3.1 D e r i v a t i v e s o f P o l y n o m i a l s a n d E x p o n e n t i a l F u n c t i o n s. The graph of ƒ=c is the line y=c, so f ª(x)=0.

Transcription

1 License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// C H A P T E R B measuring slopes at points on te sine curve, we get strong visual evience tat te erivative of te sine function is te cosine function. Differentiation Rules Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. License to: jsamuels@bmcc.cun.eu We ave seen ow to interpret erivatives as slopes an rates of cange. We ave seen ow to estimate erivatives of functions given b tables of values. We ave learne ow to grap erivatives of functions tat are efine grapicall. We ave use te efinition of a erivative to. D e r i v a t i v e s o f P o l n o m i a l s a n E p o n e n t i a l F u n c t i o n s c FIGURE Te grap of ƒ=c is te line =c, so f ª()=. = =c slope= slope= FIGURE Te grap of ƒ= is te line =, so f ª()=. calculate te erivatives of functions efine b formulas. But it woul be teious if we alwas a to use te efinition, so in tis capter we evelop rules for fining erivatives witout aving to use te efinition irectl. Tese ifferentiation rules enable us to calculate wit relative ease te erivatives of polnomials, rational functions, algebraic functions, eponential an logaritmic functions, an trigonometric an inverse trigonometric functions. We ten use tese rules to solve problems involving rates of cange an te approimation of functions. In tis section we learn ow to ifferentiate constant functions, power functions, polnomials, an eponential functions. Lets start wit te simplest of all functions, te constant function f c. Te grap of tis function is te orizontal line c, wic as slope, so we must ave f. (See Figure.) A formal proof, from te enition of a eri vative, is also eas: In Leibniz notation, we write tis rule as follows. Derivative of a Constant Function P o w e r F u n c t i o n s We net look at te functions f n, were n is a positive integer. If n, te grap of f is te line, wic as slope (see Figure ). So (You can also verif Equation from te enition of a eri vative.) We ave alrea investigate te cases n an n. In fact, in Section.9 (Eercises 9 an ) we foun tat f lim l lim l f f c Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. c c lim l 8

2 84 CHAPTER DIFFERENTIATION RULES SECTION. DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS 85 License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// Te Binomial Teorem is given on Reference Page. For n 4 we n te eri vative of f 4 as follows: Tus Comparing te equations in (), (), an (), we see a pattern emerging. It seems to be a reasonable guess tat, wen n is a positive integer, n n n. Tis turns out to be true. We prove it in two was; te secon proof uses te Binomial Teorem. Te Power Rule First Proof Te formula If n is a positive integer, ten can be verie simpl b multipling out te rigt-an sie (or b summing te secon factor as a geometric series). If f n, we can use Equation.8. for f a an te equation above to write Secon Proof f lim l n a n a n n a a n a n f a lim la na n f lim l In ning te eri vative of 4 we a to epan 4. Here we nee to epan n an we use te Binomial Teorem to o so: n n n f lim l f f lim l lim l lim l n n n f f a a lim la lim la n n a a n a n a n a n a aa n a n f f 4 4 lim l n n lim l n nn n n n n n a n a License to: jsamuels@bmcc.cun.eu n n lim l lim ln n n n because ever term ecept te rst as as a factor an terefore approaces. We illustrate te Power Rule using various notations in Eample. EXAMPLE (a) If f 6, ten f 6 5. (b) If, ten 999. (c) If t 4, ten 4t. () t Wat about power functions wit negative integer eponents? In Eercise 5 we ask ou to verif from te enition of a eri vative tat We can rewrite tis equation as an so te Power Rule is true wen n. In fact, we will sow in te net section [Eercise 44(c)] tat it ols for all negative integers. Wat if te eponent is a fraction? In Eample in Section.7 we foun, in effect, tat wic can be written as Tis sows tat te Power Rule is true even wen n. In fact, we will sow in Section.8 tat it is true for all real numbers n. Te Power Rule (General Version) nn n n n n n n n n nn s s If n is an real number, ten n n n r r r Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

3 86 CHAPTER DIFFERENTIATION RULES License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// Figure sows te function in Eample (b) an its erivative. Notice tat is not ifferentiable at ( is not efine tere). Observe tat is positive wen increases an is negative wen ecreases. _ FIGURE =#œ _ ª FIGURE 4 EXAMPLE Differentiate: (a) f (b) s SOLUTION In eac case we rewrite te function as a power of. (a) Since f, we use te Power Rule wit n : (b) EXAMPLE Fin an equation of te tangent line to te curve s at te point,. Illustrate b graping te curve an its tangent line. SOLUTION Te erivative of f s is So te slope of te tangent line at (, ) is f. Terefore, an equation of te tangent line is We grap te curve an its tangent line in Figure 4. N e w D e r i v a t i v e s f r o m O l f (s ) f s = - =œ Wen new functions are forme from ol functions b aition, subtraction, or multiplication b a constant, teir erivatives can be calculate in terms of erivatives of te ol functions. In particular, te following formula sas tat te erivative of a constant times a function is te constant times te erivative of te function. Te Constant Multiple Rule If c is a constant an f is a ifferentiable function, ten or cf c f License to: jsamuels@bmcc.cun.eu GEOMETRIC INTERPRETATION OF THE CONSTANT MULTIPLE RULE =ƒ =ƒ Multipling b c stretces te grap verticall b a factor of. All te rises ave been ouble but te runs sta te same. So te slopes are ouble, too. Using prime notation, we can write te Sum Rule as f t f t Proof Let t cf. Ten EXAMPLE 4 (a) (b) (b Law of limits) Te net rule tells us tat te erivative of a sum of functions is te sum of te erivatives. Te Sum Rule t lim l t t If f an t are bot ifferentiable, ten Proof Let F f t. Ten F F F lim l f t f t lim l l lim lim l lim l c c lim l cf f t SECTION. DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS 87 f f f t f t f f f f f f lim l cf cf t t lim l t t (b Law ) Te Sum Rule can be etene to te sum of an number of functions. For instance, using tis teorem twice, we get f t f t f t f t B writing f t as f t an appling te Sum Rule an te Constant Multiple Rule, we get te following formula. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

