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Unformatted text preview: Chapter 2 Differentiation sin x
If the function f(x) = —— Were continuous at 0, then it would be simple to compute
x the limit at 0; we would just substitute x = 0. However, since the denominator is x,‘
f (x) is not even defined at x = 0. We investigate this limit numerically and graphically. iimit Nim‘ierit‘ally limit Grapi'iieally YA
‘ihnology x 1.5 —
nnecti0n ‘ 0.95885 0.97355 . * 098507
Use the TABLE function I I 099335 and the graph to 0.99833
determine 0.99998 . cosx — 1
hm ———. 0.99998 HO X 0.99833 Explain why we cannot . 0.99335
use the continuity . 0.98507 properties to determine  0.97355
this limit. . 0.95885 We can see from both the table and the graph that . sin x
lnn = l.
Xe O x The other trigonometric limit needed in Section 2.5 is . cosx  1
11m ——— = 0.
x—>0 x
This limit is investigated in the Technology Connection located in the margin, and it is investigated algebraically in Section 2.2. Exercise if Determine whether each of the following is continuous.
1. 2. 1pute ' is x,
.cally. , and Use the graphs and functions in Exercises 1—4 to answer each of the following. 2.1 5. a) Find lim+ f(x), lim_ f(x), and lim f(x).
x—>1 xel X—>1 b) Find f(1). c) Is f continuous at x = 1? (1) Find X13132 f(x).
e) Find f(—2). f) Is f continuous at x = —2? 6. a) Find x131; g(x), Xl_i>rr11_ g(x), and g(x). b) Find g(1). c) Is g continuous at x = 1? (1) Find x1_i)rn2 g(x).
e) Find g(—2). f ) Is g continuous at x = —2? 7. 21) Find h(x).
b) Find 11(1). (2) Is 11 continuous at x = 1? (1) Find hm2 h(x).
x—>—
e) Find h(—2). f) Is It continuous at x = ~2? 8. 20 Find lim t(x).
x—>1 b) Find t(1).
c) Is t continuous at x = 1?
 (1) Find Xl_i>1nz t(x). 6) Find t( ~2). f) Is t continuous at x = —2? In Exercises 9—12, use the graphs to find the limits and answer the related questions. 4—x,
2 a 9. Consider f (x) = { forx 75 1,
forx = 1. 3) Find l11111+ f(x).
b) Find lim f(x). x—>l_ c) Find 113i f(x). Limits and Continuity: Numerically and Graphically 10. 11. (1) Find f(1).
e) Is f continuous atx = 1?
f) Is f continuous at x = 2? Consider f(x) = { 4 — x2, forx as *2,
3, forx = —2. a) Find lit—n2+ f(x).
b) Find lirn _ x>~2 c) Find x1392 f(x). d) Find f(—2).
e) Is f continuous atx = —2?
f) 15 f continuous atx = 1? 81 Refer to the graph off below to determine whether each statement is true or false. y
5
4
3
2 l J__
—2 —1 a) 333+ f(x) =f(0)
b) X1351. ﬁx) =f(0)
c) 11m+ f(x) = x1315} f(x) x—>0 d) lim+ f(x) = limﬁ f(x) x—>3 x——>3
e lirn x exists.
) x—90 > f) lim f(x) exists.
xa3 g) f is continuous at x = 0.
h) f is continuous at x = 3. 82 Chapter 2 12. Refer to the graph of g below to determine
whether each statement is true or false. 6‘) X1111; g(x) = g(2) b) x113; g(x) = $0) c) x1351. g(x) = X11321. g(x)
d) g(x) exists. e) g is continuous at x = 2. 13. Refer to the graph of f below to determine whether
each statement is true or false. b) $11112 f(x) = 0 c) x3132, f(x) = X 3:512. f(x) d) Xl_i>m#2 f(x) exists. e> .132. W) = 2 * g)f(0) = 2 h) f is continuous at x = —2. i) f is continuous at x = 0. j) f is continuous at x = —1. Differentiation 14. Refer to the graph of f below to determine whether
each statement is true or false. 21) 1111217 f(x) = 3 b) 1m;+ f(x) = o c) x1331, f(x) = x131;+ ﬁx)
(1) f(x) exists. e) f (x) exists. f) 3135 f(x) = M) g) f is continuous at x = 4.
