su u +. : s + - s us + - us + s lim us + # lim : : s + - s us + - us + s # lim. : As approaces zero, us + approaces u() because u, being ifferentiable at, is continuous at. Te two fractions approac te values of > at an u> at. In sort, u +. su u In te following eample, we ave onl numerical values wit wic to work. EXAMPLE 8 Derivative from Numerical Values Let u be te prouct of te functions u an. Fin s if us, Solution u s -, s, an s. From te Prouct Rule, in te form su u + u, Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

21 AWL/Tomas_cp7-8/9/ :6 AM Page 65. Differentiation Rules 65 we ave s us s + su s ss + ss EXAMPLE 9 Differentiating a Prouct in Two Was Fin te erivative of s + s +. Solution (a) From te Prouct Rule wit u + an +, we fin C A + B A + B D s + s + s + s (b) Tis particular prouct can be ifferentiate as well (peraps better) b multipling out te original epression for an ifferentiating te resulting polnomial: s + s Tis is in agreement wit our first calculation. Just as te erivative of te prouct of two ifferentiable functions is not te prouct of teir erivatives, te erivative of te quotient of two functions is not te quotient of teir erivatives. Wat appens instea is te Quotient Rule. RULE 6 Derivative Quotient Rule If u an are ifferentiable at an if s Z, ten te quotient u> is ifferentiable at, an u a b u - u. In function notation, gsƒ s - ƒsg s ƒs c. gs g s EXAMPLE Using te Quotient Rule Fin te erivative of t -. t + Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

22 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation Solution We appl te Quotient Rule wit u t - an t + : st + # t - st - # t t st + t + t - t + t st + t. st + su>t - us>t u a b t Proof of Rule 6 us us + s + s u a b lim : sus + - uss + s + s : lim To cange te last fraction into an equivalent one tat contains te ifference quotients for te erivatives of u an, we subtract an a ()u() in te numerator. We ten get sus + - sus + sus - uss + u a b lim s + s : s lim : us + - us s + - s - us. s + s Taking te limit in te numerator an enominator now gives te Quotient Rule. Negative Integer Powers of Te Power Rule for negative integers is te same as te rule for positive integers. RULE 7 Power Rule for Negative Integers If n is a negative integer an Z, ten n s n n -. EXAMPLE (a) - a b s s (b) a b s - s Agrees wit Eample, Section.7 Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

23 AWL/Tomas_cp7-8/9/ :6 AM Page 67. Differentiation Rules 67 Proof of Rule 7 Te proof uses te Quotient Rule. If n is a negative integer, ten n - m, were m is a positive integer. Hence, n -m > m, an n s a b m m # - m m - m - m -m - n n -. Quotient Rule wit u an m Since m 7, m s m m - Since -m n (, ) EXAMPLE Tangent to a Curve Fin an equation for te tangent to te curve A B - # A m B s m + at te point (, ) (Figure.). FIGURE. Te tangent to te curve + s> at (, ) in Eample. Te curve as a tir-quarant portion not sown ere. We see ow to grap functions like tis one in Capter. Solution Te slope of te curve is s + a b + a- b -. Te slope at is ` c Te line troug (, ) wit slope m - is - s - s Point-slope equation Te coice of wic rules to use in solving a ifferentiation problem can make a ifference in ow muc work ou ave to o. Here is an eample. EXAMPLE Coosing Wic Rule to Use Rater tan using te Quotient Rule to fin te erivative of s - s -, epan te numerator an ivie b : s - s Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

24 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation Ten use te Sum an Power Rules: s s Secon- an Higer-Orer Derivatives If ƒs is a ifferentiable function, ten its erivative ƒ s is also a function. If ƒ is also ifferentiable, ten we can ifferentiate ƒ to get a new function of enote b ƒ. So ƒ sƒ. Te function ƒ is calle te secon erivative of ƒ because it is te erivative of te first erivative. Notationall, ƒ s D sƒs D ƒs. a b Te smbol D means te operation of ifferentiation is performe twice. If 6, ten 6 5 an we ave How to Rea te Smbols for Derivatives prime ouble prime square square triple prime sn super n n to te n of b to te n n n D D to te n A 6 5 B. Tus D A 6 B. If is ifferentiable, its erivative, > > is te tir erivative of wit respect to. Te names continue as ou imagine, wit sn n sn - D n n enoting te nt erivative of wit respect to for an positive integer n. We can interpret te secon erivative as te rate of cange of te slope of te tangent to te grap of ƒs at eac point. You will see in te net capter tat te secon erivative reveals weter te grap bens upwar or ownwar from te tangent line as we move off te point of tangenc. In te net section, we interpret bot te secon an tir erivatives in terms of motion along a straigt line. EXAMPLE Fining Higer Derivatives Te first four erivatives of - + are First erivative: Secon erivative: Tir erivative: Fourt erivative: s. Te function as erivatives of all orers, te fift an later erivatives all being zero. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

25 AWL/Tomas_cp7-8/9/ :6 AM Page 69. Differentiation Rules 69 EXERCISES. Derivative Calculations In Eercises, fin te first an secon erivatives s 5t - t. w z 7-7z + z w z - - z 5. r s s + + t r u u u In Eercises 6, fin (a) b appling te Prouct Rule an (b) b multipling te factors to prouce a sum of simpler terms to ifferentiate. 5. s + a b 6. a + b a - + b Fin te erivatives of te functions in Eercises g s 8. z s - s + t - t + t -. w s - 7-s + 5. u r a + ub u s + s s - s - s - s + + Fin te erivatives of all orers of te functions in Eercises 9 an r + 7. s su - su + u + 5. w a us 5, u + z bs - z z. u s -, u s -, s. Fin te values of te following erivatives at. a. su b. u a b c. a b u. s7 - u. Suppose u an are ifferentiable functions of an tat us, s 5, u s, s -. Fin te values of te following erivatives at. a. su b. u a b c. a b u. s7 - u Slopes an Tangents b. Smallest slope Wat is te smallest slope on te curve? At wat point on te curve oes te curve ave tis slope? c. Tangents aving specifie slope Fin equations for te tangents to te curve at te points were te slope of te curve is 8.. a. Horizontal tangents Fin equations for te orizontal tangents to te curve - -. Also fin equations for te lines tat are perpenicular to tese tangents at te points of tangenc. b. Smallest slope Wat is te smallest slope on te curve? At wat point on te curve oes te curve ave tis slope? Fin an equation for te line tat is perpenicular to te curve s tangent at tis point.. Fin te tangents to Newton s serpentine (grape ere) at te origin an te point (, ). Fin te first an secon erivatives of te functions in Eercises 8.. sq - + sq +. a. Normal to a curve Fin an equation for te line perpenicular to te tangent to te curve - + at te point (, ) ƒst. s - ts + t -. ƒss q + 9. Suppose u an are functions of tat are ifferentiable at an tat. s - s - +. s - s p Using Numerical Values 8. s - t q + q - b ba q q p a t + 5t - t s + s - + (, ) 6. w sz + sz - sz + Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

