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  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Key Terms
    13. Key Equations
    14. Key Concepts
    15. Chapter Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index

Checkpoint

3.1

1414

3.2

66

3.3

f(1)=5f(1)=5

3.4

−32−32 ft/s

3.5

P(3.25)=20>0;P(3.25)=20>0; raise prices

3.6

f(x)=2xf(x)=2x

3.7

(0,+)(0,+)

3.8

a=6a=6 and b=−9b=−9

3.9

f(x)=2f(x)=2

3.10

a(t)=6ta(t)=6t

3.11

0

3.12

4x34x3

3.13

f(x)=7x6f(x)=7x6

3.14

f(x)=6x212x.f(x)=6x212x.

3.15

y=12x23y=12x23

3.16

j(x)=10x4(4x2+x)+(8x+1)(2x5)=56x6+12x5.j(x)=10x4(4x2+x)+(8x+1)(2x5)=56x6+12x5.

3.17

k(x)=13(4x3)2.k(x)=13(4x3)2.

3.18

g(x)=−7x−8.g(x)=−7x−8.

3.19

3f(x)2g(x).3f(x)2g(x).

3.20

5858

3.21

−4.4−4.4

3.22

left to right

3.23

3,300

3.24

$2

3.25

f(x)=cos2xsin2xf(x)=cos2xsin2x

3.26

cosx+xsinxcos2xcosx+xsinxcos2x

3.27

t=π3,t=2π3t=π3,t=2π3

3.28

f(x)=csc2xf(x)=csc2x

3.29

f(x)=2sec2x+3csc2xf(x)=2sec2x+3csc2x

3.30

4343

3.31

cosxcosx

3.32

cosxcosx

3.33

v(5π6)=3<0v(5π6)=3<0 and a(5π6)=−1<0.a(5π6)=−1<0. The block is speeding up.

3.34

h(x)=4(2x3+2x1)3(6x2+2)=8(3x2+1)(2x3+2x1)3h(x)=4(2x3+2x1)3(6x2+2)=8(3x2+1)(2x3+2x1)3

3.35

y=−48x88y=−48x88

3.36

h(x)=7cos(7x+2)h(x)=7cos(7x+2)

3.37

h(x)=34x(2x+3)4h(x)=34x(2x+3)4

3.38

h(x)=18x2sin5(x3)cos(x3)h(x)=18x2sin5(x3)cos(x3)

3.39

a(t)=−16sin(4t)a(t)=−16sin(4t)

3.40

2828

3.41

dydx=−3x2sin(x3)dydx=−3x2sin(x3)

3.42

g(x)=1(x+2)2g(x)=1(x+2)2

3.43

g(x)=15x4/5g(x)=15x4/5

3.44

s(t)=(2t+1)1/2s(t)=(2t+1)1/2

3.45

g(x)=11+x2g(x)=11+x2

3.46

h(x)=−36x9x2h(x)=−36x9x2

3.47

y=xy=x

3.48

dydx=520x4sec2y2ydydx=520x4sec2y2y

3.49

y=53x163y=53x163

3.50

h(x)=e2x+2xe2xh(x)=e2x+2xe2x

3.51

996

3.52

f(x)=153x+2f(x)=153x+2

3.53

9ln(3)9ln(3)

3.54

dydx=xx(1+lnx)dydx=xx(1+lnx)

3.55

y=π(tanx)π1sec2xy=π(tanx)π1sec2x

Section 3.1 Exercises

1.

44

3.

8.58.5

5.

3434

7.

0.20.2

9.

0.250.25

11.

a. −4−4 b. y=34xy=34x

13.

a. 33 b. y=3x1y=3x1

15.

a. −79−79 b. y=−79x+143y=−79x+143

17.

a. 1212 b. y=12x+14y=12x+14

19.

a. −2−2 b. y=−2x10y=−2x10

21.

55

23.

1313

25.

1414

27.

1414

29.

