the essentials of calculus - wordpress.com...221 19. a. do: xt 4, ra: yd0 220.a. do: all reals ra:...

219

3 2 24 5 3 5 151.341, 2.899 or 1.341, 2.900

1.678, 14.937 or 1.679, 14.938

6.662, 84.837 or 6.662, 84.838

x x x x x

7 5047.1 524.0517.2 544.6087.3 565.6777.4 587.264

x f x

xy

THE ESSENTIALS OF CALCULUS ANSWERS TO SELECTED EXERCISES

Assignment 1-1 page 2 1. 4

3m 3. 0m 6. 2y x 9. 4x 10. 3 0x y

11. 323 2y x 16. not collinear 18. 3 24 5 0 5,0,1x x x x

19. 22. 24. , 5 , 0,1 27. 1.732x Assignment 1-2 page 7 1. b. 11 d. 3 5x 3. 2 1x x 4. Do: 1x , Ra: 0y 8. not a function (circle) 10. a. 23y x 11. c. 1, 0x x

14. c. 48x e. 4 15. 1 12 2y x 20. a. 2x b. 2x c. 1

2x d. 2x

23. 0, 2x 24. 27. 31. 32. 34. .890 or .891x 35. Do: all reals, Ra: 4.875y 37. Do: 2.645,2.645 or 2.646,2.646 Ra: 0,2.645 or 0,2.646 Assignment 1-3 page 12 6. x-int. 3,0 , no y-int. 8. x-int. 2,0 , y-int. 0,1 12. origin 16. x-axis

25. no x-int., 26. 0,0 , 1,1 , 1, 1

y-int. 0,3 , 27. 4, 3 , 3,4

y-axis symm. 29. 18k

xy

x

y

x

y

xy

220

36. x-int. 4,0 , 38. .666 5.333 or .667 5.333x x

y-int. 0, 2 , 45. 3 122 xy x-axis symm. Assignment 1-4 page 14 1. 2 2. DNE 5. 2 6. or DNE 10. or DNE 13. 0 16. 1,2x 17. 3,4x 18. b. 0 d. 0x 19. b. or DNE g. 1 i. 1x 20. d. 3 e. 1 21. c. or DNE e. , 3 , 3,3 , 3, 22. b. DNE

22. c. 2 f. every integer 26. 1 27. 12 29. 0 31. a. 2 b. 2 c. DNE

31. d. ,0 , 0, 33. 1x 38. 41. 43. Assignment 1-5 page 19 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Problem 8 11. a. 2

33 1y x b. 123 1y x

11. c. 3y x d. x = -1 12. 1,6 13. 3,0

14. a. 2 2 2x x b. 24 4x x c. 5 15. 2, 3x 16. 2x 17. 1 3 5 1f x x

18. a.Do: all reals Ra: 0y b. x-int: 2,0 y-int: (0 , 4) e.

c. y-axis symmetry d. even

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

221

19. a. Do: 4x , Ra: 0y 20. a. Do: all reals Ra: 2y b. x-int. 4,0 , y-int. 0, 2 b. x-int: 3 2 ,0 y-int: (0 , -2) c. no symmetry d. neither c. y-axis symmetry d. even e. e. 21. 22. 23. 24. a. Do: 2x Ra: all reals 25. a. Do: 1.581,1.581 , Ra: 2.236,0

b. x-int: (0,0) y-int: (0 , 0) b. x-int. 1.581,0 , y-int. 0, 2.236 c. origin symmetry d. odd c. y-axis symm. d. even e. e. 26. a. Do: all reals 27. , 1.137 , .394 , .742 or 395 , .742. Ra: 5 750y . 28. 2 333 or 1x x. 29. 3, -6.75x b. x-int: 884 0. , , (1.884 , 0) 30. odd 31. Neither 32. -9 y-int: (0 , -5) 33. -4 34. DNE 35. -3 c. no symmetry d. neither 36. DNE 37. 2 38. -2 e. 39. 0 40. 2 41. x = 1

42. x = 2 43. 44. -1 45. DNE

46. 3 47. 0

48. x = 2

x

y

x

y

x y

x

y

x y

x

y

xy

222

Assignment 2-1 page 22 1. 2 2. 5 4. 3 5. 110 9. or DNE 11.

14 12. 4 13. 1 15. 2

16. 1 19. 1 20. 6 24. b. .166 or .167 26. DNE 30. 0 (removable), 1 (nonrem.)x x 31. 3 (removable)x 34. 3 (rem.)x 36. every even integer (nonrem.)

39. every integral multiple of 4 (nonremovable) 40. a = 2 41. 5 12 2

,a b

47. = 3 c 48. = 3 c 49. = 3 c 50. 7y x

Assignment 2-2 page 25 3. 0x (even) 5. 2x (both odd) 7. 8. DNE 9. 10. 1 12. DNE 13. 15. a. 2 b. 2 18. 23 20.

43 21. or DNE 22. 1

25. 2 27. 2x (nonremovable), 2x (removable) 29. 2x (nonremovable) 31. 2x (nonremovable) 32. 103a 33.