26 AWL/Tomas_cp7-8/9/ :6 AM Page 7 7 Capter : Differentiation. Fin te tangent to te Witc of Agnesi (grape ere) at te point (, ). 5. Suppose tat te function in te Prouct Rule as a constant value c. Wat oes te Prouct Rule ten sa? Wat oes tis sa about te Constant Multiple Rule? 8 5. Te Reciprocal Rule (, ) ow to fin te amount of meicine to wic te bo is most sensitive. 5. Quaratic tangent to ientit function Te curve a + b + c passes troug te point (, ) an is tangent to te line at te origin. Fin a, b, an c. 6. Quaratics aving a common tangent Te curves + a + b an c - ave a common tangent line at te point (, ). Fin a, b, an c. 7. a. Fin an equation for te line tat is tangent to te curve - at te point s -,. a. Te Reciprocal Rule sas tat at an point were te function () is ifferentiable an ifferent from zero, a b -. Sow tat te Reciprocal Rule is a special case of te Quotient Rule. b. Sow tat te Reciprocal Rule an te Prouct Rule togeter impl te Quotient Rule. 5. Generalizing te Prouct Rule Te Prouct Rule gives te formula T b. Grap te curve an tangent line togeter. Te tangent intersects te curve at anoter point. Use Zoom an Trace to estimate te point s coorinates. T c. Confirm our estimates of te coorinates of te secon intersection point b solving te equations for te curve an tangent simultaneousl (Solver ke). 8. a. Fin an equation for te line tat is tangent to te curve at te origin. T b. Grap te curve an tangent togeter. Te tangent intersects te curve at anoter point. Use Zoom an Trace to estimate te point s coorinates. T c. Confirm our estimates of te coorinates of te secon intersection point b solving te equations for te curve an tangent simultaneousl (Solver ke). u su u + for te erivative of te prouct u of two ifferentiable functions of. a. Wat is te analogous formula for te erivative of te prouct uw of tree ifferentiable functions of? b. Wat is te formula for te erivative of te prouct u u u u of four ifferentiable functions of? c. Wat is te formula for te erivative of a prouct u u u Á un of a finite number n of ifferentiable functions of? 5. Rational Powers > A B b writing > as # > an using te Prouct Rule. Epress our answer as a rational number times a rational power of. Work parts (b) an (c) b a similar meto. a. Fin Teor an Eamples 9. Te general polnomial of egree n as te form Ps an n + an - n - + Á + a + a + a were an Z. Fin P s. 5. Te bo s reaction to meicine Te reaction of te bo to a ose of meicine can sometimes be represente b an equation of te form R M a C M - b, were C is a positive constant an M is te amount of meicine absorbe in te bloo. If te reaction is a cange in bloo pressure, R is measure in millimeters of mercur. If te reaction is a cange in temperature, R is measure in egrees, an so on. Fin R>M. Tis erivative, as a function of M, is calle te sensitivit of te bo to te meicine. In Section.5, we will see b. Fin 5> s. c. Fin 7> s.. Wat patterns o ou see in our answers to parts (a), (b), an (c)? Rational powers are one of te topics in Section Cliner pressure If gas in a cliner is maintaine at a constant temperature T, te pressure P is relate to te volume V b a formula of te form P an nrt -, V - nb V in wic a, b, n, an R are constants. Fin P>V. (See accompaning figure.) Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

27 AWL/Tomas_cp7-8/9/ :6 AM Page 7. Differentiation Rules Te best quantit to orer One of te formulas for inventor management sas tat te average weekl cost of orering, paing for, an oling mercanise is q km Asq q + cm +, were q is te quantit ou orer wen tings run low (soes, raios, brooms, or watever te item migt be); k is te cost of placing an orer (te same, no matter ow often ou orer); c is te cost of one item (a constant); m is te number of items sol eac week (a constant); an is te weekl oling cost per item (a constant tat takes into account tings suc as space, utilities, insurance, an securit). Fin A>q an A>q. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

28 AWL/Tomas_cp7-8/9/ :6 AM Page 7. Te Derivative as a Rate of Cange. 7 Te Derivative as a Rate of Cange In Section., we initiate te stu of average an instantaneous rates of cange. In tis section, we continue our investigations of applications in wic erivatives are use to moel te rates at wic tings cange in te worl aroun us. We revisit te stu of motion along a line an eamine oter applications. It is natural to tink of cange as cange wit respect to time, but oter variables can be treate in te same wa. For eample, a psician ma want to know ow cange in osage affects te bo s response to a rug. An economist ma want to stu ow te cost of proucing steel varies wit te number of tons prouce. Instantaneous Rates of Cange If we interpret te ifference quotient sƒs + - ƒs> as te average rate of cange in ƒ over te interval from to +, we can interpret its limit as : as te rate at wic ƒ is canging at te point. DEFINITION Instantaneous Rate of Cange Te instantaneous rate of cange of ƒ wit respect to at is te erivative ƒ s lim : ƒs + - ƒs, provie te limit eists. Tus, instantaneous rates are limits of average rates. It is conventional to use te wor instantaneous even wen oes not represent time. Te wor is, owever, frequentl omitte. Wen we sa rate of cange, we mean instantaneous rate of cange. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

29 AWL/Tomas_cp7-8/9/ :6 AM Page 7 7 Capter : Differentiation EXAMPLE How a Circle s Area Canges wit Its Diameter Te area A of a circle is relate to its iameter b te equation A p D. How fast oes te area cange wit respect to te iameter wen te iameter is m? Solution Te rate of cange of te area wit respect to te iameter is A p pd # D. D Wen D m, te area is canging at rate sp> 5p m>m. Position at time t s f(t) s Motion Along a Line: Displacement, Velocit, Spee, Acceleration, an Jerk an at time t t s s f (t t) FIGURE. Te positions of a bo moving along a coorinate line at time t an sortl later at time t + t. s Suppose tat an object is moving along a coorinate line (sa an s-ais) so tat we know its position s on tat line as a function of time t: s ƒst. Te isplacement of te object over te time interval from t to t + t (Figure.) is s ƒst + t - ƒst, an te average velocit of te object over tat time interval is a ƒst + t - ƒst isplacement s. travel time t t To fin te bo s velocit at te eact instant t, we take te limit of te average velocit over te interval from t to t + t as t srinks to zero. Tis limit is te erivative of ƒ wit respect to t. DEFINITION Velocit Velocit (instantaneous velocit) is te erivative of position wit respect to time. If a bo s position at time t is s ƒst, ten te bo s velocit at time t is st EXAMPLE ƒst + t - ƒst s. lim t t t : Fining te Velocit of a Race Car Figure. sows te time-to-istance grap of a 996 Rile & Scott Mk III-Ols WSC race car. Te slope of te secant PQ is te average velocit for te -sec interval from t to t 5 sec; in tis case, it is about ft> sec or 68 mp. Te slope of te tangent at P is te speeometer reaing at t sec, about 57 ft> sec or 9 mp. Te acceleration for te perio sown is a nearl constant 8.5 ft>sec uring Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