−3−3

31.

a. (i)5.100000,(i)5.100000, (ii)5.010000,(ii)5.010000, (iii)5.001000,(iii)5.001000, (iv)5.000100,(iv)5.000100, (v)5.000010,(v)5.000010, (vi)5.000001,(vi)5.000001, (vii)4.900000,(vii)4.900000, (viii)4.990000,(viii)4.990000, (ix)4.999000,(ix)4.999000, (x)4.999900,(x)4.999900, (xi)4.999990,(xi)4.999990, (x)4.999999(x)4.999999 b. mtan=5mtan=5 c. y=5x+3y=5x+3

33.

a. (i)4.8771,(i)4.8771, (ii)4.9875(iii)4.9988,(ii)4.9875(iii)4.9988, (iv)4.9999,(iv)4.9999, (v)4.9999,(v)4.9999, (vi)4.9999(vi)4.9999 b. mtan=5mtan=5 c. y=5x+10y=5x+10

35.

a. 13;13; b. (i)0.3(i)0.3 m/s, (ii)0.3(ii)0.3 m/s, (iii)0.3(iii)0.3 m/s, (iv)0.3(iv)0.3 m/s; c. 0.3=130.3=13 m/s

37.

a. 2(h2+6h+12);2(h2+6h+12); b. (i)25.22(i)25.22 m/s, (ii)24.12(ii)24.12 m/s, (iii)24.01(iii)24.01 m/s, (iv)24(iv)24 m/s; c. 2424 m/s

39.

a. 1.25;1.25; b. 0.50.5

41.

limx0x1/30x0=limx01x2/3=limx0x1/30x0=limx01x2/3=

43.

limx111x1=01=limx1+x1x1limx111x1=01=limx1+x1x1

45.

a. (i)61.7244(i)61.7244 ft/s, (ii)61.0725(ii)61.0725 ft/s (iii)61.0072(iii)61.0072 ft/s (iv)61.0007(iv)61.0007 ft/s b. At 44 seconds the race car is traveling at a rate/velocity of 6161 ft/s.

47.

a. The vehicle represented by f(t),f(t), because it has traveled 22 feet, whereas g(t)g(t) has traveled 11 foot. b. The velocity of f(t)f(t) is constant at 11 ft/s, while the velocity of g(t)g(t) is approximately 22 ft/s. c. The vehicle represented by g(t),g(t), with a velocity of approximately 44 ft/s. d. Both have traveled 44 feet in 44 seconds.

49.

a.

The function starts in the third quadrant, passes through the x axis at x = −3, increases to a maximum around y = 20, decreases and passes through the x axis at x = 1, continues decreasing to a minimum around y = −13, and then increases through the x axis at x = 4, after which it continues increasing.


b. a1.361,2.694a1.361,2.694

51.

a. N(x)=x30N(x)=x30 b. 3.33.3 gallons. When the vehicle travels 100100 miles, it has used 3.33.3 gallons of gas. c. 130.130. The rate of gas consumption in gallons per mile that the vehicle is achieving after having traveled 100100 miles.

53.

a.

The function starts in the second quadrant and gently decreases, touches the origin, and then it increases gently.


b. −0.028,−0.16,0.16,0.028−0.028,−0.16,0.16,0.028

Section 3.2 Exercises

55.

−3−3

57.

8x8x

59.

12x12x

61.

−9x2−9x2

63.

−12x3/2−12x3/2

65.


The function starts in the third quadrant and increases to touch the origin, then decreases to a minimum at (2, −16), before increasing through the x axis at x = 3, after which it continues increasing.
67.


The function starts at (−3, 0), increases to a maximum at (−1.5, 1), decreases through the origin and to a minimum at (1.5, −1), and then increases to the x axis at x = 3.
69.

f(x)=3x2+2,a=2f(x)=3x2+2,a=2

71.

f(x)=x4,a=2f(x)=x4,a=2

73.

f(x)=ex,a=0f(x)=ex,a=0

75.

a.

The function is linear at y = 3 until it reaches (1, 3), at which point it increases as a line with slope 3.


b. limh133hlimh1+3hhlimh133hlimh1+3hh

77.

a.

The function starts in the third quadrant as a straight line and passes through the origin with slope 2; then at (1, 2) it decreases convexly as 2/x.


b. limh12hhlimh1+2x+h2xh.limh12hhlimh1+2x+h2xh.

