doc20101003103834

doc20101003103834 - 25‘ Hm x 7-f y ix 0‘1 a JIQ'L}—i...

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Unformatted text preview: 25‘) Hm x 7-f- y ix 0‘1 . _ a - JIQ'L}—i-\ —— > “W? j, '-' _ MA :24 Undead 016W" I61" u- ____ -_ 7 ~ "ii: . iter-_efi, {3(3) =fh ?{Lq ?_ ___ . 5. .r(. L!) b bid/51L beam/Sit 'r' LS ma 6 n CYGMi‘rtj’ ital": thin. . ‘ :- , J. .- I ‘ u 5),,r(g/J {L /(.’)M iv; . . 9 . .7 _ A .. .) C '9 Effie/L tzfi (Lemma én’l‘bivga. ... . 'f3*f0I7m flaw—"M nm E 11 i-mwvn .- f'w .‘I I I I Md 4 _ (IV 3 -—1 343311.11)” _15 13;},4j7gL MW at {"5 Edam 1' £2 '1 - meapr “wwmw E J? i( I ) is. bag“;sz id” [Lg/3 4-; AEEFLLL fife/w a. .43.. Q~q?7?- I __I " iAn3XJHM I x} ‘ W131 if .. ‘1‘] "K [0—3 .— . .LU‘W3..'*::-fQ—'. .i’~‘7“".c’7.. .. . 0‘ ‘= c fi-MEI 7123.2. 36s“... fro'f g t 3-fi$$§_ .‘v- ow? “J1 ...‘1.’55‘$? S-Lrwl 1431:” . I _. I I .0 ma 7 __,-_...m hat: LN” fiat“ Li gr‘m e 130‘!" I ("M63 6344 .3 mu; Ls _ 7H”) :: {/xz 41267" (W) (thaflaji) 'MW w 7",) N) “5- 2:") 7(. L «.W x; (fl;if%;%(xw11 '(3#0:<«fl(wyn ){I‘Ll}. V/an’é n‘éfl. . . 9V7“: :: .3. Wirr: «123$; l 2.2 THE DERIVATIVE AT A POINT 33 32 Chapter Two KEY CONCEPT: THE osmvanve ; f a 10. Label points it B C D F t - . . ‘. .. v “we? 3 ‘7 1 i a-JsdndFOntheg In I' ‘ ' ‘ ExampiBB Find the derivative of f is) J 1/3. at the pomt .l» _ a. 1 Km) in Figure 2.2]. mph Of y 11' The gmp‘h or y a Hm) 15 Show in Figure 121 WhiCh is larger 10 each of the following pairs? he limit of the difference quotient. so we look at (a) Point A is a 13mm on the cum: where the derivative i I . . . Solution The derivative 1s t . 2 + In my H2) g is negative. '1 __ - _ - i ' ' . - Jr (2) _ I; _ __._. i (1)) Point 8 is a point on the curve where the value of _ ‘ I _ . . _ _ ' the [auction is negative. in USWS “13 tormum for 1 And b“fil3mylnt‘:’t Ewes (C) ijolnt C is a mint on the curve where the derivative 5 y 3 Km) 1 1 1 2 —n (2 + h.) 41. IS largest 4 ' ‘2 z 1110 ~— — — 3 lim —»—_-*—— : 11111 . i d P ‘ ‘ f i ) h_n h 2 + h 2 h__0 2M2 + h) [HM-0 2M2 + h) g i ) 1:71:01) 15 4 F101!“ 011 the curve where the derivative 3 Since the limit only examines values of it close to. but not equal to, zero. we can cancel it. We get ('3) Points E and F are different points on the cum: 1 .r a I _ H1 1 where the derivative is ahoutthe same. - ' w—J——L—mmtm.utaw_i_ .L. {0:11:11 "—‘fi-f‘i. . n y ’I ' 1 2 3 :1 5 i ' -. -' .. " Figure 2.22 n—Ho 2(‘2 +11.) Figure 2.19 is —w1/4. Thus, f’ (2) z -1/il-. The slope of the tangent line in ______________¥__.__.——-——-——-d—" ~ .I :r: (:1) Average rate of change: Between :1: = 1 and ’1; 2 '5'? Or between :0 m 3 and :c m 5'? . I ' 01) ft?) or .l"(5)‘i (0 fit.) orf’(=1)'-’ Exercises and Problems for Section 2.2 Exercises 1. The table shows values of f(:1:) a 0:3 near :1: i ‘2 (to three decimal places). Use it to estimate f'(2). Difference in successive :er values 12 S h . uppose t at f(:c) is a function with f 100 x ‘l" d ' ( ) \ . a an 13. Show how to represent the followmg on Figure 2.25. Ham) 3 3. Estimate 11102). 0.998 0.996004 1.998 1.999 2.000 2.001 2.002 0005997 7.970 7.988 8.000 8.012 3.024 0.999 0.998001 ' 13. ~ - y. I 01101999 The tpnction tn Figure 2.23 has “4} = 25 and is“) 2 (it) Ht) (3,) H4) __ fig) 1.000 murmur) 1 Lo. land the coordinates of the points A B C ( ) Hr) __ “9) 0.00200 ’ ‘ ' c __.; , . 1001 1.002001 Tangemlina 5 w 2 (d) f (3) 2. (a) Make a table of values rounded to two decimal 3 0.002003 places for the function f(:t: : em [or 1'00" 1'004004 a; m 1, 1.5, 2, 2.5, and 3. Then use the table to an— swer parts (b) and (c), ’. It' f(:t!) z 0:3 wt- 4:0, estimtite f'(3} using a table with (1)) Find the average rate of change of f(:r) between 3 :L' = 1 and £1? 