h) f is continuous at x = 0. 1) 31313 f(x) = gigs f(x) j) f is continuous at x = 2. 15. Refer to the graph of f below to determine
whether each statement is true or false. 20 X133. ﬁx) =f(0) b) x133. f(x) = f(0) c) x1_i)rg+ f (x) = XEIEL ﬁx)
d)x1#1911214r f(x) = x1351. f(x)
e) f(x) exists. f) f(x) exists. g) f is continuous at x = O.
h) f is continuous at x = 2. ether Limits and Continuity: Numerically and Graphically 16. Refer to the graph of g below to determine
whether each statement is true or false. a) x1351, g(x) = g(0)
b) x1351. g(x) = g(0) c) x1351, g(x) = xlggs g(x) d) lim g(x) exists. x—>O e) g is continuous at x = 0. '1 in: Postage Function. Postal rates are $0.37 for the first
ounce and $0.23 for each additional ounce (or fraction
thereof). Formally speaking, if x is the weight of a
letter in ounces, then p(x) is the cost of mailing the letter, where p(x) = $0.37, ifO < x S 1,
p(x) = $0.60, if 1 < x S 2,
p(x) = $0.83, if2 < x S 3, and so on, up to 13 oz (at which point postal cost
also depends on distance). The graph of p is shown below. Is p continuous at 1? at 1.5? at 2? at 2.01?
18.15 p continuous at 2.99? at 3? at 3.04? at 4? 83 Using the graph of the postage function, find each of
the following limits, if it exists. 19. x131, p(x), X11311, 1900. 13311 1200 {20; X11351. p(x). 3351, p(x), i332 p(x) 21>, lim p(x), hm p(x),xl_i)1§6p(x) ,1 « x+2.6‘ x—>2.6+ 22. lim p(x) x—>3 23. lim p(x) x—>3.4 Taxicab Fares. In New York City, taxicabs charge pas~
sengers $2.00 for entering a cab and then $0.30 for
each one—fifth of a mile (or fraction thereof) that the
cab travels. (There are additional charges for slow
traffic and idle times, but these are not considered in
this problem.) Formally speaking, if x is the distance
traveled in miles, then C(x) is the cost of the taxi
fare, where C(x) = $2.00, ifx = o,
C(x) = $2.30, ifO < x < 0.2, C(x) = $2.60, no.2 s x < 0.4,
C(x) = $2.90, if0.4 s x < 0.6, and so on. The graph of C is shown below y'f‘ o—o
3.4— 3.2 — 0—0
3 _ 2.8 — 2.6 — 0W0 2.4 — y : C(x) 2.2 — 2.—  l I  i >
0.2 0.4 0.6 0.8 1 x 24. 15 C continuous at 0.1? at 0.2? at 0.25? at 0.267?
25. Is C continuous at 2.3? at 2.5? at 2.6? at 3.0? Using the graph of the taxicab fare function, find each
of the following limits, if it exists. x—>l 4 . 26. li _C(x),xgr1r/14+ C(x),xli)rri/4 C(x) v 27.x11g12w C(x), hm C(x),xl'1£1(1)‘2C(x) x—>0.2+
28. Flirt? C(x), X£1316+ C(x), $135.6 C(x)
29. lim C(x) 30. lim C(x)
x—>.0.5 x~>0.4 84 Chapter 2  Differentiation t In a certain habitat, the deer
population as a function of time (measured in years) is given in the graph of p below.