4 88 CHAPTER DIFFERENTIATION RULES License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// Tr more problems like tis one. Resources / Moule 4 / Polnomial Moels / Basic Differentiation Rules an Quiz FIGURE 5 Te curve =$-6@+4 an its orizontal tangents Te Difference Rule If f an t are bot ifferentiable, ten Te Constant Multiple Rule, te Sum Rule, an te Difference Rule can be combine wit te Power Rule to ifferentiate an polnomial, as te following eamples emonstrate. EXAMPLE EXAMPLE 6 Fin te points on te curve were te tangent line is orizontal. SOLUTION Horizontal tangents occur were te erivative is zero. We ave Tus, if or, tat is, s. So te given curve as orizontal tangents wen, s, an s. Te corresponing points are, 4, (s, 5), an (s, 5). (See Figure 5.) E p o n e n t i a l F u n c t i o n s f t f t {_œ,_5} Lets tr to compute te erivative of te eponential function f a using te enition of a erivative: f f a a f lim lim l l a a a lim l (,4) {œ,_5} a a lim l License to: jsamuels@bmcc.cun.eu In Eercise we will see tat e lies between.7 an.8. In Section 5.6 we will give a efinition of e tat will enable us to sow tat, correct to five ecimal places, e.788 Te factor oesnt epen on, so we can take it in front of te limit: Notice tat te limit is te value of te erivative of f at, tat is, Terefore, we ave sown tat if te eponential function f a is ifferentiable at, ten it is ifferentiable everwere an 4 Tis equation sas tat te rate of cange of an eponential function is proportional to te function itself. (Te slope is proportional to te eigt.) Numerical evience for te eistence of f is given in te table at te left for te cases a an a. (Values are state correct to four ecimal places. For te case a, see also Eample in Section.8.) It appears tat te limits eist an In fact, we will sow in Section 5.6 tat tese limits eist an, correct to si ecimal places, te values are Tus, from Equation 4 we ave 5 a Of all possible coices for te base a in Equation 4, te simplest ifferentiation formula occurs wen f. In view of te estimates of f for a an a, it seems reasonable tat tere is a number a between an for wic f. It is traitional to enote tis value b te letter e. (In fact, tat is ow we introuce e in Section.5.) Tus, we ave te following enition. Definition of te Number e for a, for a,.6947 SECTION. DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS a f a lim l a lim f l f f a f lim l e is te number suc tat.69 f lim. l.986. e lim l Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

5 9 CHAPTER DIFFERENTIATION RULES SECTION. DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS 9 License to: jsamuels@bmcc.cun.eu PDF Create wit eskpdf PDF Writer - Trial :: ttp:// f fª _.5.5 _ FIGURE 8 (ln,) = = FIGURE 9 Geometricall, tis means tat of all te possible eponential functions a, te function f e is te one wose tangent line at (, as a slope f tat is eactl. (See Figures 6 an 7.) FIGURE 6 = = =e If we put a e an, terefore, f in Equation 4, it becomes te following important ifferentiation formula. Derivative of te Natural Eponential Function Tus, te eponential function f e as te propert tat it is its own erivative. Te geometrical signicance of tis f act is tat te slope of a tangent line to te curve e is equal to te -coorinate of te point (see Figure 7). EXAMPLE 7 If f e, n f. Compare te graps of f an f. SOLUTION Using te Difference Rule, we ave f e e e Te function f an its erivative f are grape in Figure 8. Notice tat f as a orizontal tangent wen ; tis correspons to te fact tat f. Notice also tat, for, f is positive an f is increasing. Wen, f is negative an f is ecreasing. EXAMPLE 8 At wat point on te curve e is te tangent line parallel to te line? SOLUTION Since e, we ave e. Let te -coorinate of te point in question be a. Ten te slope of te tangent line at tat point is e a. Tis tangent line will be parallel to te line if it as te same slope, tat is,. Equating slopes, we get e a =e FIGURE 7 e e a ln Terefore, te require point is a, e a ln,. (See Figure 9.) {,e } slope= slope=e License to: jsamuels@bmcc.cun.eu. Eercises. (a) How is te number e ene? (b) Use a calculator to estimate te values of te limits an correct to two ecimal places. Wat can ou conclue about te value of e?. (a) Sketc, b an, te grap of te function f e, paing particular attention to ow te grap crosses te -ais. Wat fact allows ou to o tis? (b) Wat tpes of functions are f e an t e? Compare te ifferentiation formulas for f an t. (c) Wic of te two functions in part (b) grows more rapil wen is large? Differentiate te function.. f f s 5. f 5 6. F 4 7. f 4 8. t f t 4t 4 8. f t. 5. 5e. Vr 4 4. Rt 5t 5 r 5. Yt 6t G s e 8. s 9.. f t st st. t. s tu su su 7. a 8. ae v b v c b c v v t. u s t st s 4 t. F ( ) 5 4 s z A Be.7 lim l.8 lim l R s 7 t 6 t 4 t s. e ; 6 Fin f. Compare te graps of f an f an use tem to eplain w our answer is reasonable.. f e 5 4. f f 6. f 5 5 ; 7 8 Estimate te value of f a b zooming in on te grap of f. Ten ifferentiate f to n te e act value of f a an compare wit our estimate. 7. f, a 8. f s, a Fin an equation of te tangent line to te curve at te given point e,, 4.,, 9 ; 4 4 Fin an equation of te tangent line to te curve at te given point. Illustrate b graping te curve an te tangent line on te same screen. 4.,, 4. s, 4, 8 ; 4. (a) Use a graping calculator or computer to grap te function f in te viewing rectangle, 5 b, 5. (b) Using te grap in part (a) to estimate slopes, make a roug sketc, b an, of te grap of f. (See Eample in Section.9.) (c) Calculate f an use tis epression, wit a graping evice, to grap f. Compare wit our sketc in part (b). ; 44. (a) Use a graping calculator or computer to grap te function t e in te viewing rectangle, 4 b 8, 8. (b) Using te grap in part (a) to estimate slopes, make a roug sketc, b an, of te grap of t. (See Eample in Section.9.) (c) Calculate t an use tis epression, wit a graping evice, to grap t. Compare wit our sketc in part (b). 45. Fin te points on te curve were te tangent is orizontal. 46. For wat values of oes te grap of f ave a orizontal tangent? 47. Sow tat te curve 6 5 as no tangent line wit slope 4. ; 48. At wat point on te curve e is te tangent line parallel to te line 5? Illustrate b graping te curve an bot lines. 49. Draw a iagram to sow tat tere are two tangent lines to te parabola tat pass troug te point, 4. Fin te coorinates of te points were tese tangent lines intersect te parabola. 5. Fin equations of bot lines troug te point, tat are tangent to te parabola. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

