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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

4.1

172πcm/sec,172πcm/sec, or approximately 0.0044 cm/sec

4.2

500ft/sec500ft/sec

4.3

110rad/sec110rad/sec

4.4

−0.61ft/sec−0.61ft/sec

4.5

L(x)=2+112(x8);L(x)=2+112(x8); 2.00833

4.6

L(x)=x+π2L(x)=x+π2

4.7

L(x)=1+4xL(x)=1+4x

4.8

dy=2xex2dxdy=2xex2dx

4.9

dy=1.6,dy=1.6, Δy=1.64Δy=1.64

4.10

The volume measurement is accurate to within 21.6cm3.21.6cm3.

4.11

7.6%

4.12

x=23,x=23, x=1x=1

4.13

The absolute maximum is 33 and it occurs at x=4.x=4. The absolute minimum is −1−1 and it occurs at x=2.x=2.

4.14

c=2c=2

4.15

522522 sec

4.16

ff has a local minimum at −2−2 and a local maximum at 3.3.

4.17

ff has no local extrema because ff does not change sign at x=1.x=1.

4.18

ff is concave up over the interval (,12)(,12) and concave down over the interval (12,)(12,)

4.19

ff has a local maximum at −2−2 and a local minimum at 3.3.

4.20

Both limits are 3.3. The line y=3y=3 is a horizontal asymptote.

4.21

Let ε>0.ε>0. Let N=1ε.N=1ε. Therefore, for all x>N,x>N, we have

|31x23|=1x2<1N2=ε|31x23|=1x2<1N2=ε

Therefore, limx(31/x2)=3.limx(31/x2)=3.

4.22

Let M>0.M>0. Let N=M3.N=M3. Then, for all x>N,x>N, we have

3x2>3N2=3(M3)22=3M3=M3x2>3N2=3(M3)22=3M3=M

4.23

4.24

3535

4.25

±3±3

4.26

limxf(x)=35,limxf(x)=35, limxf(x)=−2limxf(x)=−2

4.29

y=32xy=32x

4.30

The function ff has a cusp at (0,5)(0,5) limx0f(x)=,limx0f(x)=, limx0+f(x)=.limx0+f(x)=. For end behavior, limx±f(x)=.limx±f(x)=.

4.31

The maximum area is 5000ft2.5000ft2.

4.32

V(x)=x(202x)(302x).V(x)=x(202x)(302x). The domain is [0,10].[0,10].

4.33

T(x)=x6+(15x)2+12.5T(x)=x6+(15x)2+12.5

4.34

The company should charge $75$75 per car per day.

4.35

A(x)=4x1x2.A(x)=4x1x2. The domain of consideration is [0,1].[0,1].

4.36

c(x)=259.2x+0.2x2c(x)=259.2x+0.2x2 dollars

4.37

11

4.38

00

4.39

limx0+cosx=1.limx0+cosx=1. Therefore, we cannot apply L’Hôpital’s rule. The limit of the quotient is

4.40

11

4.41

00

4.42

ee

4.43

11

4.44

The function 2x2x grows faster than x100.x100.

4.45

x10.33333333,x20.347222222x10.33333333,x20.347222222

4.46

x1=2,x2=1.75x1=2,x2=1.75

4.47

x11.842105263,x21.772826920x11.842105263,x21.772826920

4.48

x1=6,x2=8,x3=263,x4=809,x5=24227;x*=9x1=6,x2=8,x3=263,x4=809,x5=24227;x*=9

4.49

cosx+Ccosx+C

4.50

ddx(xsinx+cosx+C)=sinx+xcosxsinx=xcosxddx(xsinx+cosx+C)=sinx+xcosxsinx=xcosx

4.51

x453x3+12x27x+Cx453x3+12x27x+C

4.52

y=3x+5y=3x+5

4.53

2.93sec,64.5ft2.93sec,64.5ft

Section 4.1 Exercises

1.

88

3.

