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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 aJsdndFOntheg 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) [HM0 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 "—‘ﬁf‘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. llIint: 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 5131“, 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 iirsti (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 :evalucs
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 .‘ 1516 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 233 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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 Spring '10
 Machaut
 Derivative

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