3: 3- values of :0 near 3, spaced by 0.001. (e) Iise'iaverage rates foilgiangEfithillgpl‘fl‘Eaf mg m' 6. Graph flaz) = sin 3:, and use the graph to decide whether 5 m menus m c O U “gt: 0 ‘ '1‘ d 'L _ m the derivative of f(:1;) at rt: t 377 is positive or negative». 3' (a) NEW»: whiff #31:??? (iffifdh: bl'wo 7. For the function f(a:) n log estimate f’(1). From _ T _‘ p aces, or .l 4-) -_- EJ' "1 1” I U dse “1 the graph of j'(:i:). would you expect your estimate to ' 5 Figure 225 :1: = l. 1.5. 2, 2.3, 3. Then use this table to answer ‘ i . be greater than or less than I (1)? parts ([1) and (c). - “1 Explain your reasoning. (b) Find the average rate of cha a: m 1 and :1: z 3. (c) Use average rates of change to approximate the in- 16. For each of the following pairs of numbers use Fin- ure 2.25 to decide which is iarger. Explain your answer: 8. Estimate f’(2) for j‘{rc) : 3 9. Figure 2.20 shows the graph of j in the table with the points (1,11, 0.01, e. age of fin) between '. Match the derivativ 9L): (b) g'(_)= (a) 1’0) for? stantaneous rate of change of f(m) at :1: fi ‘2. 4. (a) z :32. Explain what’Table 2.4 tells us about \ "— (b) “3) “2) Dr “2) f“)? I L bF'd’l r1. ‘ . i. ‘ ’ ‘“ l l W” a”; o ' " " ‘tdi f’(t) Dim)? ‘ “' 1. (c) if :1' changes by 0.1 near :1: :2 1, what does f’(1) tell us about how fire) Changes? Illustrate your answer (‘1 with a sketch. g(:1: ‘ r ' ' ‘ l 17. Wlth the function If glven by Figure 2.25, arrange the lollowmg quantities in ascending order: Tangent line F. igure2.24 0: fig), fit3)- f(3)*f(2) 54 Chapter Two KEY CONCEPT: THE DERIVATIVE 18. On a copy of Figure 2.26. mark lengths that represent the quantities in parts (:1) m (d). (Pick any positive :1: and h.) (a) lb} f(:t: + ft.) (0} f(117 + it) - flat} ((1) h ((3') Using your answers to parts (aktd), show how [he f(.’L’ —I- It) - j'(:t:) it slope of a line in Figure 2.26. quantity can be represented as the Figure 2.26 19. On a copy of Figure. 2.27. mark lengths that represent the quantities in parts (a) — (d). (Pick any convenient 3:. and assume it > 0.) (a) j'(:t') (b) f(:t:+ h.) (c) flu: -1- It} — f(;i:) (d) h. (e) Using your answers to parts (a)—(tl}. show how the flu: + h.) w j'(:c) h. slope of a line on the graph. quantity can be represented as the Figure 2.27 20. Consider the function shown in Figure 2.28. (:1) Write an expression involving f for the slope of the linejoining A and B. (h) Draw the tangent line at C. Compare its slope to the slope of the line in part (a). (c) Are there any other points on the curve at which the slope of the tangent line is the same as the slope of the tangent line at C"? If so. mark them on the graph. If not. why not? Figure 2.28 21. (a) If f is even and f’(10)::{i. what is f’(—ml.0)? (b) If f is any even function and f’(0) exists. what is f' (0)? 22. Hg is an odd Function and g’(tl) z 5, what is 9’04}? 23. (:1) Estimate f’(0) iff(:c) 2 sin 3:, with a: in degrees. (h) In Example 4 on page 80, we found that the derivzu tive of sins.- at :1: = U was 1. Why do we get a dif- ferent result here? (This problem shows why radians are almost always used in calculus.) 24. Estimate the instantaneous rate of change of the function f(:lf) : :1: lna: at a; r— 1 and at a: 2 2. What do these values suggest about the concavity of the graph between 1 and 2'! 25. Estimate the derivative of flar) = sf" at :t; z 2. 26. For 3} : m 3zt:3/2 —- :15, use your calculator to con- struct a graph of y 2 fle). for 0 S a: 5 2. From your graph, estimate f’(0) and j"(1). 27. Let f(:t:) : ln(eos:t:). Use your calcuiator to approxi» mate the instantaneous rate of change of f at the point :1: = 1. Do the same thing for a: : rr/cL (Note: Be sure that your caleuiator is set ingadians.) 