13101310

5.

2323 ft/sec

7.

The distance is decreasing at 390mi/h.390mi/h.

9.

The distance between them shrinks at a rate of 132013101.5mph.132013101.5mph.

11.

9292 ft/sec

13.

It grows at a rate 4949 ft/sec

15.

The distance is increasing at (13526)26(13526)26 ft/sec

17.

5656 m/sec

19.

240π240π m2/sec

21.

12π12π cm

23.

The area is increasing at a rate (33)8ft2/sec.(33)8ft2/sec.

25.

The depth of the water decreases at 128125π128125π ft/min.

27.

The volume is decreasing at a rate of (25π)16ft3/min.(25π)16ft3/min.

29.

The water flows out at rate (2π)5m3/min.(2π)5m3/min.

31.

3232 m/sec

33.

2519π2519π ft/min

35.

245π245π ft/min

37.

The angle decreases at 4001681rad/sec.4001681rad/sec.

39.

100π/min100π/min

41.

The angle is changing at a rate of 1125rad/sec.1125rad/sec.

43.

The distance is increasing at a rate of 62.5062.50 ft/sec.

45.

The distance is decreasing at a rate of 11.9911.99 ft/sec.

Section 4.2 Exercises

47.

f(a)=0f(a)=0

49.

The linear approximation exact when y=f(x)y=f(x) is linear or constant.

51.

L(x)=1214(x2)L(x)=1214(x2)

53.

L(x)=1L(x)=1

55.

L(x)=0L(x)=0

57.

0.02

59.

1.99968751.9996875

61.

0.0015930.001593

63.

1;1; error, ~0.00005~0.00005

65.

0.97;0.97; error, ~0.0006~0.0006

67.

31600;31600; error, ~4.632×10−7~4.632×10−7

69.

dy=(cosxxsinx)dxdy=(cosxxsinx)dx

71.

dy=(x22x2(x1)2)dxdy=(x22x2(x1)2)dx

73.

dy=1(x+1)2dx,dy=1(x+1)2dx, 116116

75.

dy=9x2+12x22(x+1)3/2dx,dy=9x2+12x22(x+1)3/2dx, −0.1−0.1

77.

dy=(3x2+21x2)dx,dy=(3x2+21x2)dx, 0.20.2

79.

12xdx12xdx

81.

4πr2dr4πr2dr

83.

−1.2πcm3−1.2πcm3

85.

−100−100 ft3

Section 4.3 Exercises

91.

Answers may vary

93.

Answers will vary

95.

No; answers will vary

97.

Since the absolute maximum is the function (output) value rather than the x value, the answer is no; answers will vary

99.

When a=0a=0

101.

Absolute minimum at 3; Absolute maximum at −2.2; local minima at −2, 1; local maxima at −1, 2

103.

Absolute minima at −2, 2; absolute maxima at −2.5, 2.5; local minimum at 0; local maxima at −1, 1

105.

Answers may vary.

107.

Answers may vary.

109.

x=1x=1

111.

None

113.

x=0x=0

115.

None

117.

x=−1,1x=−1,1

119.

Absolute maximum: x=4,x=4, y=332;y=332; absolute minimum: x=1,x=1, y=3y=3

121.

Absolute minimum: x=12,x=12, y=4y=4

123.

Absolute maximum: x=2π,x=2π, y=2π;y=2π; absolute minimum: x=0,x=0, y=0y=0

125.

Absolute maximum: x=−3;x=−3; absolute minimum: −1x1,−1x1, y=2y=2

127.

Absolute maximum: x=π4,x=π4, y=2;y=2; absolute minimum: x=5π4,x=5π4, y=2y=2

129.

Absolute minimum: x=−2,x=−2, y=1y=1

131.

Absolute minimum: x=−3,x=−3, y=−135;y=−135; local maximum: x=0,x=0, y=0;y=0; local minimum: x=1,x=1, y=−7y=−7

133.