79.

a. x=1,x=1, b. x=2x=2

81.

00

83.

2x32x3

85.

f(x)=6x+2f(x)=6x+2

The function f(x) is graphed as an upward facing parabola with y intercept 4. The function f’(x) is graphed as a straight line with y intercept 2 and slope 6.
87.

f(x)=1(2x)3/2f(x)=1(2x)3/2

The function f(x) is in the first quadrant and has asymptotes at x = 0 and y = 0. The function f’(x) is in the fourth quadrant and has asymptotes at x = 0 and y = 0.
89.

f(x)=3x2f(x)=3x2

The function f(x) starts is the graph of the cubic function shifted up by 1. The function f’(x) is the graph of a parabola that is slightly steeper than the normal squared function.
91.

a. Average rate at which customers spent on concessions in thousands per customer. b. Rate (in thousands per customer) at which xx customers spent money on concessions in thousands per customer.

93.

a. Average grade received on the test with an average study time between two values. b. Rate (in percentage points per hour) at which the grade on the test increased or decreased for a given average study time of xx hours.

95.

a. Average change of atmospheric pressure between two different altitudes. b. Rate (torr per foot) at which atmospheric pressure is increasing or decreasing at xx feet.

97.

a. The rate (in degrees per foot) at which temperature is increasing or decreasing for a given height x.x. b. The rate of change of temperature as altitude changes at 10001000 feet is −0.1−0.1 degrees per foot.

99.

a. The rate at which the number of people who have come down with the flu is changing tt weeks after the initial outbreak. b. The rate is increasing sharply up to the third week, at which point it slows down and then becomes constant.

101.
Time (seconds) h(t)(m/s)h(t)(m/s)
00 22
11 22
22 5.55.5
33 10.510.5
44 9.59.5
55 77
103.

G(t)=2.858t+0.0857G(t)=2.858t+0.0857

This graph has the points (0, 0), (1, 2), (2, 4), (3, 13), (4, 25), and (5, 32). There is a quadratic line fit to the points with y intercept near 0.


This graph has a straight line with y intercept near 0 and slope slightly less than 3.
105.

H(t)=0,G(t)=2.858andf(t)=1.222t+5.912H(t)=0,G(t)=2.858andf(t)=1.222t+5.912 represent the acceleration of the rocket, with units of meters per second squared (m/s2).(m/s2).

Section 3.3 Exercises

107.

f(x)=15x21f(x)=15x21

109.

f(x)=32x3+18xf(x)=32x3+18x

111.

f(x)=270x4+39(x+1)2f(x)=270x4+39(x+1)2

113.

f(x)=−5x2f(x)=−5x2

115.

f(x)=4x4+2x22xx4f(x)=4x4+2x22xx4

117.

f(x)=x218x+64(x27x+1)2f(x)=x218x+64(x27x+1)2

119.


The graph y is a slightly curving line with y intercept at 1. The line T(x) is straight with y intercept 3 and slope 1/2.


T(x)=12x+3T(x)=12x+3

121.


The graph y is a two crescents with the crescent in the third quadrant sloping gently from (−3, −1) to (−1, −5) and the other crescent sloping more sharply from (0.8, −5) to (3, 0.2). The straight line T(x) is drawn through (0, −5) with slope 4.


T(x)=4x5T(x)=4x5

123.

h(x)=3x2f(x)+x3f(x)h(x)=3x2f(x)+x3f(x)

125.

h(x)=3f(x)(g(x)+2)3f(x)g(x)(g(x)+2)2h(x)=3f(x)(g(x)+2)3f(x)g(x)(g(x)+2)2

127.

169169

129.

Undefined

131.

a. 2,2, b. does not exist, c. 2.52.5

133.

a. 23, b. y=23x28y=23x28

The graph is a slightly deformed cubic function passing through the origin. The tangent line is drawn through (0, −28) with slope 23.
135.

a. 3, b. y=3x+2y=3x+2

The graph starts in the third quadrant, increases quickly and passes through the x axis near −0.9, then increases at a lower rate, passes through (0, 2), increases to (1, 5), and then decreases quickly and passes through the x axis near 1.2.
137.

y=−7x3y=−7x3

139.

y=−5x+7y=−5x+7

141.

y=32x+152y=32x+152

143.

y=−3x2+9x1y=−3x2+9x1

145.