30 AWL/Tomas_cp7-8/9/ :6 AM Page 7. Te Derivative as a Rate of Cange 7 s 8 7 Distance (ft) 6 5 Secant slope is average velocit for interval from t to t 5. Q Tangent slope is speeometer reaing at t (instantaneous velocit). P t Elapse time (sec) FIGURE. Te time-to-istance grap for Eample. Te slope of te tangent line at P is te instantaneous velocit at t sec. eac secon, wic is about.89g, were g is te acceleration ue to gravit. Te race car s top spee is an estimate 9 mp. (Source: Roa an Track, Marc 997.) Besies telling ow fast an object is moving, its velocit tells te irection of motion. Wen te object is moving forwar (s increasing), te velocit is positive; wen te bo is moving backwar (s ecreasing), te velocit is negative (Figure.). s s s f (t) s f (t) s t s t t s increasing: positive slope so moving forwar t s ecreasing: negative slope so moving backwar FIGURE. For motion s ƒst along a straigt line, s/t is positive wen s increases an negative wen s ecreases. If we rive to a frien s ouse an back at mp, sa, te speeometer will sow on te wa over but it will not sow - on te wa back, even toug our istance from ome is ecreasing. Te speeometer alwas sows spee, wic is te absolute value of velocit. Spee measures te rate of progress regarless of irection. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

31 AWL/Tomas_cp7-8/9/ :6 AM Page 7 7 Capter : Differentiation DEFINITION Spee Spee is te absolute value of velocit. s Spee ƒ st ƒ ` ` t EXAMPLE Horizontal Motion Figure.5 sows te velocit ƒ st of a particle moving on a coorinate line. Te particle moves forwar for te first sec, moves backwar for te net sec, stans still for a secon, an moves forwar again. Te particle acieves its greatest spee at time t, wile moving backwar. MOVES FORWARD FORWARD AGAIN ( ) ( ) f '(t) Spees up Stea ( const) Slows own Spees up Stans still ( ) t (sec) Greatest spee Spees up Slows own MOVES BACKWARD ( ) FIGURE.5 Te velocit grap for Eample. HISTORICAL BIOGRAPHY Bernar Bolzano (78 88) Te rate at wic a bo s velocit canges is te bo s acceleration. Te acceleration measures ow quickl te bo picks up or loses spee. A suen cange in acceleration is calle a jerk. Wen a rie in a car or a bus is jerk, it is not tat te accelerations involve are necessaril large but tat te canges in acceleration are abrupt. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

32 AWL/Tomas_cp7-8/9/ :6 AM Page 75. Te Derivative as a Rate of Cange 75 DEFINITIONS Acceleration, Jerk Acceleration is te erivative of velocit wit respect to time. If a bo s position at time t is s ƒst, ten te bo s acceleration at time t is s. t t Jerk is te erivative of acceleration wit respect to time: ast jst s a. t t Near te surface of te Eart all boies fall wit te same constant acceleration. Galileo s eperiments wit free fall (Eample, Section.) lea to te equation s gt, were s is istance an g is te acceleration ue to Eart s gravit. Tis equation ols in a vacuum, were tere is no air resistance, an closel moels te fall of ense, eav objects, suc as rocks or steel tools, for te first few secons of teir fall, before air resistance starts to slow tem own. Te value of g in te equation s s>gt epens on te units use to measure t an s. Wit t in secons (te usual unit), te value of g etermine b measurement at sea level is approimatel ft>sec (feet per secon square) in Englis units, an g 9.8 m>sec (meters per secon square) in metric units. (Tese gravitational constants epen on te istance from Eart s center of mass, an are sligtl lower on top of Mt. Everest, for eample.) Te jerk of te constant acceleration of gravit sg ft>sec is zero: j t (secons) t s (meters) t 5 sg. t An object oes not eibit jerkiness uring free fall. EXAMPLE Moeling Free Fall Figure.6 sows te free fall of a eav ball bearing release from rest at time t sec. 5 t (a) How man meters oes te ball fall in te first sec? (b) Wat is its velocit, spee, an acceleration ten? 5 Solution 5 (a) Te metric free-fall equation is s.9t. During te first sec, te ball falls t ss.9s 9.6 m. 5 (b) At an time t, velocit is te erivative of position: FIGURE.6 A ball bearing falling from rest (Eample ). st s st s.9t 9.8t. t Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

33 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation At t, te velocit is s s 9.6 m>sec sma in te ownwar (increasing s) irection. Te spee at t is Heigt (ft) Spee ƒ s ƒ 9.6 m>sec. t? 56 Te acceleration at an time t is ast st s st 9.8 m>sec. At t, te acceleration is 9.8 m>sec. EXAMPLE 5 A namite blast blows a eav rock straigt up wit a launc velocit of 6 ft> sec (about 9 mp) (Figure.7a). It reaces a eigt of s 6t - 6t ft after t sec. s (a) How ig oes te rock go? (b) Wat are te velocit an spee of te rock wen it is 56 ft above te groun on te wa up? On te wa own? (c) Wat is te acceleration of te rock at an time t uring its fligt (after te blast)? () Wen oes te rock it te groun again? (a) s, s 6t 6t 6 6 Moeling Vertical Motion Solution 5 t s 6 t t (b) FIGURE.7 (a) Te rock in Eample 5. (b) Te graps of s an as functions of time; s is largest wen s/t. Te grap of s is not te pat of te rock: It is a plot of eigt versus time. Te slope of te plot is te rock s velocit, grape ere as a straigt line. (a) In te coorinate sstem we ave cosen, s measures eigt from te groun up, so te velocit is positive on te wa up an negative on te wa own. Te instant te rock is at its igest point is te one instant uring te fligt wen te velocit is. To fin te maimum eigt, all we nee to o is to fin wen an evaluate s at tis time. At an time t, te velocit is s s6t - 6t 6 - t ft>sec. t t Te velocit is zero wen 6 - t or t 5 sec. Te rock s eigt at t 5 sec is sma ss5 6s5-6s5 8 - ft. See Figure.7b. (b) To fin te rock s velocit at 56 ft on te wa up an again on te wa own, we first fin te two values of t for wic sst 6t - 6t 56. To solve tis equation, we write 6t - 6t st - t + 6 st - st - 8 t sec, t 8 sec. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