Local maximum: x=122,x=122, y=342;y=342; local minimum: x=1+22,x=1+22, y=3+42y=3+42

135.

Absolute maximum: x=22,x=22, y=32;y=32; absolute minimum: x=22,x=22, y=32y=32

137.

Local maximum: x=−2,x=−2, y=59;y=59; local minimum: x=1,x=1, y=−130y=−130

139.

Absolute maximum: x=0,x=0, y=1;y=1; absolute minimum: x=−2,2,x=−2,2, y=0y=0

141.

h=924549m,h=924549m, t=30049st=30049s

143.

The global minimum was in 1848, when no gold was produced.

145.

Absolute minima: x=0,x=0, x=2,x=2, y=1;y=1; local maximum at x=1,x=1, y=2y=2

147.

No maxima/minima if aa is odd, minimum at x=1x=1 if aa is even

Section 4.4 Exercises

149.

One example is f(x)=|x|+3,−2x2f(x)=|x|+3,−2x2

151.

Yes, but the Mean Value Theorem still does not apply

153.

(,0),(0,)(,0),(0,)

155.

(,−2),(2,)(,−2),(2,)

157.

2 points

159.

5 points

161.

c=233c=233

163.

c=12,1,32c=12,1,32

165.

c=1c=1

167.

Not differentiable

169.

Not differentiable

171.

Yes

173.

The Mean Value Theorem does not apply since the function is discontinuous at x=14,34,54,74.x=14,34,54,74.

175.

Yes

177.

The Mean Value Theorem does not apply; discontinuous at x=0.x=0.

179.

Yes

181.

The Mean Value Theorem does not apply; not differentiable at x=0.x=0.

183.

b=±2cb=±2c

185.

c=±1πcos−1(π2),c=±1πcos−1(π2), c=±0.1533c=±0.1533

187.

The Mean Value Theorem does not apply.

189.

12c+12c3=5212880;12c+12c3=5212880; c=3.133,5.867c=3.133,5.867

191.

Yes

193.

It is constant.

Section 4.5 Exercises

195.

It is not a local maximum/minimum because ff does not change sign

197.

No

199.

False; for example, y=x.y=x.

201.

Increasing for −2<x<−1−2<x<−1 and x>2;x>2; decreasing for x<−2x<−2 and −1<x<2−1<x<2

203.

Decreasing for x<1,x<1, increasing for x>1x>1

205.

Decreasing for −2<x<−1−2<x<−1 and 1<x<2;1<x<2; increasing for −1<x<1−1<x<1 and x<−2x<−2 and x>2x>2

207.

a. Increasing over −2<x<−1,0<x<1,x>2,−2<x<−1,0<x<1,x>2, decreasing over x<−2,x<−2, −1<x<0,1<x<2;−1<x<0,1<x<2; b. maxima at x=−1x=−1 and x=1,x=1, minima at x=−2x=−2 and x=0x=0 and x=2x=2

209.

a. Increasing over x>0,x>0, decreasing over x<0;x<0; b. Minimum at x=0x=0

211.

Concave up on all x,x, no inflection points

213.

Concave up on all x,x, no inflection points

215.

Concave up for x<0x<0 and x>1,x>1, concave down for 0<x<1,0<x<1, inflection points at x=0x=0 and x=1x=1

217.

Answers will vary

219.

Answers will vary

221.

a. Increasing over π2<x<π2,π2<x<π2, decreasing over x<π2,x>π2x<π2,x>π2 b. Local maximum at x=π2;x=π2; local minimum at x=π2x=π2

223.

a. Concave up for x>43,x>43, concave down for x<43x<43 b. Inflection point at x=43x=43

225.

a. Increasing over x<0x<0 and x>4,x>4, decreasing over 0<x<40<x<4 b. Maximum at x=0,x=0, minimum at x=4x=4 c. Concave up for x>2,x>2, concave down for x<2x<2 d. Infection point at x=2x=2