6 9 CHAPTER DIFFERENTIATION RULES License to: 5. Te normal line to a curve C at a point P is, b enition, te line tat passes troug P an is perpenicular to te tangent line to C at P. Fin an equation of te normal line to te parabola at te point (, ). Sketc te parabola an its normal line. PDF Create wit eskpdf PDF Writer - Trial :: ttp:// 5. Were oes te normal line to te parabola at te point (, ) intersect te parabola a secon time? Illustrate wit a sketc. 5. Use te enition of a eri vative to sow tat if f, ten f. (Tis proves te Power Rule for te case n.) 54. Fin te parabola wit equation a b wose tangent line at (, ) as equation. 55. Let f Is f ifferentiable at? Sketc te graps of f an f. 56. At wat numbers is te following function t ifferentiable? t Give a formula for t an sketc te graps of t an t. 57. (a) For wat values of is te function ifferentiable? Fin a formula for f. (b) Sketc te graps of f an f. if if if if if f 9. T e P r o u c t a n Q u o t i e n t R u l e s 58. Were is te function ifferentiable? Give a formula for an sketc te graps of an. 59. For wat values of a an b is te line b tangent to te parabola a wen? 6. Let Fin te values of m an b tat make f ifferentiable everwere. 6. Fin a cubic function f wose grap as orizontal tangents at te points, 6 an,. 6. A tangent line is rawn to te perbola c at a point P. (a) Sow tat te mipoint of te line segment cut from tis tangent line b te coorinate aes is P. (b) Sow tat te triangle forme b te tangent line an te coorinate aes alwas as te same area, no matter were P is locate on te perbola. 6. Evaluate lim. l m b if if a b c 64. Draw a iagram sowing two perpenicular lines tat intersect on te -ais an are bot tangent to te parabola. Were o tese lines intersect? Te formulas of tis section enable us to ifferentiate new functions forme from ol functions b multiplication or ivision. T e P r o u c t R u l e ft B analog wit te Sum an Difference Rules, one migt be tempte to guess, as Leibniz i tree centuries ago, tat te erivative of a prouct is te prouct of te erivatives. We can see, owever, tat tis guess is wrong b looking at a particular eample. Let f Î uî ÎuÎ an t. Ten te Power Rule gives f an t. But, so u u Îu Îu ft. Tus, ft f t. Te correct formula was iscovere b Leibniz (soon after is false start) an is calle te Prouct Rule. Before stating te Prouct Rule, lets see ow we migt iscover it. We start b assuming tat u f an v t are bot positive ifferentiable functions. Ten we can interpret te prouct uv as an area of a rectangle (see Figure ). If canges b an amount, ten te corresponing canges in u an v are FIGURE Te geometr of te Prouct Rule u f f v t t License to: jsamuels@bmcc.cun.eu Recall tat in Leibniz notation te efinition of a erivative can be written as lim l In prime notation: ft ft tf Figure sows te graps of te function f of Eample an its erivative f. Notice tat f is positive wen f is increasing an negative wen f is ecreasing. _.5 f FIGURE fª _ an te new value of te prouct, u uv v, can be interprete as te area of te large rectangle in Figure (provie tat u an v appen to be positive). Te cange in te area of te rectangle is If we ivie b, we get If we now let l, we get te erivative of uv: uv u uv v uv u v v u u v uv uv uv lim lim l l u v u v v u u lim l v u v u v v v u uv u v (Notice tat ul as l since f is ifferentiable an terefore continuous.) Altoug we starte b assuming (for te geometric interpretation) tat all te quantities are positive, we notice tat Equation is alwas true. (Te algebra is vali weter u, v, u, an v are positive or negative.) So we ave prove Equation, known as te Prouct Rule, for all ifferentiable functions u an v. Te Prouct Rule If f an t are bot ifferentiable, ten In wors, te Prouct Rule sas tat te erivative of a prouct of two functions is te r st function times te erivative of te secon function plus te secon function times te erivative of te r st function. EXAMPLE If f e, n f. te sum of te tree sae areas SOLUTION B te Prouct Rule, we ave u v u v v u u v lim l lim l f t f t t f f e e e e e e SECTION. THE PRODUCT AND QUOTIENT RULES 9 u lim l v Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