yA
13 — H
115— y=pm
12 — e——=—O 9—0
11.5 —
11 — 0—0 9—0 10.5
log—G
I  l I 0.5 1 1.5 2 >
t ' 31. Identify each point where the population func tion is discontinuous. 32. At each point where the function is not continu
ous, identify an event that might have occurred
in the population to cause the discontinuity. 33. Find thpp p(t). “' 34. Find blip? p(t). , , The population of bears in a cer
tain region is given by the graph of p below. Time t
is measured in months. yA
35 ‘— o————o
34 — y = ptt) o——°
33 — o————o
32— o———o 9—0
31 — e—o
30;—=0
l   _ i >
0.2 0.4 0.6 0.8 1 t 35. Identify each point where the population func—
tion is discontinuous. 36. At each point where the function is not continu~
ous, identify an event that might have occurred
in the population to cause the discontinuity. 37. Find lim p(t). t—>0.6Jr 38. Find $1316 p(t). ‘ ln psychology one often takes a cer
tain amount of time t to learn a task. Suppose that the
goal is to do a task perfectly and that you are practicing
the ability to master it. After a certain time period, what (W tW tW is known to psychologists as an “I’ve got it!” experience 1
occurs, and you are able to perform the task perfectly 39. At what point do you think the “I’ve got it!” experi
ence happens on the learning curve below? 40. Why do you think the curve below is constant for
inputs t Z 20?
N(t)A
H 100  o—————
8 90 —
E 80 —
3, % 7o —
E E _
LE 8 60
o ,_. 50 —
5% w—
,0
E: 30 —
:3
Z 20 :/j
10 —
_ ____________L9
10 20 30 t Time (in hours) Using the graph above, find each of the following lim
its, if it exists.
41. [1136+ N(t),t11)12r6w N(t),t1i)r121O N(t) 42. [Egg N(t), [Egg N(t),tl_1_)r§10 N(t) 43. Is N continuous at 20? at 30?
44. Is N continuous at 10? at 26? 45. Discuss three ways in which a function may not be
continuous at a point a. Draw graphs to illustrate
your discussion. Use the Continuity Properties C1— C5 to justify that the
function is continuous. Then give the limit using the
fact that the function is continuous. 46. f(x) = x2 + 5x — 5; 1mg f(x)
x%
47. f(x) = 3x3 + 2x2 —~ 9x + 4; lim1 f(x)
x——> x x
48. = ,f 75 1; 1'
g(x) x  l orx x—1>n11x — 1
49 () x2 + 9x — 7
. x =—————,
’3 x+ 2
2
+ _.
forxsté ~2‘,limx——9x—Z
x91 x+ 2
7T 7T
50. tanx,for——<x<——; lim tanx
2 2 x—>7r/4 51.cotx,for0<x<7r; lim cotx x%7r/3
7T 7T ,
52. secx, for —— <x < —; hm secx
2 2 raw/6 53. csc x, for O < x < 77; lim csc (x)
xew 4 )t be
ate It the
he 2.2 Limits: Aigebraicaliy 85 OBJECTIVE I Find limits using algebraic methods. the limit. (542}ﬂx) = VXZ + 2x + 4 (Hint: f is a composition 1. x3 + x2 — x w 1
L/ of functions); 59. Xirnl ——————————x3 + 1
55. f(x) = Vsinx,for0 <x < 77; lim Vsinx 1 _ COSX
‘ Wm 60. lim ————
56. g(x) = sinzx; lim sinzx ’HO 3‘
I“? 77/4 1 tan h
57. g(x) = cos(2x + 3n);xhr7rT1/6 cos(2x + 37r) 61. Illino h
62 1, sinx — x
"it? in": n fanny n '* U N;  x1310 x3 '
In Exercises 58—63, use a grapher to determine the 63' hm gal}:
limits. Make a table for each and draw the graph. 1160 h
5814 x2+3x—4O
' 13% x2 + 4x — 2.2 Limits: Algebraically Using Limit Principles If a function is continuous at a, we can substitute to find Examing 4 Find lim sz — 3x + 2.
x—>0 ,goiniion Using the Continuity Principles, we have shown
that polynomials such as x2 — 3x + 2 are continuous for all
values of x. When we restrict x to values for which x2 — 3x + 2 is nonnegative, it follows from Principle C5 that sz — 3x + 2 is continuous. Since x2 — 3x + 2 is
nonnegative when x = 0, we can substitute to find the limit: lir%Vx2—3x+2=VO2—3O+2
X%
=\/i. aaaaiata 2 Find lim sin(x2 ~ 4).