227.

a. Increasing over x<0x<0 and x>6011,x>6011, decreasing over 0<x<60110<x<6011 b. Minimum at x=6011x=6011 c. Concave down for x<5411,x<5411, concave up for x>5411x>5411 d. Inflection point at x=5411x=5411

229.

a. Increasing over x>12,x>12, decreasing over x<12x<12 b. Minimum at x=12x=12 c. Concave up for all xx d. No inflection points

231.

a. Increases over 14<x<34,14<x<34, decreases over x>34x>34 and x<14x<14 b. Minimum at x=14,x=14, maximum at x=34x=34 c. Concave up for 34<x<14,34<x<14, concave down for x<34x<34 and x>14x>14 d. Inflection points at x=34,x=14x=34,x=14

233.

a. Increasing for all xx b. No local minimum or maximum c. Concave up for x>0,x>0, concave down for x<0x<0 d. Inflection point at x=0x=0

235.

a. Increasing for all xx where defined b. No local minima or maxima c. Concave up for x<1;x<1; concave down for x>1x>1 d. No inflection points in domain

237.

a. Increasing over π4<x<3π4,π4<x<3π4, decreasing over x>3π4,x<π4x>3π4,x<π4 b. Minimum at x=π4,x=π4, maximum at x=3π4x=3π4 c. Concave up for π2<x<π2,π2<x<π2, concave down for x<π2,x>π2x<π2,x>π2 d. Infection points at x=±π2x=±π2

239.

a. Increasing over x>4,x>4, decreasing over 0<x<40<x<4 b. Minimum at x=4x=4 c. Concave up for 0<x<823,0<x<823, concave down for x>823x>823 d. Inflection point at x=823x=823

241.

f>0,f>0,f<0f>0,f>0,f<0

243.

f>0,f<0,f<0f>0,f<0,f<0

245.

f>0,f>0,f>0f>0,f>0,f>0

247.

True, by the Mean Value Theorem

249.

True, examine derivative

Section 4.6 Exercises

251.

x=1x=1

253.

x=−1,x=2x=−1,x=2

255.

x=0x=0

257.

Yes, there is a vertical asymptote

259.

Yes, there is vertical asymptote

261.

00

263.

265.

1717

267.

−2−2

269.

−4−4

271.

Horizontal: none, vertical: x=0x=0

273.

Horizontal: none, vertical: x=±2x=±2

275.

Horizontal: none, vertical: none

277.

Horizontal: y=0,y=0, vertical: x=±1x=±1

279.

Horizontal: y=0,y=0, vertical: x=0x=0 and x=−1x=−1

281.

Horizontal: y=1,y=1, vertical: x=1x=1

283.

Horizontal: none, vertical: none

285.

Answers will vary, for example: y=2xx1y=2xx1

287.

Answers will vary, for example: y=4xx+1y=4xx+1

289.

y=0y=0

291.

293.

y=3y=3

295.


The function starts in the third quadrant, increases to pass through (−1, 0), increases to a maximum and y intercept at 4, decreases to touch (2, 0), and then increases to (4, 20).
297.


An upward-facing parabola with minimum between x = 0 and x = −1 with y intercept between 0 and 1.
299.


This graph starts at (−2, 4) and decreases in a convex way to (1, 0). Then the graph starts again at (4, 0) and increases in a convex way to (6, 3).
301.


This graph has vertical asymptote at x = 0. The first part of the function occurs in the second and third quadrants and starts in the third quadrant just below (−2π, 0), increases and passes through the x axis at −3π/2, reaches a maximum and then decreases through the x axis at −π/2 before approaching the asymptote. On the other side of the asymptote, the function starts in the first quadrant, decreases quickly to pass through π/2, decreases to a local minimum and then increases through (3π/2, 0) before staying just above (2π, 0).
303.