34 AWL/Tomas_cp7-8/9/ :6 AM Page 77. Te Derivative as a Rate of Cange Te rock is 56 ft above te groun sec after te eplosion an again 8 sec after te eplosion. Te rock s velocities at tese times are (ollars) Slope marginal cost c() (tons/week) 77 s 6 - s ft>sec. s8 6 - s ft>sec. At bot instants, te rock s spee is 96 ft> sec. Since s 7, te rock is moving upwar (s is increasing) at t sec; it is moving ownwar (s is ecreasing) at t 8 because s8 6. FIGURE.8 Weekl steel prouction: c() is te cost of proucing tons per week. Te cost of proucing an aitional tons is cs + - cs. (c) At an time uring its fligt following te eplosion, te rock s acceleration is a constant a s6 - t - ft>sec. t t Te acceleration is alwas ownwar. As te rock rises, it slows own; as it falls, it spees up. () Te rock its te groun at te positive time t for wic s. Te equation 6t - 6t factors to give 6t s - t, so it as solutions t an t. At t, te blast occurre an te rock was trown upwar. It returne to te groun sec later. Derivatives in Economics Engineers use te terms velocit an acceleration to refer to te erivatives of functions escribing motion. Economists, too, ave a specialize vocabular for rates of cange an erivatives. Te call tem marginals. In a manufacturing operation, te cost of prouction c() is a function of, te number of units prouce. Te marginal cost of prouction is te rate of cange of cost wit respect to level of prouction, so it is c>. Suppose tat c() represents te ollars neee to prouce tons of steel in one week. It costs more to prouce + units per week, an te cost ifference, ivie b, is te average cost of proucing eac aitional ton: c() cs + - cs average cost of eac of te aitional tons of steel prouce. c c Te limit of tis ratio as : is te marginal cost of proucing more steel per week wen te current weekl prouction is tons (Figure.8). cs + - cs c lim marginal cost of prouction. : Sometimes te marginal cost of prouction is loosel efine to be te etra cost of proucing one unit: FIGURE.9 Te marginal cost c> is approimatel te etra cost c of proucing more unit. cs + - cs c, wic is approimate b te value of c> at. Tis approimation is acceptable if te slope of te grap of c oes not cange quickl near. Ten te ifference quotient will be close to its limit c>, wic is te rise in te tangent line if (Figure.9). Te approimation works best for large values of. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

35 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation Economists often represent a total cost function b a cubic polnomial cs a + b + g + were represents fie costs suc as rent, eat, equipment capitalization, an management costs. Te oter terms represent variable costs suc as te costs of raw materials, taes, an labor. Fie costs are inepenent of te number of units prouce, wereas variable costs epen on te quantit prouce. A cubic polnomial is usuall complicate enoug to capture te cost beavior on a relevant quantit interval. EXAMPLE 6 Marginal Cost an Marginal Revenue Suppose tat it costs cs ollars to prouce raiators wen 8 to raiators are prouce an tat rs - + gives te ollar revenue from selling raiators. Your sop currentl prouces raiators a a. About ow muc etra will it cost to prouce one more raiator a a, an wat is our estimate increase in revenue for selling raiators a a? Solution Te cost of proucing one more raiator a a wen are prouce is about c s : c s A B c s s - s Te aitional cost will be about $95. Te marginal revenue is r s A - + B Te marginal revenue function estimates te increase in revenue tat will result from selling one aitional unit. If ou currentl sell raiators a a, ou can epect our revenue to increase b about r s s - 6s + $5 if ou increase sales to raiators a a. EXAMPLE 7 Marginal Ta Rate To get some feel for te language of marginal rates, consier marginal ta rates. If our marginal income ta rate is 8% an our income increases b $, ou can epect to pa an etra $8 in taes. Tis oes not mean tat ou pa 8% of our entire income in taes. It just means tat at our current income level I, te rate of increase of taes T wit respect to income is T>I.8. You will pa $.8 out of ever etra ollar ou earn in taes. Of course, if ou earn a lot more, ou ma lan in a iger ta bracket an our marginal rate will increase. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

36 AWL/Tomas_cp7-8/9/ :6 AM Page 79. Te Derivative as a Rate of Cange 79 Sensitivit to Cange Wen a small cange in prouces a large cange in te value of a function ƒ(), we sa tat te function is relativel sensitive to canges in. Te erivative ƒ s is a measure of tis sensitivit. EXAMPLE 8 Genetic Data an Sensitivit to Cange Te Austrian monk Gregor Joann Menel (8 88), working wit garen peas an oter plants, provie te first scientific eplanation of briization. His careful recors sowe tat if p (a number between an ) is te frequenc of te gene for smoot skin in peas (ominant) an s - p is te frequenc of te gene for wrinkle skin in peas, ten te proportion of smoot-skinne peas in te net generation will be ps - p + p p - p. Te grap of versus p in Figure.a suggests tat te value of is more sensitive to a cange in p wen p is small tan wen p is large. Inee, tis fact is borne out b te erivative grap in Figure.b, wic sows tat >p is close to wen p is near an close to wen p is near. /p p p p p (a) p p (b) FIGURE. (a) Te grap of p - p, escribing te proportion of smoot-skinne peas. (b) Te grap of >p (Eample 8). Te implication for genetics is tat introucing a few more ominant genes into a igl recessive population (were te frequenc of wrinkle skin peas is small) will ave a more ramatic effect on later generations tan will a similar increase in a igl ominant population. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

37 AWL/Tomas_cp7-8/9/ :6 AM Page 79. Te Derivative as a Rate of Cange 79 EXERCISES. Motion Along a Coorinate Line Eercises 6 give te positions s ƒst of a bo moving on a coorinate line, wit s in meters an t in secons. a. Fin te bo s isplacement an average velocit for te given time interval. b. Fin te bo s spee an acceleration at te enpoints of te interval. c. Wen, if ever, uring te interval oes te bo cange irection?. s t - t +,. s 6t - t, t t 6 Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

38 AWL/Tomas_cp7-8/9/ :6 AM Page 8 8 Capter : Differentiation. s - t + t - t,. s st > - t + t, 5. s t, t 6. s 5, t + 5 t t t 5 - t 7. Particle motion At time t, te position of a bo moving along te s-ais is s t - 6t + 9t m. a. Fin te bo s acceleration eac time te velocit is zero. b. Fin te bo s spee eac time te acceleration is zero. c. Fin te total istance travele b te bo from t to t. 8. Particle motion At time t Ú, te velocit of a bo moving along te s-ais is t - t +. a. Fin te bo s acceleration eac time te velocit is zero. b. Wen is te bo moving forwar? Backwar? c. Wen is te bo s velocit increasing? Decreasing? Free-Fall Applications eigt above groun t sec into te fall woul ave been s 79-6t. a. Wat woul ave been te ball s velocit, spee, an acceleration at time t? b. About ow long woul it ave taken te ball to it te groun? c. Wat woul ave been te ball s velocit at te moment of impact?. Galileo s free-fall formula Galileo evelope a formula for a bo s velocit uring free fall b rolling balls from rest own increasingl steep incline planks an looking for a limiting formula tat woul preict a ball s beavior wen te plank was vertical an te ball fell freel; see part (a) of te accompaning figure. He foun tat, for an given angle of te plank, te ball s velocit t sec into motion was a constant multiple of t. Tat is, te velocit was given b a formula of te form kt. Te value of te constant k epene on te inclination of te plank. In moern notation part (b) of te figure wit istance in meters an time in secons, wat Galileo etermine b eperiment was tat, for an given angle u, te ball s velocit t sec into te roll was 9.8ssin ut m>sec. 9. Free fall on Mars an Jupiter Te equations for free fall at te surfaces of Mars an Jupiter (s in meters, t in secons) are s.86t on Mars an s.t on Jupiter. How long oes it take a rock falling from rest to reac a velocit of 7.8 m> sec (about km> ) on eac planet? Free-fall position. Lunar projectile motion A rock trown verticall upwar from te surface of te moon at a velocit of m> sec (about 86 km> ) reaces a eigt of s t -.8t meters in t sec. a. Fin te rock s velocit an acceleration at time t. (Te acceleration in tis case is te acceleration of gravit on te moon.)? θ (b) (a) b. How long oes it take te rock to reac its igest point? a. Wat is te equation for te ball s velocit uring free fall? c. How ig oes te rock go? b. Builing on our work in part (a), wat constant acceleration oes a freel falling bo eperience near te surface of Eart?. How long oes it take te rock to reac alf its maimum eigt? e. How long is te rock aloft?. Fining g on a small airless planet Eplorers on a small airless planet use a spring gun to launc a ball bearing verticall upwar from te surface at a launc velocit of 5 m> sec. Because te acceleration of gravit at te planet s surface was gs m>sec, te eplorers epecte te ball bearing to reac a eigt of s 5t - s>gs t meters t sec later. Te ball bearing reace its maimum eigt sec after being launce. Wat was te value of gs?. Speeing bullet A 5-caliber bullet fire straigt up from te surface of te moon woul reac a eigt of s 8t -.6t feet after t sec. On Eart, in te absence of air, its eigt woul be s 8t - 6t ft after t sec. How long will te bullet be aloft in eac case? How ig will te bullet go?. Free fall from te Tower of Pisa Ha Galileo roppe a cannonball from te Tower of Pisa, 79 ft above te groun, te ball s Conclusions About Motion from Graps 5. Te accompaning figure sows te velocit s>t ƒst (m> sec) of a bo moving along a coorinate line. (m/sec) f (t) 6 8 t (sec) a. Wen oes te bo reverse irection? b. Wen (approimatel) is te bo moving at a constant spee? c. Grap te bo s spee for t.. Grap te acceleration, were efine. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