2092 505mm By the Continuity Principles, we know that x2 — 4 is continuous. There
fore, the composition sin (x2 — 4) is also continuous since the sine function is con~
tinuous. To find the limit, we simply substitute x = 2,. lim2 sin(x2 — 4) = sin(22 — 4) = sin(O) = 0.
X'ﬁ Using the fact that many of the usual functions from algebra and trigonome
try are continuous, we can compute many limits by simply evaluating the func
tion at the point in question. It is also possible to use Limit Principles to compute
limits. These principles can be used when we are uncertain of the continuity of
the function. l the , and .15. 2.2 Find the limit using the algebraic method. Verify using
the numerical or graphical method. 1. lim (x2 —~ 3) x—>0 x
5. lim (2x + 5)
x—93 . x2 — 25
7. hm i
x+~5 x + 5 2. lim (x2 + 4)
x—>1 4. lim :4:
x—>O x 6. lim (5 — 3x)
x—>4 2
— l6
8. lim £————
X~>4 x+ 4
—2
10. lim —
x—>—5 x
2.. _
I x x 20
12. hm —— x944 x + 4 Find the limit. Use the algebraic method. 13. 11m5 V3 x2 — 17
X% 15. lim (x + sin x) x—>7r/4
, l+sinx
17.l1m ——.—
x—>01—smx
. 1
19.)}1_I§12x_2
3 2—4 +2
21. mill—i—
x—>27x ~5x+3
4 2+5 —7
22. lim 353;
xa13x —2x+ 1
2+ —6
23.11m¥———
x—>2 x *4
2
“16
24.lim x xa‘txz—x—IZ 25. Ilirno (6x2 + 6x11 + 2112)
1% 26. lim (10x + 511) 11%0 . —2x — h
27' 113%) x2(x + 102 28. lim —_5—
h~>0 x(x + h) tan X 29. lim
x—>0 x 30. lim x csc x
x~>0 14. 111112 ‘\/x2 + 5
X6 16. 11m (cosx + tanx)
x—>7r/6 . 1 + cosx
18.11m —————
26—20 cosx 20 l' 1
. 1m ——————
x—>1(x — l)2 2.2 Limits: Algebraically 91 Find the limit, if it exists. ‘ sinxsin h , sinxcos h — sinx
31. hm — 32. llm ——
h~90 h 1160 h
, x2+3x .x2—2x4
33. 11m 2——.—. 34. 11m 2—
x—>0x — 2x x—>Ox + 3x
. XV; . x + 2c2
35. xlgggc + x2 36. lino xv;
x — 2
37. lim —————
X_’2 x2 — x — 2
‘ x2 — 1
38. km
X~>~1 x + 1
39 1‘ x2 _ 9
. 1m
x—>3 2x — 6
3 2 + 5 — 2
40.11111 ———x x H72 x2 — 3x — 10 '5. N )7. . '. A 4 3‘32 ,. r
l t H u l 1.1 l; 4,7 t. 0 ll Further Use 011110 114811? Failure. In Section 2.1, we
discussed how to use the TABLE feature to find limits.
Consider . Vl+x—1
l1m——————.
x—>0 x Input—output tables for this function are shown below.
The table on the left uses TblStart =  l and ATbl =
0.5. By using smaller and smaller step values and begin—
ning closer to O, we can refine the table and obtain a better estimate of the limit. On the right is an input—
output table with TblStart = ~0.03 and ATbl = 0.01. 0.503807 . 0.585786 0.502525
0 ERROR * 0.501256
0.5 0.449490 ERROR
1 ' 0.414214 0.498756
1.5 0.387426 0.497525
2 0.366025 0.490305 92 Chapter 2  Differentiation It appears that the limit is 0.5. We can verify this
by graphing \/1+x—1 y: x and tracing the curve near x = O, zooming in on that portion of the curve. 41.
y = W~ 1
x 42.