28. The population, Flt), of China,2 in billions, can be up proxitnated by Po):1auntmnr, where f. is the number of years since the start of 2000. Ac- cording to this model. how fast was the population grow— ing at the start of 2000 and at the start of 2007? Give your answers in millions of people per year. . On October 17, 2006, in an article called “US Popula- tion Reaches 300 Million.“ the BBC reported that the U5 gains 1 person every 1 l seconds. if f(t) is the US p011? ulation in millions f. years after October 17. 2006, find f(0) anti f'(0). ' 30. (:1) Graph f(;:) z éar2 and g(:t:) : _f-'(:r;) + 3 on It 1: same set of axesthat can you say about the slope-‘1 of the tangent lines to the two graphs at the POW :1: = 0‘? :t' = 2'? Any point :t: : :L'u'? ' (b) Explain why adding a constant value. tion does not change the value of the slope “I. . graph at any point. ll-Iint: Let g(:t:) : f0") "5' and calculate the difference quotients for f and 5' 2www.unescap.orgt’slau'datafapif."indett.asp, accessed May 1. 2007. 31. Suppose Table 2.3 on page Si is continued with smaller values of it. A particular calculator gives the results in Table 2.5. (Your calculator may give slightly different results.) Comment on the values of the difference quotient in Table 2.5. In particular. why is the last value of (2h —- l)/lt zero? What do you expect the calculated value of (2" "— l}/it to be when ft 2 10””? Table 2.5 Qttcxtirmabie values of rt'fflcrcncc quotients of?" near 17 I 0 Difference quotient: (2" m 1)/h. 0.693l7l2 0.693147 0.693] 0.69 0 - Use algebra to evaluate the limits in Problems 31—37. *3 I '-‘ WE : _ 3 a. “[11 W 33. liln h—n h .IlvHU h litn M11 35. an. h—- a h h fl 0 h 2.3 THE DERIVATIVE FUNCTION 85 _ :1 + It. —— 2 . 36. —7I~— [Hintz Multiply by M4 —|— h. + ‘2 in nu- merator and denominatorJ 37. “m I. fl “7- LI Find the derivatives in Problems 38m43 algebraically. 33’ f0“) 3 5-131“, at :r: m 10 39. f(.'1,') z m3 ata: 2 w—‘3 40. g(t) 2 {,2 .H at t z m} 1.3 +5 mm = 1 41. J'(:t:) 42. g(;t:) = 1/37 at :1: m 2 43. g(:) :72. find _r]’(2) For Problems 4447, find the equation of the line tangent to the Function at the given point. 44- f0“) “"—-" 5:1?2 at :1: m 10 45. f(:t:) m a?” at :1: = w2 46. f(:t:) 2 :r at :t: = 20 47. flat) 2 1/3:2 at (l, l) in the previous section we looked at the derivative of a function at a fixed point. Now we consider what happens at a variety of points. The derivative generally takes on different values at different points and is itselfa function. First, remember that the derivative of a function at a point tells us the rate at which the value of the function is changing at that point. Geometrically, we can think of the derivative as the slope of the curve or of the tangent line at the point. Xample 1 Estimate the derivative of the function f(:t:) graphed in Figure 2.29 at :1: 2 ~2. ~1. 0. 1. 2‘ 3 4 5 Slope at tangent z f’(-l) w 2 ' Figure 2.29: Estimating the derivative graphically as the slope of the tangent line 2.3 THE DEHJVATIVE FUNCTION 91 90 Chapter Two KEY CONCEPT: THE DERIVATIVE Probiems Since in taking the limit as h. we 0, we consider values of it near. but not equal to. zero, we can cancel It giving 23. In each case, graph a smooth curve whose slope meets 33. the condition. - ,2 - ,. '2 a n q f’(ie) : lim 3'?) h + 3J1” + h : liin (3.7," + Bath. + it"). hi“ it ii-rsti (a) Everywhere positive and increasing graduaily, (b) Everywhere positive and decreasing gradually. (c) Everywhere negative and increasing gradually (be- coming less negative). ((1) Everywhere negative and decreasing graduaily (be- coming more negative). As it H—a U, the value of (3min. -i— 11.2) we 0 so