39 AWL/Tomas_cp7-8/9/ :6 AM Page 8. Te Derivative as a Rate of Cange 6. A particle P moves on te number line sown in part (a) of te accompaning figure. Part (b) sows te position of P as a function of time t. P s (cm) f. Wen was te rocket s acceleration greatest? g. Wen was te acceleration constant? Wat was its value ten (to te nearest integer)? 8. Te accompaning figure sows te velocit ƒst of a particle moving on a coorinate line. (a) s (cm) f(t) s f (t) t (sec) t (sec) (6, ) a. Wen oes te particle move forwar? Move backwar? Spee up? Slow own? (b) b. Wen is te particle s acceleration positive? Negative? Zero? c. Wen oes te particle move at its greatest spee? a. Wen is P moving to te left? Moving to te rigt? Staning still? b. Grap te particle s velocit an spee (were efine). 7. Launcing a rocket Wen a moel rocket is launce, te propellant burns for a few secons, accelerating te rocket upwar. After burnout, te rocket coasts upwar for a wile an ten begins to fall. A small eplosive carge pops out a paracute sortl after te rocket starts own. Te paracute slows te rocket to keep it from breaking wen it lans. Te figure ere sows velocit ata from te fligt of te moel rocket. Use te ata to answer te following.. Wen oes te particle stan still for more tan an instant? 9. Two falling balls Te multiflas potograp in te accompaning figure sows two balls falling from rest. Te vertical rulers are marke in centimeters. Use te equation s 9t (te freefall equation for s in centimeters an t in secons) to answer te following questions. a. How fast was te rocket climbing wen te engine stoppe? b. For ow man secons i te engine burn? Velocit (ft/sec) Time after launc (sec) c. Wen i te rocket reac its igest point? Wat was its velocit ten?. Wen i te paracute pop out? How fast was te rocket falling ten? e. How long i te rocket fall before te paracute opene? Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

40 AWL/Tomas_cp7-8/9/ :6 AM Page 8 8 Capter : Differentiation a. How long i it take te balls to fall te first 6 cm? Wat was teir average velocit for te perio? A b. How fast were te balls falling wen te reace te 6-cm mark? Wat was teir acceleration ten? c. About ow fast was te ligt flasing (flases per secon)?. A traveling truck Te accompaning grap sows te position s of a truck traveling on a igwa. Te truck starts at t an returns 5 later at t 5. t B a. Use te tecnique escribe in Section., Eample, to grap te truck s velocit s>t for t 5. Ten repeat te process, wit te velocit curve, to grap te truck s acceleration >t. C b. Suppose tat s 5t - t. Grap s>t an s>t an compare our graps wit tose in part (a). FIGURE. Te graps for Eercise. Position, s (km) 5 Economics. Marginal cost Suppose tat te ollar cost of proucing wasing macines is cs a. Fin te average cost per macine of proucing te first wasing macines. b. Fin te marginal cost wen wasing macines are prouce. 5 Elapse time, t (r) 5. Te graps in Figure. sow te position s, velocit s>t, an acceleration a s>t of a bo moving along a coorinate line as functions of time t. Wic grap is wic? Give reasons for our answers. c. Sow tat te marginal cost wen wasing macines are prouce is approimatel te cost of proucing one more wasing macine after te first ave been mae, b calculating te latter cost irectl.. Marginal revenue Suppose tat te revenue from selling wasing macines is rs, a - b ollars. A a. Fin te marginal revenue wen macines are prouce. B C t b. Use te function r s to estimate te increase in revenue tat will result from increasing prouction from macines a week to macines a week. c. Fin te limit of r s as : q. How woul ou interpret tis number? Aitional Applications FIGURE. Te graps for Eercise.. Te graps in Figure. sow te position s, te velocit s>t, an te acceleration a s>t of a bo moving along te coorinate line as functions of time t. Wic grap is wic? Give reasons for our answers. 5. Bacterium population Wen a bactericie was ae to a nutrient brot in wic bacteria were growing, te bacterium population continue to grow for a wile, but ten stoppe growing an began to ecline. Te size of te population at time t (ours) was b 6 + t - t. Fin te growt rates at a. t ours. b. t 5 ours. c. t ours. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