This graph has vertical asymptotes at x = ±π/2. The graph is symmetric about the y axis, so describing the left hand side will be sufficient. The function starts at (−π, 0) and decreases quickly to the asymptote. Then it starts on the other side of the asymptote in the second quadrant and decreases to the the origin.
305.


This function starts at (−2π, 0), increases to near (−3π/2, 25), decreases through (−π, 0), achieves a local minimum and then increases through the origin. On the other side of the origin, the graph is the same but flipped, that is, it is congruent to the other half by a rotation of 180 degrees.
307.

The degree of Q(x)Q(x) must be greater than the degree of P(x)P(x).

309.

limx1f(x)=-∞andlimx1g(x)=limx1f(x)=-∞andlimx1g(x)=

Section 4.7 Exercises

311.

The critical points can be the minima, maxima, or neither.

313.

False; y=x2y=x2 has a minimum only

315.

h=623h=623 in.

317.

11

319.

100ft by100ft100ft by100ft

321.

40ft by40ft40ft by40ft

323.

19.73ft.19.73ft.

325.

84bpm84bpm

327.

T(θ)=40θ3v+40cosθvT(θ)=40θ3v+40cosθv

329.

v=bav=ba

331.

approximately 34.02mph34.02mph

333.

44

335.

00

337.

Maximal: x=5,y=5;x=5,y=5; minimal: x=0,y=10x=0,y=10 and y=0,x=10y=0,x=10

339.

Maximal: x=1,y=9;x=1,y=9; minimal: none

341.

4π334π33

343.

66

345.

r=2,h=4r=2,h=4

347.

(2,1)(2,1)

349.

(0.8351,0.6974)(0.8351,0.6974)

351.

A=20r2r212πr2A=20r2r212πr2

353.

C(x)=5x2+32xC(x)=5x2+32x

355.

P(x)=(50x)(800+25x50)P(x)=(50x)(800+25x50)

Section 4.8 Exercises

357.

359.

12a12a

361.

1nan11nan1

363.

Cannot apply directly; use logarithms

365.

Cannot apply directly; rewrite as limx0x3limx0x3

367.

66

369.

−2−2

371.

−1−1

373.

nn

375.

1212

377.

1212

379.

11

381.

1616

383.

11

385.

00

387.

00

389.

−1−1

391.

393.

00

395.

1e1e

397.

00

399.

11

401.

00

403.

tan(1)tan(1)

405.

22

Section 4.9 Exercises

407.

F(xn)=xnxn3+2xn+13xn2+2F(xn)=xnxn3+2xn+13xn2+2

409.

F(xn)=xnexnexnF(xn)=xnexnexn

411.

|c|>0.5|c|>0.5 fails, |c|0.5|c|0.5 works

413.

c=1f(xn)c=1f(xn)

415.

a. x1=1225,x2=312625;x1=1225,x2=312625; b. x1=−4,x2=−40x1=−4,x2=−40

417.

a. x1=1.291,x2=0.8801;x1=1.291,x2=0.8801; b. x1=0.7071,x2=1.189x1=0.7071,x2=1.189

419.

a. x1=2625,x2=1224625;x1=2625,x2=1224625; b. x1=4,x2=18x1=4,x2=18

421.

a. x1=610,x2=610;x1=610,x2=610; b. x1=2,x2=2x1=2,x2=2

423.

3.1623or3.16233.1623or3.1623

425.

0,−1or10,−1or1

427.

00

429.

0.5188or1.29060.5188or1.2906

431.

00

433.

4.4934.493

435.

0.159,3.1460.159,3.146

437.

We need ff to be twice continuously differentiable.

439.

x=0x=0

441.

x=−1x=−1

443.

x=5.619x=5.619

445.

x=−1.326x=−1.326

447.

There is no solution to the equation.

449.

It enters a cycle.

451.

00

453.

−0.3513−0.3513

455.

Newton: 1111 iterations, secant: 1616 iterations

457.

Newton: three iterations, secant: six iterations

459.