7 94 CHAPTER DIFFERENTIATION RULES SECTION. THE PRODUCT AND QUOTIENT RULES 95 License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// EXAMPLE Differentiate te function f t st t. SOLUTION Using te Prouct Rule, we ave SOLUTION If we rst use te la ws of eponents to rewrite f t, ten we can procee irectl witout using te Prouct Rule. wic is equivalent to te answer given in Solution. Eample sows tat it is sometimes easier to simplif a prouct of functions tan to use te Prouct Rule. In Eample, owever, te Prouct Rule is te onl possible meto. EXAMPLE If f s t, were t4 an t4, n f 4. SOLUTION Appling te Prouct Rule, we get So f t st t t t t st st t t st t st f t st tst t t f t t t f [s t] s t t [s] s t t s t t s f 4 s4 t4 t4 6.5 s4 EXAMPLE 4 A telepone compan wants to estimate te number of new resiential pone lines tat it will nee to install uring te upcoming mont. At te beginning of Januar te compan a, subscribers, eac of wom a. pone lines, on average. Te compan estimate tat its subscribersip was increasing at te rate of montl. B polling its eisting subscribers, te compan foun tat eac intene to install an average of. new pone lines b te en of Januar. Estimate te number of new lines te compan will ave to install in Januar b calculating te rate of increase of lines at te beginning of te mont. SOLUTION Let st be te number of subscribers an let nt be te number of pone lines per subscriber at time t, were t is measure in monts an t correspons to te beginning of Januar. Ten te total number of lines is given b Lt stnt an we want to n L. Accoring to te Prouct Rule, we ave Lt t stnt st t nt nt t st t st License to: jsamuels@bmcc.cun.eu We are given tat s, an n.. Te compans estimates concerning rates of increase are tat s an n.. Terefore, Te compan will nee to install approimatel new pone lines in Januar. Notice tat te two terms arising from te Prouct Rule come from ifferent sources ol subscribers an new subscribers. One contribution to L is te number of eisting subscribers (,) times te rate at wic te orer new lines (about. per subscriber montl). A secon contribution is te average number of lines per subscriber (. at te beginning of te mont) times te rate of increase of subscribers ( montl). T e Q u o t i e n t R u l e We n a rule for if ferentiating te quotient of two ifferentiable functions u f an v t in muc te same wa tat we foun te Prouct Rule. If, u, an v cange b amounts, u, an v, ten te corresponing cange in te quotient uv is so L sn ns u v u u v v u v vu uv vv v v u v u v u uv lim lim l l vv v As l, vl also, because t is ifferentiable an terefore continuous. Tus, using te Limit Laws, we get v u v lim l,.. u v lim l v u lim l v v Te Quotient Rule If f an t are ifferentiable, ten u uv uv v vv v f t t f f t t In wors, te Quotient Rule sas tat te erivative of a quotient is te enominator times te erivative of te numerator minus te numerator times te erivative of te enominator, all ivie b te square of te enominator. Te Quotient Rule an te oter ifferentiation formulas enable us to compute te erivative of an rational function, as te net eample illustrates. v u v u v Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

8 96 CHAPTER DIFFERENTIATION RULES License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// We can use a graping evice to ceck tat te answer to Eample 5 is plausible. Figure sows te graps of te function of Eample 5 an its erivative. Notice tat wen grows rapil (near ), is large. An wen grows slowl, is near. ª.5 _4 4 FIGURE.5 _.5 = + _.5 FIGURE 4 EXAMPLE 5 Let. 6 Ten EXAMPLE 6 Fin an equation of te tangent line to te curve e at te point, e. SOLUTION Accoring to te Quotient Rule, we ave So te slope of te tangent line at, e is Tis means tat te tangent line at, e is orizontal an its equation is e. [See Figure 4. Notice tat te function is increasing an crosses its tangent line at, e.] NOTE Dont use te Quotient Rule ever time ou see a quotient. Sometimes its easier to rewrite a quotient rst to put it in a form tat is simpler for te purpose of if feren- tiation. For instance, altoug it is possible to ifferentiate te function using te Quotient Rule, it is muc easier to perform te ivision rst an write te function as before ifferentiating e e e e e F s F License to: jsamuels@bmcc.cun.eu Table of Differentiation Formulas. Eercises We summarize te ifferentiation formulas we ave learne so far as follows. c cf cf ft ft tf. Fin te erivative of in two was: b using te Prouct Rule an b performing te multiplication rst. Do our answers agree?. Fin te erivative of te function in two was: b using te Quotient Rule an b simplifing rst. So w tat our answers are equivalent. Wic meto o ou prefer? Differentiate.. f e 4. t s e 9. V 4. Yu u u u 5 u.. Rt t e t ( st). t 4. t t 5. r re r 6. s ke s 7. v vsv 8. z w w ce w v s s. f c F F 4 5 s s e e 7. t 8. f t t 4 t t t t 4 a b. f c n n n f t f t f t SECTION. THE PRODUCT AND QUOTIENT RULES 97 tf ft t e e f t f t 6 Fin an equation of te tangent line to te given curve at te specie point..,, 4. s, 4,.4, 6. e 5. e,,, e 7. (a) Te curve is calle a witc of Maria Agnesi. Fin an equation of te tangent line to tis curve at te point (, ). ; (b) Illustrate part (a) b graping te curve an te tangent line on te same screen. 8. (a) Te curve is calle a serpentine. Fin an equation of te tangent line to tis curve at te point,.. ; (b) Illustrate part (a) b graping te curve an te tangent line on te same screen. 9. (a) If f e, n f. ; (b) Ceck to see tat our answer to part (a) is reasonable b comparing te graps of f an f.. (a) If f, n f. ; (b) Ceck to see tat our answer to part (a) is reasonable b comparing te graps of f an f.. Suppose tat f 5, f 5 6, t5, an t5. Fin te following values. (a) ft5 (b) ft5 (c) tf 5. If f 4, t, f 6, an t 5, n te following numbers. (a) f t (b) ft (c) () f f t f t. If f e t, were t an t 5, n f. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