43.
Limit=0.5 44.
45.
We see that 46
\/ 1 + — 1 ' lim ————x—" = 0.5.
Xﬂfo x 47
This can be verified algebraically. (Hint: Multiply by 1, ' . \/ 1 + x + 1
usm —————. g m + 1 48 OBJECTIVES I Compute an average rate of change.
I Find a simplified difference quotient. feature and start with ATbl = 0.1. Then move to 0.01,
0.001, and 0.0001. When you think you know the limit, graph and use the TRACE feature to verify your assertion. Then try to verify algebraically. y‘ In Exercises 41—48, find the limit. Use the TABLE 2~
. a 4
11m————
a~>'Z\/a2+5—3
, W—l
lim———
x—>1 x—l V3—x—\/§ lim
x—>0 X
hm ______.V4+x— V‘i—X x—>O x
x — W
x ~ 1
hm ___.__V7+2x—W x—>0 x . 2—\/§
lim—— x—>4 4—x . 7— V49—x2
l1m———— x—>O x lim
x—>1 2.3 Average Rates of Change Let’s say that a car travels 110 mi in 2 hr. Its average rate of
change (speed) is 110 mi/Z hr, or 55 mi/hr (55 mph). On the
other hand, suppose that you are on the freeway and you
begin accelerating. Glancing at the speedometer, you see that
at that instant your instantaneous rate of change is 55 mph.
These are two quite different concepts. The first you are prob— ably familiar with. The second involves ideas of limits and
calculus. To understand instantaneous rate of change, we first use this section to de—
velop a solid understanding of average rate of change. Photosynthesis is the conversion of light energy to chemical energy that is
stored in glucose or other organic compounds.1 In the process, oxygen is produced.
The following graph approximates the amount of oxygen a plant produces by pho
tosynthesis. Time 0 is taken to be 8 AM. The rate of photosynthesis depends on a
number of factors, including the amount of light and temperature. 1N. A. Campbell and].B. Reece, Biology, 6th ed. (Benjamin Cummings, New York 2002). 98 Chapter 2  Differentiation Eiercisf‘e’éetflgéH For the functions in each of Exercises 1—12, (a) find a) The average growth rate of a typical boy during
a simplified form of the difference quotient and his first year of life. (Your answer should be in (b) complete the following table. pounds per month.)
b) The average growth rate of a typical boy during his second year of life. c) The average growth rate of a typical boy during
his third year of life. d) The average growth rate of a typical boy during
his first 3 yr of life. e) When does the graph indicate that a boy’s
growth rate is greatest during his first 3 yr of life? 1. f(x) = 7x2 2. f(x) = 5x2
3. f(x) = —7x2 4. f(x) = —5x2
5. f(x) = 7x3 6. f(x) = 5x3 fl
5 4
. = — 8. = —
7 ﬁx) x xx) X
9. f(x) = ~2x + 5 10. f(x) = 2x + 3
11 f(x) = x2 _ x 12 f(x) = X2 + X a, a as: Earthy. Use the graph in Exercise 13 to
' ' estimate: a) The average growth rate of a typical boy during
his first 9 mo of life. (Your answer should be in pounds per month.) 13. Gaby. The median weight of boys b) The average growth rate of a typical boy during
IS given in the graph below. Use the graph to his ﬁrst 6 m0 of life
estimate:2 ' .
c) The average growth rate of a typical boy during his first 3 mo of life. Jig 30 v tW d) Based on your answers in parts (a)—(c) and
3 25 the graph, estimate what the average growth
g rate of a typical boy should be the first few
3;) 20 " weeks of his life. g 10 < "U E 5 ._ Age in months 2Centers for Disease Control. Developed by the National Center
for Health Statistics in collaboration with the National Center
for Chronic Disease Prevention and Health Promotion (2000). 13 to during
:1 be in during
during 1nd
wth
3W 15. Growth oj‘a Baby. Use the following graph to estimate:
M
26
V)
g 255
8
D; 25
1:1
'2 24.5
a» 2.