Newton: five iterations, secant: eight iterations

461.

E=4.071E=4.071

463.

4.394%4.394%

Section 4.10 Exercises

465.

F(x)=15x2+4x+3F(x)=15x2+4x+3

467.

F(x)=2xex+x2exF(x)=2xex+x2ex

469.

F(x)=exF(x)=ex

471.

F(x)=exx3cos(x)+CF(x)=exx3cos(x)+C

473.

F(x)=x22x2cos(2x)+CF(x)=x22x2cos(2x)+C

475.

F(x)=12x2+4x3+CF(x)=12x2+4x3+C

477.

F(x)=25(x)5+CF(x)=25(x)5+C

479.

F(x)=32x2/3+CF(x)=32x2/3+C

481.

F(x)=x+tan(x)+CF(x)=x+tan(x)+C

483.

F(x)=13sin3(x)+CF(x)=13sin3(x)+C

485.

F(x)=12cot(x)1x+CF(x)=12cot(x)1x+C

487.

F(x)=secx4cscx+CF(x)=secx4cscx+C

489.

F(x)=18e−4xcosx+CF(x)=18e−4xcosx+C

491.

cosx+Ccosx+C

493.

3x2x+C3x2x+C

495.

83x3/2+45x5/4+C83x3/2+45x5/4+C

497.

14x2x12x2+C14x2x12x2+C

499.

f(x)=12x2+32f(x)=12x2+32

501.

f(x)=sinx+tanx+1f(x)=sinx+tanx+1

503.

f(x)=16x32x+136f(x)=16x32x+136

505.

Answers may vary; one possible answer is f(x)=exf(x)=ex

507.

Answers may vary; one possible answer is f(x)=sinxf(x)=sinx

509.

5.8675.867 sec

511.

7.3337.333 sec

513.

13.7513.75 ft/sec2

515.

F(x)=13x3+2xF(x)=13x3+2x

517.

F(x)=x2cosx+1F(x)=x2cosx+1

519.

F(x)=1(x+1)+1F(x)=1(x+1)+1

521.

True

523.

False

Chapter Review Exercises

525.

True, by Mean Value Theorem

527.

True

529.

Increasing: (−2,0)(4,),(−2,0)(4,), decreasing: (,−2)(0,4)(,−2)(0,4)

531.

L(x)=1716+12(1+4π)(x14)L(x)=1716+12(1+4π)(x14)

533.

Critical point: x=3π4,x=3π4, absolute minimum: x=0,x=0, absolute maximum: x=πx=π

535.

Increasing: (−1,0)(3,),(−1,0)(3,), decreasing: (,−1)(0,3),(,−1)(0,3), concave up: (,13(213))(13(2+13),),(,13(213))(13(2+13),), concave down: (13(213),13(2+13))(13(213),13(2+13))

537.

Increasing: (14,),(14,), decreasing: (0,14),(0,14), concave up: (0,),(0,), concave down: nowhere

539.

33

541.

1π1π

543.

x1=−1,x2=−1x1=−1,x2=−1

545.

F(x)=2x3/23+1x+CF(x)=2x3/23+1x+C

547.


This graph has vertical asymptotes at x = 0 and x = −1. The first part of the function occurs in the third quadrant with a horizontal asymptote at y = 0. The function decreases quickly from near (−5, 0) to near the vertical asymptote (−1, ∞). On the other side of the asymptote, the function is roughly U-shaped and pointed down in the third quadrant between x = −1 and x = 0 with maximum near (−0.4, −6). On the other side of the x = 0 asympotote, the function decreases from its vertical asymptote near (0, ∞) and to approach the horizontal asymptote y = 0.


Inflection points: none; critical points: x=13;x=13; zeros: none; vertical asymptotes: x=−1,x=−1, x=0;x=0; horizontal asymptote: y=0y=0

549.

The height is decreasing at a rate of 0.1250.125 m/sec

551.

x=abx=ab feet

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