9 98 CHAPTER DIFFERENTIATION RULES License to: 4. If 4 an, n PDF Create wit eskpdf PDF Writer - Trial :: ttp:// 5. If f an t are te functions wose graps are sown, let u f t an v f t. (a) Fin u. (b) Fin v5. 6. Let P FG an Q FG, were F an G are te functions wose graps are sown. (a) Fin P. (b) Fin Q7. 7. If t is a ifferentiable function, n an e pression for te erivative of eac of te following functions. (a) t (b) (c) t t 8. If f is a ifferentiable function, n an e pression for te erivative of eac of te following functions. (a) f (c) f f (b) () f f s 9. In tis eercise we estimate te rate at wic te total personal income is rising in te Ricmon-Petersburg, Virginia, metropolitan area. In 999, te population of tis area was 96,4, an te population was increasing at rougl 9 people per ear. Te average annual income was $,59 per capita, an tis average was increasing at about $4 per ear (a little F G g above te national average of about $5 earl). Use te Prouct Rule an tese gures to estimate te rate at wic total personal income was rising in te Ricmon-Petersburg area in 999. Eplain te meaning of eac term in te Prouct Rule. 4. A manufacturer prouces bolts of a fabric wit a e wit. Te quantit q of tis fabric (measure in ars) tat is sol is a function of te selling price p (in ollars per ar), so we can write q f p. Ten te total revenue earne wit selling price p is Rp pf p. (a) Wat oes it mean to sa tat f, an f 5? (b) Assuming te values in part (a), n R an interpret our answer. 4. How man tangent lines to te curve ) pass troug te point,? At wic points o tese tangent lines touc te curve? 4. Fin equations of te tangent lines to te curve tat are parallel to te line. 4. (a) Use te Prouct Rule twice to prove tat if f, t, an are ifferentiable, ten ft f t ft ft (b) Taking f t in part (a), sow tat f f f (c) Use part (b) to ifferentiate e. 44. (a) If t is ifferentiable, te Reciprocal Rule sas tat Use te Quotient Rule to prove te Reciprocal Rule. (b) Use te Reciprocal Rule to ifferentiate te function in Eercise 9. (c) Use te Reciprocal Rule to verif tat te Power Rule is vali for negative integers, tat is, for all positive integers n. t t t n n n License to: jsamuels@bmcc.cun.eu. R a t e s o f C a n g e i n t e N a t u r a l a n S o c i a l S c i e n c e s FIGURE P{,fl} Q{, } Î Î m PQ average rate of cange m=fª( )=instantaneous rate of cange Resources / Moule 4 / Polnomial Moels / Start of Polnomial Moels SECTION. RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES 99 Recall from Section.8 tat if f, ten te erivative can be interprete as te rate of cange of wit respect to. In tis section we eamine some of te applications of tis iea to psics, cemistr, biolog, economics, an oter sciences. Lets recall from Section.7 te basic iea bein rates of cange. If canges from to, ten te cange in is an te corresponing cange in is Te ifference quotient is te average rate of cange of wit respect to over te interval, an can be interprete as te slope of te secant line PQ in Figure. Its limit as l is te erivative f, wic can terefore be interprete as te instantaneous rate of cange of wit respect to or te slope of te tangent line at P, f. Using Leibniz notation, we write te process in te form Wenever te function f as a specic interpretation in one of te sciences, its erivative will ave a specic interpretation as a rate of cange. (As we iscusse in Section.7, te units for are te units for ivie b te units for.) We now look at some of tese interpretations in te natural an social sciences. P s i c s f f f f lim l If s f t is te position function of a particle tat is moving in a straigt line, ten st represents te average velocit over a time perio t, an v st represents te instantaneous velocit (te rate of cange of isplacement wit respect to time). Tis was iscusse in Sections.7 an.8, but now tat we know te ifferentiation formulas, we are able to solve velocit problems more easil. EXAMPLE Te position of a particle is given b te equation s f t t 6t 9t were t is measure in secons an s in meters. (a) Fin te velocit at time t. (b) Wat is te velocit after s? After 4 s? (c) Wen is te particle at rest? Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

10 CHAPTER DIFFERENTIATION RULES SECTION. RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// t= s= t= s= FIGURE t= s=4 s () Wen is te particle moving forwar (tat is, in te positive irection)? (e) Draw a iagram to represent te motion of te particle. (f) Fin te total istance travele b te particle uring te rst v e secons. SOLUTION (a) Te velocit function is te erivative of te position function. (b) Te velocit after s means te instantaneous velocit wen t, tat is, Te velocit after 4 s is (c) Te particle is at rest wen vt, tat is, t t 9 t 4t t t an tis is true wen t or t. Tus, te particle is at rest after s an after s. () Te particle moves in te positive irection wen vt, tat is, Tis inequalit is true wen bot factors are positive t or wen bot factors are negative t. Tus, te particle moves in te positive irection in te time intervals t an t. It moves backwar (in te negative irection) wen t. (e) Using te information from part (), we make a scematic sketc as sown in Figure of te motion of te particle back an fort along a line (te s-ais). (f) Because of wat we learne in parts () an (e), we nee to calculate te istances travele uring te time intervals [, ], [, ], an [, 5] separatel. Te istance travele in te rst secon is From t to t te istance travele is From t to t 5 te istance travele is Te total istance is m. s f t t 6t 9t vt s t t t 9 v s 9 ms t t v ms t t 9 t t f f 4 4 m f f 4 4 m f 5 f m EXAMPLE If a ro or piece of wire is omogeneous, ten its linear ensit is uniform an is ene as te mass per unit lengt an measure in kilograms per ml License to: jsamuels@bmcc.cun.eu FIGURE 4 FIGURE meter. Suppose, owever, tat te ro is not omogeneous but tat its mass measure from its left en to a point is m f, as sown in Figure. Te mass of te part of te ro tat lies between an is given b m f f, so te average ensit of tat part of te ro is If we now let l (tat is, l), we are computing te average ensit over smaller an smaller intervals. Te linear ensit at is te limit of tese average ensities as l; tat is, te linear ensit is te rate of cange of mass wit respect to lengt. Smbolicall, Tus, te linear ensit of te ro is te erivative of mass wit respect to lengt. For instance, if m f s, were is measure in meters an m in kilograms, ten te average ensit of te part of te ro given b. is wile te ensit rigt at is Tis part of te ro as mass ƒ. average ensit m f f m f. f s..48 kgm.. m lim l m m.5 kgm s EXAMPLE A current eists wenever electric carges move. Figure 4 sows part of a wire an electrons moving troug a sae plane surface. If Q is te net carge tat passes troug tis surface uring a time perio t, ten te average current uring tis time interval is ene as average current Q t If we take te limit of tis average current over smaller an smaller time intervals, we get wat is calle te current I at a given time t: Q I lim Q tl t t Q Q t t Tus, te current is te rate at wic carge o ws troug a surface. It is measure in units of carge per unit time (often coulombs per secon, calle amperes). Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