f"(.1:) m Illn}](3£i:2 + 3:1:h + it?) 2 33:2. W W ompute the derivatives of power functions of the t 24. Forf(:i:) = In :L', construct tables, rounded to four deci— e Binomial Theorem to show the power rule tor a mals.'near a: = 1, :i: $2,1- '2 5, and :1: = 10. Use the tables to estimate I'LL), f’(2}, f'(5). and WHO). Then guess a general formula [or j"(:i:). The previous two examples show how to c form j'[;i:) : 21:”, when n is 2 or 3. We can use th positive integer n: . Given the numerical values shown, [ind approximate vai- ties for the derivative of f(;i:) at each of the anvalues gwcn‘ Wham 15 the we of change Of posmve' 37. A vehicle movmg along a straight road has distance f0) la L" This result is in fact valid for any real value of it. a v ‘ ‘ ‘ i I when? 15 It negative? WI:ch does) the rule at Changa or from its starting point at time t. Which of the graphs in MIL) hebm to e grembl‘ Figure 2.33 could be f’(t) for the foliowing scenarios? Exercises and Problems for Section 2.3 (Assume the scales on the vertical axes are ail the same.) Exercises . . .c. (a) A bus on a popular route, With no tral'fic (b) A car with no traffic and all green lights (c) A car in heavy traffic conditions 1. (3) Estimate f’(‘2) using the values off in the table. (b) For what values of re does f’{:i:) appear to be posi- tive? Negative? .1: 0 2 4 6 8 10 ll Her) 10 18 24 2i .J) l8 l5 q 2. Find approximate values for f'(:r) at each of the :e-valucs given in the following table. 26. Values of :i: and g(:c) are given in the table. For what value 0i:.’lil!~i g'{:e) closest to 3‘? ill l“) I (an Figure 2.33 For Exercises 3—12, graph the derivative of the given func- 38. A child inflates a balloon, admires it for a while and then lets the air out at a constant rate. If l/(t) gives the volume of the balloontat time t, then Figure 2.34 shows V’(t) as a function of t. At what time does the child: tions. 1n Exercises 13—44. find a formula for the derivative using the .3 1e. Confirm it usinu difference quotients. - ' H power m a (:1) Begin to inflate the balloon? L. Hm) 30- /\ (a) Finish inflatinI “r . r, __ ,2 J _ t gthe balioon. I3. L:(:L') : 1/3: 14. Ms») W 1/?L H : __~_.\_\_"_ :1: (0 Begin to let the airom? ii“ fl Find a formula for the derivatives of the functions in Exer :L' / 1 2\ (d) What WOUld the graph 0f WU} 100k like ifihe Ciliid . \ I“) had alternated between pinching and releasing the .‘ 15-16 usin difference uotients. - I uses g q open end of the bailoon, instead oi letting the air out at a constant rate? 15. 9(a) : 23:2 w 3 16. m,(:1:) :z 1/(ii: + 'l.) I V’(i) For Exercises 17$" sketch the graph of Mrs), anti use ti“ ....—. graph to sketch the graph of j" (:i:). 17. f(:c} : 5a: Figure 2.34 19. fire) = 21. 2 cos :i: 92 Chapter Two KEY CONCEPT: THE DERIVATIVE 39 Finure 2.35 shows a graph of voltage across an electri- 43. The population of a herd of deer is modeled by cal: capacitor as a function of time. The current is pro- portional to the derivative of the voltage; the constant of proportionality is positive. Sketch a graph of the current as a function of time. Pnlzamm44monu(maw.i) ‘) where t is measured in years from January l . voila e ‘ ‘ ' g (a) How does this population vary With time? Sketch a graph of PU) for one year. (b) Use the graph to decide when in the year the popula- tion is a maximum. What is that maximum? Is there a minimum? if so, when? - Figure 2.35 (c) Use the graph to decide when the population is growing fastest. When is it decreasing fastest? 40. in the graph of f in Figure 2.36, at which of the labeled ((1) Estimate rough” how fast fl“: popumnon ls Chung "’Dlvamcs is ing on the first of July. (it) f(:i:) greatest? (h) f(:i:) least? (c) f'(:i:)greatest? ((1) film) least? 