41 AWL/Tomas_cp7-8/9/ :6 AM Page 8 8. Te Derivative as a Rate of Cange 6. Draining a tank Te number of gallons of water in a tank t minutes after te tank as starte to rain is Qst s - t. How fast is te water running out at te en of min? Wat is te average rate at wic te water flows out uring te first min? T 7. Draining a tank It takes ours to rain a storage tank b opening te valve at te bottom. Te ept of flui in te tank t ours after te valve is opene is given b te formula t 6 a b m. a. Fin te rate >t (m> ) at wic te tank is raining at time t. b. Wen is te flui level in te tank falling fastest? Slowest? Wat are te values of >t at tese times? c. Grap an >t togeter an iscuss te beavior of in relation to te signs an values of >t. 8. Inflating a balloon Te volume V s>pr of a sperical balloon canges wit te raius. T Eercises give te position function s ƒst of a bo moving along te s-ais as a function of time t. Grap ƒ togeter wit te velocit function st s>t ƒ st an te acceleration function ast s>t ƒ st. Comment on te bo s beavior in relation to te signs an values of an a. Inclue in our commentar suc topics as te following: a. Wen is te bo momentaril at rest? b. Wen oes it move to te left (own) or to te rigt (up)? c. Wen oes it cange irection?. Wen oes it spee up an slow own? e. Wen is it moving fastest (igest spee)? Slowest? f. Wen is it fartest from te ais origin?. s t - 6t, t.5 (a eav object fire straigt up from Eart s surface at ft> sec). s t - t +, t 5 a. At wat rate sft>ft oes te volume cange wit respect to te raius wen r ft?. s t - 6t + 7t, b. B approimatel ow muc oes te volume increase wen te raius canges from to. ft? 5. Torougbre racing A raceorse is running a -furlong race. (A furlong is ars, altoug we will use furlongs an secons as our units in tis eercise.) As te orse passes eac furlong marker (F ), a stewar recors te time elapse (t) since te beginning of te race, as sown in te table: 9. Airplane takeoff Suppose tat te istance an aircraft travels along a runwa before takeoff is given b D s>9t, were D is measure in meters from te starting point an t is measure in secons from te time te brakes are release. Te aircraft will become airborne wen its spee reaces km>. How long will it take to become airborne, an wat istance will it travel in tat time?. Volcanic lava fountains Altoug te November 959 Kilauea Iki eruption on te islan of Hawaii began wit a line of fountains along te wall of te crater, activit was later confine to a single vent in te crater s floor, wic at one point sot lava 9 ft straigt into te air (a worl recor). Wat was te lava s eit velocit in feet per secon? In miles per our? (Hint: If is te eit velocit of a particle of lava, its eigt t sec later will be s t - 6t ft. Begin b fining te time at wic s>t. Neglect air resistance.) t. s - 7t + 6t - t, t F t a. How long oes it take te orse to finis te race? b. Wat is te average spee of te orse over te first 5 furlongs? c. Wat is te approimate spee of te orse as it passes te -furlong marker?. During wic portion of te race is te orse running te fastest? e. During wic portion of te race is te orse accelerating te fastest? Coprigt Ä m SÚI;[) ] Õ Ï»"C½Í 7iš îã}[ z+á >Ê zê LÕn

42 AWL/Tomas_cp7-8/9/ :6 AM Page 8. Derivatives of Trigonometric Functions. 8 Derivatives of Trigonometric Functions Man of te penomena we want information about are approimatel perioic (electromagnetic fiels, eart rtms, ties, weater). Te erivatives of sines an cosines pla a ke role in escribing perioic canges. Tis section sows ow to ifferentiate te si basic trigonometric functions. Derivative of te Sine Function To calculate te erivative of ƒs sin, for measure in raians, we combine te limits in Eample 5a an Teorem 7 in Section. wit te angle sum ientit for te sine: sin s + sin cos + cos sin. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

43 AWL/Tomas_cp7-8/9/ :6 AM Page 8 8 Capter : Differentiation If ƒs sin, ten ƒs + - ƒs : sin s + - sin lim : ssin cos + cos sin - sin lim : sin scos - + cos sin lim : ƒ s lim lim asin # : Derivative efinition Sine angle sum ientit cos - sin b + lim acos # b : cos - sin + cos # lim : sin # + cos # cos. sin # lim : Eample 5(a) an Teorem 7, Section. Te erivative of te sine function is te cosine function: ssin cos. EXAMPLE Derivatives Involving te Sine (a) - sin : A sin B Difference Rule - cos. (b) sin : A sin B + sin Prouct Rule cos + sin. (c) sin : # A sin B - sin # cos - sin. Quotient Rule Derivative of te Cosine Function Wit te elp of te angle sum formula for te cosine, cos s + cos cos - sin sin, Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

44 AWL/Tomas_cp7-8/9/ :6 AM Page 85. Derivatives of Trigonometric Functions 85 we ave cos s + - cos scos lim : cos scos cos - sin sin - cos : lim ' lim ' sin : Cosine angle sum ientit cos scos - - sin sin lim cos # Derivative efinition : cos - sin - lim sin # : cos - sin - sin # lim : : cos # lim FIGURE. Te curve - sin as te grap of te slopes of te tangents to te curve cos. cos # - sin # - sin. Eample 5(a) an Teorem 7, Section. Te erivative of te cosine function is te negative of te sine function: scos - sin Figure. sows a wa to visualize tis result. EXAMPLE Derivatives Involving te Cosine (a) 5 + cos : s5 + A cos B Sum Rule 5 - sin. (b) sin cos : sin A cos B + cos A sin B Prouct Rule sin s - sin + cos scos cos - sin. (c) cos : - sin A - sin B A cos B - cos A - sin B s - sin s - sin s -sin - cos s - cos s - sin - sin s - sin. - sin Quotient Rule Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle sin + cos

45 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation Simple Harmonic Motion Te motion of a bo bobbing freel up an own on te en of a spring or bungee cor is an eample of simple armonic motion. Te net eample escribes a case in wic tere are no opposing forces suc as friction or buoanc to slow te motion own. 5 Rest position 5 Position at t EXAMPLE A bo anging from a spring (Figure.) is stretce 5 units beon its rest position an release at time t to bob up an own. Its position at an later time t is s 5 cos t. Wat are its velocit an acceleration at time t? s FIGURE. A bo anging from a vertical spring an ten isplace oscillates above an below its rest position. Its motion is escribe b trigonometric functions (Eample ). We ave Position: Solution s 5 cos t s Velocit: s5 cos t - 5 sin t t t Acceleration: a s - 5 sin t - 5 cos t. t t Notice ow muc we can learn from tese equations: s,. 5 sin t s 5 cos t. Motion on a Spring 5 t. FIGURE.5 Te graps of te position an velocit of te bo in Eample.. As time passes, te weigt moves own an up between s - 5 an s 5 on te s-ais. Te amplitue of te motion is 5. Te perio of te motion is p. Te velocit - 5 sin t attains its greatest magnitue, 5, wen cos t, as te graps sow in Figure.5. Hence, te spee of te weigt, ƒ ƒ 5 ƒ sin t ƒ, is greatest wen cos t, tat is, wen s (te rest position). Te spee of te weigt is zero wen sin t. Tis occurs wen s 5 cos t ; 5, at te enpoints of te interval of motion. Te acceleration value is alwas te eact opposite of te position value. Wen te weigt is above te rest position, gravit is pulling it back own; wen te weigt is below te rest position, te spring is pulling it back up. Te acceleration, a - 5 cos t, is zero onl at te rest position, were cos t an te force of gravit an te force from te spring offset eac oter. Wen te weigt is anwere else, te two forces are unequal an acceleration is nonzero. Te acceleration is greatest in magnitue at te points fartest from te rest position, were cos t ;. EXAMPLE Jerk Te jerk of te simple armonic motion in Eample is j a s - 5 cos t 5 sin t. t t It as its greatest magnitue wen sin t ;, not at te etremes of te isplacement but at te rest position, were te acceleration canges irection an sign. Derivatives of te Oter Basic Trigonometric Functions Because sin an cos are ifferentiable functions of, te relate functions sin tan cos, cot cos, sin sec cos, an Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle csc sin