11 CHAPTER DIFFERENTIATION RULES SECTION. RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// Velocit, ensit, an current are not te onl rates of cange tat are important in psics. Oters inclue power (te rate at wic work is one), te rate of eat o w, temperature graient (te rate of cange of temperature wit respect to position), an te rate of eca of a raioactive substance in nuclear psics. C e m i s t r EXAMPLE 4 A cemical reaction results in te formation of one or more substances (calle proucts) from one or more starting materials (calle reactants). For instance, te equation inicates tat two molecules of rogen an one molecule of ogen form two molecules of water. Lets consier te reaction were A an B are te reactants an C is te prouct. Te concentration of a reactant A is te number of moles ( mole 6. molecules) per liter an is enote b A. Te concentration varies uring a reaction, so A, B, an C are all functions of time t. Te average rate of reaction of te prouct C over a time interval t t t is But cemists are more intereste in te instantaneous rate of reaction, wic is obtaine b taking te limit of te average rate of reaction as te time interval t approaces : Since te concentration of te prouct increases as te reaction procees, te erivative Ct will be positive. (You can see intuitivel tat te slope of te tangent to te grap of an increasing function is positive.) Tus, te rate of reaction of C is positive. Te concentrations of te reactants, owever, ecrease uring te reaction, so, to make te rates of reaction of A an B positive numbers, we put minus signs in front of te erivatives At an Bt. Since A an B eac ecrease at te same rate tat C increases, we ave More generall, it turns out tat for a reaction of te form we ave rate of reaction C t a C rate of reaction lim C tl t t A t C t H O l HO aa bb l cc D b A B l C Ct Ct B t t t A t c C t B t D t License to: jsamuels@bmcc.cun.eu Te rate of reaction can be etermine b grapical metos (see Eercise ). In some cases we can use te rate of reaction to n e plicit formulas for te concentrations as functions of time (see Eercises 9.). EXAMPLE 5 One of te quantities of interest in termonamics is compressibilit. If a given substance is kept at a constant temperature, ten its volume V epens on its pressure P. We can consier te rate of cange of volume wit respect to pressurenamel, te erivative VP. As P increases, V ecreases, so VP. Te compressibilit is ene b introucing a minus sign an i viing tis erivative b te volume V: Tus, measures ow fast, per unit volume, te volume of a substance ecreases as te pressure on it increases at constant temperature. For instance, te volume V (in cubic meters) of a sample of air at 5C was foun to be relate to te pressure P (in kilopascals) b te equation Te rate of cange of V wit respect to P wen P 5 kpa is Te compressibilit at tat pressure is B i o l o g isotermal compressibilit V V 5. P P5 P V V 5. P P m kpa V.. m kpam P P EXAMPLE 6 Let n f t be te number of iniviuals in an animal or plant population at time t. Te cange in te population size between te times t t an t t is n f t f t, an so te average rate of growt uring te time perio t t t is average rate of growt n f t f t t t t Te instantaneous rate of growt is obtaine from tis average rate of growt b letting te time perio t approac : n growt rate lim n tl t t V P Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

12 4 CHAPTER DIFFERENTIATION RULES License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// FIGURE 5 A smoot curve approimating a growt function Strictl speaking, tis is not quite accurate because te actual grap of a population function n f t woul be a step function tat is iscontinuous wenever a birt or eat occurs an, terefore, not ifferentiable. However, for a large animal or plant population, we can replace te grap b a smoot approimating curve as in Figure 5. To be more specic, consier a population of bacteria in a omogeneous nutrient meium. Suppose tat b sampling te population at certain intervals it is etermine tat te population oubles ever our. If te initial population is n an te time t is measure in ours, ten an, in general, n f f n f f n f f n f t t n Te population function is n n t. In Section. we iscusse erivatives of eponential functions an foun tat.69 So te rate of growt of te bacteria population at time t is n t t nt n.69 t For eample, suppose tat we start wit an initial population of n bacteria. Ten te rate of growt after 4 ours is n.69 t t4 4 4 Tis means tat, after 4 ours, te bacteria population is growing at a rate of about bacteria per our. t License to: jsamuels@bmcc.cun.eu FIGURE 6 Bloo flow in an arter SECTION. RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES 5 EXAMPLE 7 Wen we consier te o w of bloo troug a bloo vessel, suc as a vein or arter, we can take te sape of te bloo vessel to be a clinrical tube wit raius R an lengt l as illustrate in Figure 6. Because of friction at te walls of te tube, te velocit v of te bloo is greatest along te central ais of te tube an ecreases as te istance r from te ais increases until v becomes at te wall. Te relationsip between v an r is given b te law of laminar o w iscovere b te Frenc psician Jean-Louis-Marie Poiseuille in 84. Tis states tat were is te viscosit of te bloo an P is te pressure ifference between te ens of te tube. If P an l are constant, ten v is a function of r wit omain, R. [For more etaile information, see W. Nicols an M. ORourke (es.), McDonals Bloo Flow in Arteries: Teoretic, Eperimental, an Clinical Principles, 4t e. (New York: Ofor Universit Press, 998).] Te average rate of cange of te velocit as we move from r r outwar to r r is given b an if we let rl, we obtain te velocit graient, tat is, te instantaneous rate of cange of velocit wit respect to r: Using Equation, we obtain v v vr vr r r r v v velocit graient lim rl r r v r P Pr r 4l l For one of te smaller uman arteries we can take, R.8 cm, l cm, an P 4 nescm, wic gives v r r At r. cm te bloo is o wing at a spee of R r v cms l P 4l R r.7 Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