44. The graph in Figure 2.39 shows the accumulated federal debt since 1970. Sketch the derivative of this function. What does it represent? debt (trillions ol dollars) 1i} WW year 41. Figure 2.37 is the graph of f’, the derivative ofa function 1975 1985 1995 2005 f. On what intervalts) is the function f . Figure 2.39 (a) increasing? 0)) Decreasmg? a ‘ 1 45. Draw the graph of a continuous function ’y : f(a:) that satisfies the following three conditions. Figure 2.37: Graph Of ff‘ mt f 46. Draw the graph of a continuous function y fl ICE) that satisfies the following three conditions: 42. The derivative of f is the spike function in Figure 2.38. ‘ " t‘ :r 3 What can you say about the graph of f? f (.L) > D for 1 < i < f’(:a) < 0 fora: < 1 and :i: > 3 ‘ f'(:c) : D ata: z 1 ands: :2 3 '47. if lim f(:i:) m 50 and _f'(:1:) is positive for all :i:. Whil- _ your Ell—“CEO - - U . v r . is 11111 f’ (.12)? (Assume this limit extsts.) Explain Ill—"CID answer with a picture. 48. Using a graph, explain why if f(:e) is an even funL'ElOIh- then f’(a‘) is odd. . u . ‘ - ‘ Figure 2-33 49. Using a graph, explain why if 9(a) is an odd leIlLi' then g'(a:) is even. xeyzwiiewrizw. eleven/m- 2.4 INTERPRETATIONS OF THE DERIVATIVE 93 4 lNTERPRETATIONS or THE DERIVATIVE We have seen the derivative interpreted as a slope and as a rate of change. In this section, we see other interpretations. The purpose of these examples is not to make a catalog of interpretations but to illustrate the process of obtaining them. An Alternative Notation for the Derivative So far we have used the notation j” to stand for the derivative of the function j". An alternative notation for derivatives was introduced by the German mathematician Wilheim Gottfried Leibniz (l646—17l6). If the variable 3/ depends on the variable :i:, that is, if y:[email protected] then he wrote rip/(la: for the derivative, so dym f'(:r). Leibniz‘s notation is quite suggestive if we think of the letter d in ply/do: as standing for “small difference in . . . The notation rig/ole: reminds us that the derivative is a limit of ratios of the form Difference in (ii—values Difference in ;i:-values' The notation ply/the suggests the units for the derivative: the units for y divided by the units for :c. The separate entities (ly and dry officially have no independent meaning: they are all part of one notation. in fact, a good way to view the notation sly/do: is to think of (ll/do: as a single symbol meaning “the derivative with respect to :i; of . . .." So sly/rim can be viewed as 3}), meaning “the derivative with respect to :i: of 3;.” On the other hand, many scientists and mathematicians think of (13} and do: as separate entities representing “infinitesimaliy” small differences in y and 3:, even though it is difficult to say exactly how smail “infinitesimal” is. Although not formally correct, it can be helpful to think of rig/airs as a small change in y divided by a small change in For example, récall that if s 2 If is the position of a moving object at time t, then '1} 2 f’(i;) is the velocity of the object at time .6. Writing (f. i:— ’ (It reminds us that ii is a velocity, since the notation suggests a distance, ds, over a time, (It, and we know that distance over time is velocity. Similarly, we recognize (Ly M ,, H (In: — J i as the 510136 0f the graph 0ft! 2 flat) since slope is vertical rise, nit}, over horizontal run, dry. The disadvantage of Leibniz‘s notation is that it is awkward to specify the :i:-vaiue at which we are evaluating the derivative. To specify f’(2), for example, we have to write M (In: my, ' ...
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