1212112121 or 0.0992 ft/s

147.

a. −2t42t3+200t+50(t3+50)2−2t42t3+200t+50(t3+50)2 b. −0.02395−0.02395 mg/L-hr, −0.01344 mg/L-hr, −0.003566 mg/L-hr, −0.001579 mg/L-hr c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.

149.

a. F(d)=−2Gm1m2d3F(d)=−2Gm1m2d3 b. −1.33×10−7−1.33×10−7 N/m

Section 3.4 Exercises

151.

a. v(t)=6t230t+36,a(t)=12t30;v(t)=6t230t+36,a(t)=12t30; b. speeds up (2,2.5)(3,),(2,2.5)(3,), slows down (0,2)(2.5,3)(0,2)(2.5,3)

153.

a. 464ft/s2464ft/s2 b. −32ft/s2−32ft/s2

155.

a. 5 ft/s b. 9 ft/s

157.

a. 84 ft/s, −84 ft/s b. 84 ft/s c. 258s258s d. −32ft/s2−32ft/s2 in both cases e. 18(25+965)s18(25+965)s f. −4965ft/s−4965ft/s

159.

a. Velocity is positive on (0,1.5)(6,7),(0,1.5)(6,7), negative on (1.5,2)(5,6),(1.5,2)(5,6), and zero on (2,5).(2,5). b.

The graph is a straight line from (0, 2) to (2, −1), then is discontinuous with a straight line from (2, 0) to (5, 0), and then is discontinuous with a straight line from (5, −4) to (7, 4).


c. Acceleration is positive on (5,7),(5,7), negative on (0,2),(0,2), and zero on (2,5).(2,5). d. The object is speeding up on (6,7)(1.5,2)(6,7)(1.5,2) and slowing down on (0,1.5)(5,6).(0,1.5)(5,6).

161.

a. R(x)=10x0.001x2R(x)=10x0.001x2 b. R(x)=100.002xR(x)=100.002x c. $6 per item, $0 per item

163.

a. C(x)=65C(x)=65 b. R(x)=143x0.03x2,R(x)=1430.06xR(x)=143x0.03x2,R(x)=1430.06x c. 83,−97.83,−97. At a production level of 1000 cordless drills, revenue is increasing at a rate of $83 per drill; at a production level of 4000 cordless drills, revenue is decreasing at a rate of $97 per drill. d. P(x)=−0.03x2+78x75000,P(x)=−0.06x+78P(x)=−0.03x2+78x75000,P(x)=−0.06x+78 e. 18,−162.18,−162. At a production level of 1000 cordless drills, profit is increasing at a rate of $18 per drill; at a production level of 4000 cordless drills, profit is decreasing at a rate of $162 per drill.

165.

a. N(t)=3000(−4t2+400(t2+100)2)N(t)=3000(−4t2+400(t2+100)2) b. 120,0,−14.4,−9.6120,0,−14.4,−9.6 c. The bacteria population increases from time 0 to 10 hours; afterwards, the bacteria population decreases. d. 0,−6,0.384,0.432.0,−6,0.384,0.432. The rate at which the bacteria is increasing is decreasing during the first 10 hours. Afterwards, the bacteria population is decreasing at a decreasing rate.

167.

a. P(t)=0.03983+0.4280P(t)=0.03983+0.4280 b. P(t)=0.03983.P(t)=0.03983. The population is increasing. c. P(t)=0.P(t)=0. The rate at which the population is increasing is constant.

169.

a. p(t)=−0.6071x2+0.4357x0.3571p(t)=−0.6071x2+0.4357x0.3571 b. p(t)=−1.214x+0.4357.p(t)=−1.214x+0.4357. This is the velocity of the sensor. c. p(t)=−1.214.p(t)=−1.214. This is the acceleration of the sensor; it is a constant acceleration downward.