13 6 CHAPTER DIFFERENTIATION RULES SECTION. RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES 7 License to: jsamuels@bmcc.cun.eu PDF Create wit eskpdf PDF Writer - Trial :: ttp:// an te velocit graient at tat point is To get a feeling for wat tis statement means, lets cange our units from centimeters to micrometers ( cm, m). Ten te raius of te arter is 8 m. Te velocit at te central ais is,85 ms, wic ecreases to, ms at a istance of r m. Te fact tat vr 74 (ms)m means tat, wen r m, te velocit is ecreasing at a rate of about 74 ms for eac micrometer tat we procee awa from te center. E c o n o m i c s EXAMPLE 8 Suppose C is te total cost tat a compan incurs in proucing units of a certain commoit. Te function C is calle a cost function. If te number of items prouce is increase from to, te aitional cost is C C C, an te average rate of cange of te cost is Te limit of tis quantit as l, tat is, te instantaneous rate of cange of cost wit respect to te number of items prouce, is calle te marginal cost b economists: [Since often takes on onl integer values, it ma not make literal sense to let approac, but we can alwas replace C b a smoot approimating function as in Eample 6.] Taking an n large (so tat is small compare to n), we ave Tus, te marginal cost of proucing n units is approimatel equal to te cost of proucing one more unit [te n st unit]. It is often appropriate to represent a total cost function b a polnomial were a represents te overea cost (rent, eat, maintenance) an te oter terms represent te cost of raw materials, labor, an so on. (Te cost of raw materials ma be proportional to, but labor costs migt epen partl on iger powers of because of overtime costs an inefciencies in volve in large-scale operations.) For instance, suppose a compan as estimate tat te cost (in ollars) of proucing items is Ten te marginal cost function is v cmscm r r..7 C C C C C C C marginal cost lim l Cn Cn Cn C a b c C, 5. C 5. License to: jsamuels@bmcc.cun.eu Te marginal cost at te prouction level of 5 items is Tis gives te rate at wic costs are increasing wit respect to te prouction level wen 5 an preicts te cost of te 5st item. Te actual cost of proucing te 5st item is Notice tat C5 C5 C5. Economists also stu marginal eman, marginal revenue, an marginal prot, wic are te erivatives of te eman, revenue, an prot functions. Tese will be consiere in Capter 4 after we ave evelope tecniques for ning te maimum an minimum values of functions. O t e r S c i e n c e s Rates of cange occur in all te sciences. A geologist is intereste in knowing te rate at wic an intrue bo of molten rock cools b conuction of eat into surrouning rocks. An engineer wants to know te rate at wic water o ws into or out of a reservoir. An urban geograper is intereste in te rate of cange of te population ensit in a cit as te istance from te cit center increases. A meteorologist is concerne wit te rate of cange of atmosperic pressure wit respect to eigt (see Eercise 7 in Section 9.4). In pscolog, tose intereste in learning teor stu te so-calle learning curve, wic graps te performance Pt of someone learning a skill as a function of te training time t. Of particular interest is te rate at wic performance improves as time passes, tat is, Pt. In sociolog, ifferential calculus is use in analzing te sprea of rumors (or innovations or fas or fasions). If pt enotes te proportion of a population tat knows a rumor b time t, ten te erivative pt represents te rate of sprea of te rumor (see Eercise 7 in Section.5). S u m m a r C5 5.5 $5item C5 C5, 55.5 $5., 55.5 Velocit, ensit, current, power, an temperature graient in psics, rate of reaction an compressibilit in cemistr, rate of growt an bloo velocit graient in biolog, marginal cost an marginal prot in economics, rate of eat o w in geolog, rate of improvement of performance in pscolog, rate of sprea of a rumor in sociologtese are all special cases of a single matematical concept, te erivative. Tis is an illustration of te fact tat part of te power of matematics lies in its abstractness. A single abstract matematical concept (suc as te erivative) can ave ifferent interpretations in eac of te sciences. Wen we evelop te properties of te matematical concept once an for all, we can ten turn aroun an appl tese results to all of te sciences. Tis is muc more efcient tan e veloping properties of special concepts in eac separate science. Te Frenc matematician Josep Fourier (768 8) put it succinctl: Matematics compares te most i verse penomena an iscovers te secret analogies tat unite tem. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.