OJ
3
a 23.5
g
z 23 
22.5 )1
16 17 18 x
Age in months 12 13 14 15 a) The average growth rate of a typical boy between
ages 12 mo and 18 mo. (Your answer should be
in pounds per month.) b) The average growth rate of a typical boy between
ages 12 mo and 14 mo. c) The average growth rate of a typical boy between
ages 12 mo and 13 mo. tW (1) Based on your answers in parts (a)—(c) and the
graph, estimate the average growth rate of a typi—
cal boy when he is 12 mo old. 16. ilfittpei’ciim‘iﬁ Dining (111 Himss. The temperature T,
in degrees Fahrenheit, of a patient during an illness
is shown in the following graph, where t is the
time, in days. TA
(6 102.5)
A 103— ’
6 102— (4,102) (8,102)
"U
.E
z 101— (3,101)
H
:3
g 100_ (9.100)
g (2, 99,5) (11, 98.6)
a 99—
98.6 (1, 98.6) (10, 98.6)
1 1 1 1 1 _1_ 1 1 1 1 1 >
1234567891011t Time (in days) a) Find the average rate of change of T as t changes
from 1 to 10. Using this rate of change, would
you know that the person was sick? b) Find the rate of change of T with respect to t, as
t changes from 1 to 2; from 2 to 3; from 3 to 4;
from 4 to 5; from 5 to 6; from 6 to 7; from 7 to
8; from 8 to 9; from 9 to 10; from 10 to 11. tW 17. UN 18. 19. 20. 2.3 Average Rates of Change 99 c) When do you think the temperature began to
rise? reached its peak? began to subside? was
back to normal? d) Explain your answers to part (c). Memmy, The total number of words M(t) that a
person can memorize in time t, in minutes, is
shown in the following graph. M Number of words memorized >.
0 8 16 24 323640 1 Time (in minutes) a) Find the average rate of change of M as t
changes from O to 8; from 8 to 16; from 16 to
24; from 24 to 32; from 32 to 36. b) Why do the average rates of change become 0
after 24 min? Average Velocity. A car is at a distance 5, in miles,
from its starting point in t hours, given by s(t) = 10t2. 21) Find 5(2) and 5(5).
b) Find 5(5) — 5(2). What does this represent? c) Find the average rate of change of distance with
respect to time as t changes from t1 = 2 to
t; = 5. This is known as average velocity,
or speed. Average Velocity. An object is dropped from a
certain height. It is known that it will fall a
distance 5, in feet, in t seconds, given by s(t) = 16t2. a) How far will the object fall in 3 see? b) How far will the object fall in 5 see? c) What is the average rate of change of distance
with respect to time during the period from 3 to
5 see? This is also the average velocity. Gas Mileage. At the beginning of a trip, the odome
ter on a car reads 30,680 and the car has a full tank 100 Chapter 2 Differentiation of gas. At the end of the trip. the odometer reads
30,970. It takes 15 gal of gas to refill the tank. a) What is the average rate of change of the number
of miles with respect to the number of gallons? b) What is the average rate of consumption (that is,
the rate of change of the number of miles with respect to the number of gallons)? each of two countries at time t, in years. U)
0
O Population (in thousands) (3, 200) of
100 (1, 125) v I  I 
0 1 2 3 4 Time (in years) HY 2) Find the average rate of change of each
population (the number of people in the population) with respect to time t as t changes
from 0 to 4. This is often called an average growth rate. tW b) If the calculation in part (a) were the only one made, would we detect the fact that the populations were growing differently? Explain. c) Find the average rates of change of each population as t changes from 0 to 1; from 1 to 2; from 2 to 3; from 3 to 4. UN d) For which population does the statement “the
population grew by 125 million each year”
convey the least information about what really took place? Explain. fr to 2000 is approximated in the graph shown below. Use this graph to answer Exercises 22 and 23.3 3California Department of Fish and Game. . , . 7.3mm; The two curves shown in the
following figure describe the number o...
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