171.

a.

The graph is a straight line drawn through the origin with slope 1/2.


b. f(x)=a.f(x)=a. The more increase in prey, the more growth for predators. c. f(x)=0.f(x)=0. As the amount of prey increases, the rate at which the predator population growth increases is constant. d. This equation assumes that if there is more prey, the predator is able to increase consumption linearly. This assumption is unphysical because we would expect there to be some saturation point at which there is too much prey for the predator to consume adequately.

173.

a.

The graph increases from the origin quickly at first and then slowly to (10, 0.4).


b. f(x)=2axn2(n2+x2)2.f(x)=2axn2(n2+x2)2. When the amount of prey increases, the predator growth increases. c. f(x)=2an2(n23x2)(n2+x2)3.f(x)=2an2(n23x2)(n2+x2)3. When the amount of prey is extremely small, the rate at which predator growth is increasing is increasing, but when the amount of prey reaches above a certain threshold, the rate at which predator growth is increasing begins to decrease. d. At lower levels of prey, the prey is more easily able to avoid detection by the predator, so fewer prey individuals are consumed, resulting in less predator growth.

Section 3.5 Exercises

175.

dydx=2xsecxtanxdydx=2xsecxtanx

177.

dydx=2xcotxx2csc2xdydx=2xcotxx2csc2x

179.

dydx=xsecxtanxsecxx2dydx=xsecxtanxsecxx2

181.

dydx=(1sinx)(1sinx)cosx(x+cosx)dydx=(1sinx)(1sinx)cosx(x+cosx)

183.

dydx=2csc2x(1+cotx)2dydx=2csc2x(1+cotx)2

185.

y=xy=x

The graph shows negative sin(x) and the straight line T(x) with slope −1 and y intercept 0.
187.

y=x+23π2y=x+23π2

The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 – 3π)/2.
189.

y=xy=x

The graph shows the function as starting at (−1, 3), decreasing to the origin, continuing to slowly decrease to about (1, −0.5), at which point it decreases very quickly.
191.

3cosxxsinx3cosxxsinx

193.

12sinx12sinx

195.

2cscx(csc2x+cot2x)2cscx(csc2x+cot2x)

197.

(2n+1)π4,wherenis an integer(2n+1)π4,wherenis an integer

199.

(π4,1),(3π4,−1)(π4,1),(3π4,−1)

201.

a=0,b=3a=0,b=3

203.

y=5cos(x),y=5cos(x), increasing on (0,π2),(3π2,5π2),(0,π2),(3π2,5π2), and (7π2,12)(7π2,12)

209.

3sinx3sinx

211.

5cosx5cosx

213.

720x75tan(x)sec3(x)tan3(x)sec(x)720x75tan(x)sec3(x)tan3(x)sec(x)

Section 3.6 Exercises

215.

18u2·7=18(7x4)2·718u2·7=18(7x4)2·7

217.

sinu·−18=sin(x8)·−18sinu·−18=sin(x8)·−18

219.

8x2424u+3=4x124x224x+38x2424u+3=4x124x224x+3

221.

a. u=3x2+1;u=3x2+1; b. 18x(3x2+1)218x(3x2+1)2

223.

a. f(u)=u7,u=x7+7x;f(u)=u7,u=x7+7x; b. 7(x7+7x)6·(177x2)7(x7+7x)6·(177x2)

225.

a. f(u)=cscu,u=πx+1;f(u)=cscu,u=πx+1; b. πcsc(πx+1)·cot(πx+1)πcsc(πx+1)·cot(πx+1)

227.

a. f(u)=−6u−3,u=sinx,f(u)=−6u−3,u=sinx, b. 18sin−4x·cosx18sin−4x·cosx

229.

4(52x)34(52x)3

231.

6(2x3x2+6x+1)2(3x2x+3)6(2x3x2+6x+1)2(3x2x+3)

233.

−3(tanx+sinx)−4·(sec2x+cosx)−3(tanx+sinx)−4·(sec2x+cosx)

235.

−7cos(cos7x)·sin7x−7cos(cos7x)·sin7x

237.