14 8 CHAPTER DIFFERENTIATION RULES License to: PDF Create wit eskpdf PDF Writer - Trial :: ttp:// Eercises 6 A particle moves accoring to a law of motion s f t, t, were t is measure in secons an s in feet. (a) Fin te velocit at time t. (b) Wat is te velocit after s? (c) Wen is te particle at rest? () Wen is te particle moving in te positive irection? (e) Fin te total istance travele uring te rst 8 s. (f) Draw a iagram like Figure to illustrate te motion of te particle.. f t t t. f t t 9t 5t. f t t t 6t 4. f t t 4 4t 5. s t 6. s st t 5t 9 t 7. Te position function of a particle is given b s t 4.5t 7t t Wen oes te particle reac a velocit of 5 ms? 8. If a ball is given a pus so tat it as an initial velocit of 5 ms own a certain incline plane, ten te istance it as rolle after t secons is s 5t t. (a) Fin te velocit after s. (b) How long oes it take for te velocit to reac 5 ms? 9. If a stone is trown verticall upwar from te surface of te moon wit a velocit of ms, its eigt (in meters) after t secons is t.8t. (a) Wat is te velocit of te stone after s? (b) Wat is te velocit of te stone after it as risen 5 m?. If a ball is trown verticall upwar wit a velocit of 8 fts, ten its eigt after t secons is s 8t 6t. (a) Wat is te maimum eigt reace b te ball? (b) Wat is te velocit of te ball wen it is 96 ft above te groun on its wa up? On its wa own?. (a) A compan makes computer cips from square wafers of silicon. It wants to keep te sie lengt of a wafer ver close to 5 mm an it wants to know ow te area A of a wafer canges wen te sie lengt canges. Fin A5 an eplain its meaning in tis situation. (b) Sow tat te rate of cange of te area of a square wit respect to its sie lengt is alf its perimeter. Tr to eplain geometricall w tis is true b rawing a square wose sie lengt is increase b an amount. How can ou approimate te resulting cange in area A if is small?. (a) Soium clorate crstals are eas to grow in te sape of cubes b allowing a solution of water an soium clorate to evaporate slowl. If V is te volume of suc a cube wit sie lengt, calculate V wen mm an eplain its meaning. (b) Sow tat te rate of cange of te volume of a cube wit respect to its ege lengt is equal to alf te surface area of te cube. Eplain geometricall w tis result is true b arguing b analog wit Eercise (b).. (a) Fin te average rate of cange of te area of a circle wit respect to its raius r as r canges from (i) to (ii) to.5 (iii) to. (b) Fin te instantaneous rate of cange wen r. (c) Sow tat te rate of cange of te area of a circle wit respect to its raius (at an r) is equal to te circumference of te circle. Tr to eplain geometricall w tis is true b rawing a circle wose raius is increase b an amount r. How can ou approimate te resulting cange in area A if r is small? 4. A stone is roppe into a lake, creating a circular ripple tat travels outwar at a spee of 6 cms. Fin te rate at wic te area witin te circle is increasing after (a) s, (b) s, an (c) 5 s. Wat can ou conclue? 5. A sperical balloon is being inate. Fin te rate of increase of te surface area S 4r wit respect to te raius r wen r is (a) ft, (b) ft, an (c) ft. Wat conclusion can ou make? 6. (a) Te volume of a growing sperical cell is V 4 r, were te raius r is measure in micrometers ( m 6 m). Fin te average rate of cange of V wit respect to r wen r canges from (i) 5 to 8 m (ii) 5 to 6 m (iii) 5 to 5. m (b) Fin te instantaneous rate of cange of V wit respect to r wen r 5 m. (c) Sow tat te rate of cange of te volume of a spere wit respect to its raius is equal to its surface area. Eplain geometricall w tis result is true. Argue b analog wit Eercise (c). 7. Te mass of te part of a metal ro tat lies between its left en an a point meters to te rigt is kg. Fin te linear ensit (see Eample ) wen is (a) m, (b) m, an (c) m. Were is te ensit te igest? Te lowest? 8. If a tank ols 5 gallons of water, wic rains from te bottom of te tank in 4 minutes, ten Torricellis Law gives te volume V of water remaining in te tank after t minutes as V 5 t t 4 4 Fin te rate at wic water is raining from te tank after (a) 5 min, (b) min, (c) min, an () 4 min. At wat time is te water o wing out te fastest? Te slowest? Summarize our nings. 9. Te quantit of carge Q in coulombs (C) tat as passe troug a point in a wire up to time t (measure in secons) is were G is te gravitational constant an r is te istance between te boies. (a) Fin Fr an eplain its meaning. Wat oes te minus sign inicate? (b) Suppose it is known tat Eart attracts an object wit a force tat ecreases at te rate of Nkm wen r, km. How fast oes tis force cange wen r, km?. Boles Law states tat wen a sample of gas is compresse at a constant temperature, te prouct of te pressure an te volume remains constant: PV C. (a) Fin te rate of cange of volume wit respect to pressure. (b) A sample of gas is in a container at low pressure an is steail compresse at constant temperature for minutes. Is te volume ecreasing more rapil at te beginning or te en of te minutes? Eplain. (c) Prove tat te isotermal compressibilit (see Eample 5) is given b. P. Te ata in te table concern te lactonization of rovaleric aci at 5C. Te give te concentration Ct of tis aci in moles per liter after t minutes. (a) Fin te average rate of reaction for te following time intervals: (i) t 6 (ii) t 4 (iii) t (b) Plot te points from te table an raw a smoot curve troug tem as an approimation to te grap of te concentration function. Ten raw te tangent at t an use it to estimate te instantaneous rate of reaction wen t. ;. Te table gives te population of te worl in te t centur. SECTION. RATES OF CHANGE IN THE NATURAL AND SOCIAL SCIENCES 9 License to: jsamuels@bmcc.cun.eu given b Qt t t 6t. Fin te current wen (a) Estimate te rate of population growt in 9 an in 98 (a) t.5 s an (b) t s. [See Eample. Te unit of current is an ampere ( A Cs).] At wat time is te current (b) Use a graping calculator or computer to n a cubic func- b averaging te slopes of two secant lines. lowest? tion (a tir-egree polnomial) tat moels te ata. (See. Newtons Law of Gravitation sas tat te magnitue F of te Section..) force eerte b a bo of mass m on a bo of mass M is (c) Use our moel in part (b) to n a moel for te rate of population growt in te t centur. F GmM () Use part (c) to estimate te rates of growt in 9 an r 98. Compare wit our estimates in part (a). (e) Estimate te rate of growt in 985. t C(t) Population Population Year (in millions) Year (in millions) ; 4. Te table sows ow te average age of rst marriage of Japanese women varie in te last alf of te t centur. t At (a) Use a graping calculator or computer to moel tese ata wit a fourt-egree polnomial. (b) Use part (a) to n a moel for At. (c) Estimate te rate of cange of marriage age for women in 99. () Grap te ata points an te moels for A an A. 5. If, in Eample 4, one molecule of te prouct C is forme from one molecule of te reactant A an one molecule of te reactant B, an te initial concentrations of A an B ave a common value A B a molesl, ten C a ktakt were k is a constant. (a) Fin te rate of reaction at time t. (b) Sow tat if C, ten ka t (c) Wat appens to te concentration as tl? () Wat appens to te rate of reaction as tl? (e) Wat o te results of parts (c) an () mean in practical terms? 6. Suppose tat a bacteria population starts wit 5 bacteria an triples ever our. (a) Wat is te population after ours? After 4 ours? After t ours? (b) Use (5) in Section. to estimate te rate of increase of te bacteria population after 6 ours. 7. Refer to te law of laminar o w given in Eample 7. Consier a bloo vessel wit raius. cm, lengt cm, pressure ifference nescm, an viscosit. (a) Fin te velocit of te bloo along te centerline r, at raius r.5 cm, an at te wall r R. cm. t At.7 Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part. Coprigt 5 Tomson Learning, Inc. All Rigts Reserve. Ma not be copie, scanne, or uplicate, in wole or in part.