46 AWL/Tomas_cp7-8/9/ :6 AM Page 87. Derivatives of Trigonometric Functions 87 are ifferentiable at ever value of at wic te are efine. Teir erivatives, calculate from te Quotient Rule, are given b te following formulas. Notice te negative signs in te erivative formulas for te cofunctions. Derivatives of te Oter Trigonometric Functions stan sec ssec sec tan scot - csc scsc - csc cot To sow a tpical calculation, we erive te erivative of te tangent function. Te oter erivations are left to Eercise 5. EXAMPLE 5 Fin (tan )>. Solution sin A tan B a cos b cos A sin B - sin A cos B cos Quotient Rule cos cos - sin s - sin cos cos + sin cos sec cos EXAMPLE 6 Fin if sec. Solution sec sec tan ssec tan sec A tan B + tan A sec B sec ssec + tan ssec tan sec + sec tan Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle Prouct Rule

47 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation Te ifferentiabilit of te trigonometric functions trougout teir omains gives anoter proof of teir continuit at ever point in teir omains (Teorem, Section.). So we can calculate limits of algebraic combinations an composites of trigonometric functions b irect substitution. EXAMPLE 7 lim : Fining a Trigonometric Limit + sec + + sec - - cos sp - tan cos sp - tan cos sp - Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

48 AWL/Tomas_cp7-8/9/ :6 AM Page Capter : Differentiation EXERCISES. Derivatives 6. Fin s > if In Eercises, fin >. a. - sin cos. + 5 sin. csc cot - 5. ssec + tan ssec - tan 7. cot + cot 8. cos + sin cos. + cos 9. cos + tan. sin + cos - sin 8. tan, -p> 6 6 p> - p>,, p> 9. sec, -p> 6 6 p> -p> p - p>, p>. s t - sec t + + csc t - csc t -p> p - p,, p>. + cos, In Eercises 6, fin s>t. 5. s In Eercises 7, grap te curves over te given intervals, togeter wit teir tangents at te given values of. Label eac curve an tangent wit its equation. - p>, p>. cos - sin - cos. s tan t - t Tangent Lines 7. sin, 6. ssin + cos sec b. 9 cos. 6. s sin t - cos t T Do te graps of te functions in Eercises ave an orizontal tangents in te interval p? If so, were? If not, w not? Visualize our finings b graping te functions wit a graper.. + sin In Eercises 7, fin r>u. 7. r - u sin u 8. r u sin u + cos u 9. r sec u csc u. r s + sec u sin u. + sin. - cot. + cos In Eercises, fin p>q.. p 5 + cot q. p sin q + cos q cos q. p s + csc q cos q. p tan q + tan q 5. Fin if a. csc. 5. Fin all points on te curve tan, - p> 6 6 p>, were te tangent line is parallel to te line. Sketc te curve an tangent(s) togeter, labeling eac wit its equation. 6. Fin all points on te curve cot, 6 6 p, were te tangent line is parallel to te line -. Sketc te curve an tangent(s) togeter, labeling eac wit its equation. b. sec. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

49 AWL/Tomas_cp7-8/9/ :6 AM Page 89. Derivatives of Trigonometric Functions In Eercises 7 an 8, fin an equation for (a) te tangent to te curve at P an (b) te orizontal tangent to te curve at Q Is tere a value of b tat will make g s e 8. + b, 6 cos, Ú continuous at? Differentiable at? Give reasons for our answers. Q P, Fin 999> 999 scos. 5. Derive te formula for te erivative wit respect to of cot csc a. sec. P, b. csc. T 5. Grap cos for -p p. On te same screen, grap Q T 5. Grap - sin for - p p. On te same screen, grap Trigonometric Limits 9. lim sin a - b : : - p>6 T 5. Centere ifference quotients Te centere ifference quotient + cos sp csc. lim sec ccos + p tan a : ƒs + - ƒs - p b - sec is use to approimate ƒ s in numerical work because () its limit as : equals ƒ s wen ƒ s eists, an () it usuall gives a better approimation of ƒ s for a given value of tan Fermat s ifference quotient p + tan b. lim sin a tan - sec :. lim tan a t:. lim cos a u : cos s + - cos for,.5,., an.. Ten, in a new winow, tr -, -.5, an -.. Wat appens as : +? As : -? Wat penomenon is being illustrate ere? Fin te limits in Eercises 9. lim sin s + - sin for,.5,., an.. Ten, in a new winow, tr -, -.5, an -.. Wat appens as : +? As : -? Wat penomenon is being illustrate ere? 兹 csc cot. c. cot. sin t t b ƒs + - ƒs. pu b sin u See te accompaning figure. Simple Harmonic Motion Slope f '() Te equations in Eercises 5 an 6 give te position s ƒst of a bo moving on a coorinate line (s in meters, t in secons). Fin te bo s velocit, spee, acceleration, an jerk at time t p> sec. 6. s sin t + cos t 5. s - sin t B f( ) f() A Slope Teor an Eamples f( ) f( ) f () 7. Is tere a value of c tat will make sin, ƒs L c, Slope C Z continuous at? Give reasons for our answer. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

50 AWL/Tomas_cp7-8/9/ :6 AM Page 9 9 Capter : Differentiation a. To see ow rapil te centere ifference quotient for ƒs sin converges to ƒ s cos, grap cos togeter wit sin s + - sin s - over te interval [- p, p] for,.5, an.. Compare te results wit tose obtaine in Eercise 5 for te same values of. b. To see ow rapil te centere ifference quotient for ƒs cos converges to ƒ s - sin, grap - sin togeter wit T 57. Eploring (sin k) / Grap ssin >, ssin >, an ssin > togeter over te interval -. Were oes eac grap appear to cross te -ais? Do te graps reall intersect te ais? Wat woul ou epect te graps of ssin 5> an ssin s - > to o as :? W? Wat about te grap of ssin k> for oter values of k? Give reasons for our answers. T 58. Raians versus egrees: egree moe erivatives Wat appens to te erivatives of sin an cos if is measure in egrees instea of raians? To fin out, take te following steps. cos s + - cos s - over te interval [- p, p] for,.5, an.. Compare te results wit tose obtaine in Eercise 5 for te same values of. 5. A caution about centere ifference quotients of Eercise 5.) Te quotient T 56. Slopes on te grap of te cotangent function Grap cot an its erivative togeter for 6 6 p. Does te grap of te cotangent function appear to ave a smallest slope? A largest slope? Is te slope ever positive? Give reasons for our answers. (Continuation ƒs + - ƒs - ma ave a limit as : wen ƒ as no erivative at. As a case in point, take ƒs ƒ ƒ an calculate ƒ + ƒ - ƒ - ƒ lim. : As ou will see, te limit eists even toug ƒs ƒ ƒ as no erivative at. Moral: Before using a centere ifference quotient, be sure te erivative eists. T 55. Slopes on te grap of te tangent function Grap tan an its erivative togeter on s -p>, p>. Does te grap of te tangent function appear to ave a smallest slope? a largest slope? Is te slope ever negative? Give reasons for our answers. a. Wit our graping calculator or computer graper in egree moe, grap sin ƒs an estimate lim: ƒs. Compare our estimate wit p>8. Is tere an reason to believe te limit soul be p>8? b. Wit our graper still in egree moe, estimate cos - lim. : c. Now go back to te erivation of te formula for te erivative of sin in te tet an carr out te steps of te erivation using egree-moe limits. Wat formula o ou obtain for te erivative?. Work troug te erivation of te formula for te erivative of cos using egree-moe limits. Wat formula o ou obtain for te erivative? e. Te isavantages of te egree-moe formulas become apparent as ou start taking erivatives of iger orer. Tr it. Wat are te secon an tir egree-moe erivatives of sin an cos? Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