−12cot2(4x+1)·csc2(4x+1)−12cot2(4x+1)·csc2(4x+1)

239.

10341034

241.

y=−12xy=−12x

243.

x=±6x=±6

245.

10

247.

1818

249.

−4−4

251.

−12−12

253.

a. 200343200343 m/s, b. 60024016002401 m/s2, c. The train is slowing down since velocity and acceleration have opposite signs.

255.

a. C(x)=0.0003x20.04x+3C(x)=0.0003x20.04x+3 b. dCdt=100·(0.0003x20.04x+3)dCdt=100·(0.0003x20.04x+3) c. Approximately $90,300 per week

257.

a. dSdt=8πr2(t+1)3dSdt=8πr2(t+1)3 b. The volume is decreasing at a rate of π36π36 ft3/min.

259.

~2.3~2.3 ft/hr

Section 3.7 Exercises

261.

a.

A curved line starting at (−3, 0) and passing through (−2, 1) and (1, 2). There is another curved line that is symmetric with this about the line x = y. That is, it starts at (0, −3) and passes through (1, −2) and (2, 1).


b. (f−1)(1)~2(f−1)(1)~2

263.

a.

A quarter circle starting at (0, 4) and ending at (4, 0).


b. (f−1)(1)~1/3(f−1)(1)~1/3

265.

a. 6, b. x=f−1(y)=(y+32)1/3,x=f−1(y)=(y+32)1/3, c. 1616

267.

a. 1,1, b. x=f−1(y)=sin−1y,x=f−1(y)=sin−1y, c. 11

269.

1515

271.

1313

273.

11

275.

a. 4,4, b. y=4xy=4x

277.

a. 196,196, b. y=113x+1813y=113x+1813

279.

2x1x42x1x4

281.

−11x2−11x2

283.

3(1+tan−1x)21+x23(1+tan−1x)21+x2

285.

−1(1+x2)(tan−1x)2−1(1+x2)(tan−1x)2

287.

x(5x2)4x2x(5x2)4x2

289.

−1−1

291.

1212

293.

110110

295.

a. v(t)=11+t2v(t)=11+t2 b. a(t)=−2t(1+t2)2a(t)=−2t(1+t2)2 c. (a)0.2,0.06,0.03;(b)0.16,−0.028,−0.0088(a)0.2,0.06,0.03;(b)0.16,−0.028,−0.0088 d. The hockey puck is decelerating/slowing down at 2, 4, and 6 seconds.

297.

−0.0168−0.0168 radians per foot

299.

a. dθdx=10100+x2401600+x2dθdx=10100+x2401600+x2 b. 18325,9340,424745,018325,9340,424745,0 c. As a person moves farther away from the screen, the viewing angle is increasing, which implies that as he or she moves farther away, his or her screen vision is widening. d. 5412905,3500,19829945,913605412905,3500,19829945,91360 e. As the person moves beyond 20 feet from the screen, the viewing angle is decreasing. The optimal distance the person should stand for maximizing the viewing angle is 20 feet.

Section 3.8 Exercises

301.

dydx=−2xydydx=−2xy

303.

dydx=x3yy2xdydx=x3yy2x

305.

dydx=yy2x+4x+4xdydx=yy2x+4x+4x

307.

dydx=y2cos(xy)2ysin(xy)xycosxydydx=y2cos(xy)2ysin(xy)xycosxy

309.

dydx=−3x2yy3x3+3xy2dydx=−3x2yy3x3+3xy2

311.


The graph has a crescent in each of the four quadrants. There is a straight line marked T(x) with slope −1/2 and y intercept 2.


y=−12x+2y=−12x+2

313.


The graph has two curves, one in the first quadrant and one in the fourth quadrant. They are symmetric about the x axis. The curve in the first quadrant goes from (0.3, 5) to (1.5, 3.5) to (5, 4). There is a straight line marked T(x) with slope 1/(π + 12) and y intercept −(3π + 38)/(π + 12).


y=1π+12x3π+38π+12y=1π+12x3π+38π+12

315.


The graph starts in the third quadrant near (−5, 0), remains near 0 until x = −4, at which point it decreases until it reaches near (0, −5). There is an asymptote at x = 0. The graph begins again near (0, 5) decreases to (1, 0) and then increases a little bit before decreasing to be near (5, 0). There is a straight line marked T(x) that coincides with y = 0.


y=0y=0

317.

a. y=x+2y=x+2 b. (3,−1)(3,−1)

319.

a. (±7,0)(±7,0) b. −2−2 c. They are parallel since the slope is the same at both intercepts.

321.

y=x+1y=x+1

323.

a. −0.5926−0.5926 b. When $81 is spent on labor and $16 is spent on capital, the amount spent on capital is decreasing by $0.5926 per $1 spent on labor.

325.

−8−8

327.

−2.67−2.67

329.

y=11x2y=11x2

Section 3.9 Exercises

331.

2xex+x2ex2xex+x2ex

333.

ex3lnx(3x2lnx+x2)ex3lnx(3x2lnx+x2)

335.

4(ex+ex)24(ex+ex)2

337.

24x+2·ln2+8x24x+2·ln2+8x

339.

πxπ1·πx+xπ·πxlnππxπ1·πx+xπ·πxlnπ

341.

52(5x7)52(5x7)

343.

tanxln10tanxln10

345.

2x·ln2·log37x24+2x·2xln7ln32x·ln2·log37x24+2x·2xln7ln3

347.

(sin2x)4x[4·ln(sin2x)+8x·cot2x](sin2x)4x[4·ln(sin2x)+8x·cot2x]

349.

xlog2x·2lnxxln2xlog2x·2lnxxln2

351.

xcotx·[csc2x·lnx+cotxx]xcotx·[csc2x·lnx+cotxx]

353.

x−1/2(x2+3)2/3(3x4)4·[−12x+4x3(x2+3)+123x4]x−1/2(x2+3)2/3(3x4)4·[−12x+4x3(x2+3)+123x4]

355.


The function starts at (−3, 0), decreases slightly and then increases through the origin and increases to (1.25, 10). There is a straight line marked T(x) with slope −1/(5 + 5 ln 5) and y intercept 5 + 1/(5 + 5 ln 5).


y=−15+5ln5x+(5+15+5ln5)y=−15+5ln5x+(5+15+5ln5)

357.

a. x=e~2.718x=e~2.718 b. (e,),(0,e)(e,),(0,e)

359.

a. P=500,000(1.05)tP=500,000(1.05)t individuals b. P(t)=24395·(1.05)tP(t)=24395·(1.05)t individuals per year c. 39,73739,737 individuals per year

361.

a. At the beginning of 1960 there were 5.3 thousand cases of the disease in New York City. At the beginning of 1963 there were approximately 723 cases of the disease in the United States. b. At the beginning of 1960 the number of cases of the disease was decreasing at rate of −4.611−4.611 thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of −0.2808−0.2808 thousand per year.

363.

p=35741(1.045)tp=35741(1.045)t

365.
Years since 1790 PP
0 69.25
10 107.5
20 167.0
30 259.4
40 402.8
50 625.5
60 971.4
70 1508.5

Chapter Review Exercises

367.

False.

369.

False

371.

12x+412x+4

373.

9x2+8x39x2+8x3

375.

esinxcosxesinxcosx

377.

xsec2(x)+2xcos(x)+tan(x)x2sin(x)xsec2(x)+2xcos(x)+tan(x)x2sin(x)

379.

14(x1x2+sin−1(x))14(x1x2+sin−1(x))

381.

cosx·(lnx+1)xln(x)sinxcosx·(lnx+1)xln(x)sinx

383.

4x(ln4)2+2sinx+4xcosxx2sinx4x(ln4)2+2sinx+4xcosxx2sinx

385.

T=(2+e)x2T=(2+e)x2

387.


The function is the straight line y = −4 until x = 0, at which point it becomes a straight line starting at the origin with slope 2. There is no value assigned for this function at x = 0.
389.

w(3)=2.9π6.w(3)=2.9π6. At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.

391.

−7.5.−7.5. The wind speed is decreasing at a rate of 7.5 mph/hr

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