51 AWL/Tomas_cp7-8/9/ :6 AM Page 9 9 Capter : Differentiation.5 Te Cain Rule an Parametric Equations We know ow to ifferentiate ƒsu sin u an u gs -, but ow o we ifferentiate a composite like Fs ƒsgs sin s -? Te ifferentiation formulas we ave stuie so far o not tell us ow to calculate F s. So ow o we fin te erivative of F ƒ g? Te answer is, wit te Cain Rule, wic sas tat te erivative of te composite of two ifferentiable functions is te prouct of teir erivatives evaluate at appropriate points. Te Cain Rule is one of te most important an wiel use rules of ifferentiation. Tis section escribes te rule an ow to use it. We ten appl te rule to escribe curves in te plane an teir tangent lines in anoter wa. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

52 AWL/Tomas_cp7-8/9/ :6 AM Page 9.5 Te Cain Rule an Parametric Equations 9 Derivative of a Composite Function We begin wit eamples. EXAMPLE Relating Derivatives s is te composite of te functions u an u. How are te erivatives of tese functions relate? Te function Solution, Since C: turns We ave B: u turns an u. #, we see tat u #. u A: turns FIGURE.6 Wen gear A makes turns, gear B makes u turns an gear C makes turns. B comparing circumferences or counting teet, we see tat u> (C turns one-alf turn for eac B turn) an u (B turns tree times for A s one), so >. Tus, > > s>s s>usu>., u Is it an accient tat u #? u If we tink of te erivative as a rate of cange, our intuition allows us to see tat tis relationsip is reasonable. If ƒsu canges alf as fast as u an u gs canges tree times as fast as, ten we epect to cange > times as fast as. Tis effect is muc like tat of a multiple gear train (Figure.6). EXAMPLE Te function s + is te composite of u an u +. Calculating erivatives, we see tat u # u # 6 u s + # Calculating te erivative from te epane formula, we get A B 6 +. Once again, u #. u Te erivative of te composite function ƒ(g ()) at is te erivative of ƒ at g() times te erivative of g at. Tis is known as te Cain Rule (Figure.7). Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

53 AWL/Tomas_cp7-8/9/ :6 AM Page 9 9 Capter : Differentiation Composite f g Rate of cange at is f'(g()) g'(). g f Rate of cange at is g'(). Rate of cange at g() is f '( g()). u g() f (u) f(g()) FIGURE.7 Rates of cange multipl: Te erivative of ƒ g at is te erivative of ƒ at g () times te erivative of g at. THEOREM Te Cain Rule If ƒ(u) is ifferentiable at te point u gs an g() is ifferentiable at, ten te composite function sƒ gs ƒsgs is ifferentiable at, an sƒ g s ƒ sgs # g s. In Leibniz s notation, if ƒsu an u gs, ten u #, u were >u is evaluate at u gs. Intuitive Proof of te Cain Rule: Let u be te cange in u corresponing to a cange of in, tat is u gs + - gs Ten te corresponing cange in is ƒsu + u - ƒsu. It woul be tempting to write u # u () an take te limit as : : lim : u # lim : u # lim u lim : u : # lim u lim u: u : u. u (Note tat u : as : since g is continuous.) Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

54 AWL/Tomas_cp7-8/9/ :6 AM Page 9.5 Te Cain Rule an Parametric Equations 9 Te onl flaw in tis reasoning is tat in Equation () it migt appen tat u (even wen Z ) an, of course, we can t ivie b. Te proof requires a ifferent approac to overcome tis flaw, an we give a precise proof in Section.8. EXAMPLE Appling te Cain Rule An object moves along te -ais so tat its position at an time t Ú is given b st cos st +. Fin te velocit of te object as a function of t. We know tat te velocit is >t. In tis instance, is a composite function: cos su an u t +. We ave Solution - sin su u cossu u t. t u t + B te Cain Rule, # u t u t - sin su # t - sin st + # t - t sin st +. evaluate at u u As we see from Eample, a ifficult wit te Leibniz notation is tat it oesn t state specificall were te erivatives are suppose to be evaluate. Outsie-Insie Rule It sometimes elps to tink about te Cain Rule tis wa: If ƒsgs, ten ƒ sgs # g s. In wors, ifferentiate te outsie function ƒ an evaluate it at te insie function g() left alone; ten multipl b te erivative of te insie function. EXAMPLE Differentiating from te Outsie In Differentiate sin s + wit respect to. Solution sin ( + ) cos ( + ) # ( + ) (+)+* (+)+* (+)+* insie insie erivative of left alone te insie Repeate Use of te Cain Rule We sometimes ave to use te Cain Rule two or more times to fin a erivative. Here is an eample. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle

55 AWL/Tomas_cp7-8/9/ :6 AM Page 9 9 Capter : Differentiation HISTORICAL BIOGRAPHY EXAMPLE 5 A Tree-Link Cain Joann Bernoulli (667 78) Fin te erivative of gst tan s5 - sin t. Solution Notice ere tat te tangent is a function of 5 - sin t, wereas te sine is a function of t, wic is itself a function of t. Terefore, b te Cain Rule, g st A tan A 5 - sin t B B t A 5 - sin t B t sec s5 - sin t # a - cos t # A t B b t sec s5 - sin t # Derivative of tan u wit u 5 - sin t Derivative of 5 - sin u wit u t sec s5 - sin t # s - cos t # - scos t sec s5 - sin t. Te Cain Rule wit Powers of a Function If ƒ is a ifferentiable function of u an if u is a ifferentiable function of, ten substituting ƒsu into te Cain Rule formula u # u leas to te formula u ƒsu ƒ su. Here s an eample of ow it works: If n is a positive or negative integer an ƒsu u n, te Power Rules (Rules an 7) tell us tat ƒ su nu n -. If u is a ifferentiable function of, ten we can use te Cain Rule to eten tis to te Power Cain Rule: n u u nu n -. EXAMPLE 6 (a) A u n B nu n - u Appling te Power Cain Rule s5-7 7s5-6 A 5 - B 7s5-6s5 # - 7s5-6s5 - (b) Power Cain Rule wit u 5 -, n 7 b s - - a - - s - - s - Power Cain Rule wit u -, n - - s - -s s - In part (b) we coul also ave foun te erivative wit te Quotient Rule. Coprigt 5 Pearson Eucation, Inc., publising as Pearson Aison-Wesle