2 23 33 2 or 7 4y x y x

35. Do: 1x , Ra: 1y 37. 1.414, 2x 41. 3 42. DNE 50. 2x (removable), 2x (nonremovable) 51. nonremovable discontinuity at every integral multiple of 12

Assignment 2-3 page 30 1. Do: all reals 2. Do: 0,1x 3. Do: , 1 , 1, EB: Horiz. Asympt. Hole: (1,0 ) (even) HA: 1y 0y VA: x = 0 (odd)

4. x-int: 2,0 (odd) 6. Do: 2x 9. x-int: 0,0 (odd)

1,0 (even) VA: 2x (odd) y-int 0,0

y-int: 0,2 x-int: 2,0 Start Pt: 121,

EB: like 3y x y-int: 0, 1 Symm: origin EB: HA 1y

11. Do: 1x 12. Do: 3, 2x VA: 1x (odd) VA: 3x (odd) Hole: 1,0 (odd) Holes: 1

52, , 3,0 (odd)

y-int: 0, 1 x-int: 12

,0 (even), y-int: 19

0,

EB: like 2y x EB: HA 4y

15. 14

17. 14

20. 5 22. DNE 24. 2 25. or DNE 26. 37

30. 0

2

3

2 1 33

red

x xf x

x

21 1

1redx x

f xx

223

1.449,1.898 or 1.449,1.899 ,

3.449, 7.898 or 3.449, 7.899

Assignment 2-4 page 33

1. 3f x 2. 2f x x 3. 211

f xx

4. 23 12f t t

5. 0f x 6. 2 4f 7. 2 4f 8. 19

3f 9. 1f is undefined

10. 2 1y x 11. 23dydx

x 12. 4 , 4 16f x x f 13. 4 16f

15. 32

18. 52x 21. 32x 22.

23x

Assignment 2-5 page 37 1. 0y 3. 23g x x 5. 4 3f t t 6. 2 2

3 5f x x 7. 1 6f

9. 1 3h 10. 2 12f 11. 21

yx

12. 382f x xx

14. 23 1g x x 15. 3

1 22

y x 17. 1f t , 0t 19. 32

3y

x

20. 32

9y

x 22. 14

4f 23. 12

2f 25. 5 3 0f

28. 3 4 2y x 30. 12

2 1y x 31. 2y 32. 1x 33. 32

0,x

35. ARC = 84 36. 3 81f 38. 1 1y 39. x = 1 (sharp turn) 40. 1x (vert. asymp.), x = 0 (hole) 43. x = 0 (vert. tang.) 44. x = 1 (sharp turn) Assignment 2-6 page 40 1. 7 2. 6 3. 1

6 4. DNE 5. 4 6. or DNE 7. 2 8. 1

4 9. 0

10. 0 11. 118

12. 14

13. 12

14. 4 15. or DNE 16. 0

17. or DNE 18. 1 19. 5 20. 5 21. DNE 22. 1 23. 0 24. 0 25. DNE 26. 1 27. 3 28. 1 29. 0 30. or DNE 31. 1x 32. 2, 1x 33. 34. 2 35. DNE 36. 6 37. 3 38. 2 39. 2x 40. 3, 4a b 41. ,0 , 0,1 , 1, 42.

43. 44. 45. 46.

47. 48. 49. y-axis 50. neither 51.

2x

0,1 , 1,0

23

3 1y x 1 12 2

5 3 or 3 1y x y x

1x 32

3 1y x 31 2f x x

x

y

224

52. 53. 54. 55. Do: VA: (odd)

x-int: (odd)

Hole: y-int:

EB: HA 56. 57. 58. a. AROC = 15 b.

59. 60. a. Yes, c = -2, b. No, c. Yes, c = 4 61.

62. 63.

64. a. always continuous b. c. always differentiable

65. a. x = 1 b. c. x = 1

66. a. always continuous b. c. x = 1

67. a. x = -2, 2 b. x = -2, 0, 2 68. a. D: x ≠ -1, 1, 2, 3 2

2

2 1

1 2red

x xf

x x

Holes: (3, 12.5), (1,0) odd VA: x = -1 odd, x = 2 even EB: y = 1 x-int (-2,0) even

y-int: (0,-1) 68. b. D: x ≠ 3 VA: x = 3 EB: 1y , 1y y-int: (0, -2/3)

68. c. D: (-∞,-3), (3,∞) VA: x = 3, x = -3 EB: 1y , 1y x-int: (4,0) Assignment 3-1 page 45 1. b. d. e. f. h. 22.5 m

2. a. sec b. d. 3. b.

4. c. slowing down (the slope of S is decreasing) 5. a. disp. = 50 cm 5. b. d. t = 0,2 min e. TD = 58 cm f.

5. g. avg. speed = 6. a. c.

1x 1x

1redx

yx

0,0

12

1, 0,0

1y

6 1f x x 3 81f 3 51f12

12 8y x 26 6f x x21 3g x x 5f x x

6 1, 15, 1

x xf x

x6 1, 15, 1

x xf x

x6 1, 14, 1

x xf x

x

2 54 msec

V2

3 72 msec

A 1 sect 1.5 33 msecV

5t 1 5 ftsec

V 5 13 ftsec

V 2.791, 1.791t sec

4 24 cmmin

V avg 10cmmin

V

585

cmmin

17

4t 1 4t

x

y

x

y

x

y

x

y

225

7. a. b. 8.

9. 10. 11. 12.

13. 14. 17. 18. AROC = 4 Assignment 3-2 page 48

1. 3. 4.

6. 8. 9. 11.

12. 13. 14. AROC 16. a. g x x f x f x

17. 18. 19. moving left on 22. t = 5 sec, ft

24. 0 26. TD = 66.5 m 27. 667.447 or 667.448

29. b. d. f. just before t = 2 sec


1. 3. 4.

6. 7. 8.

9. 11. 13. 14.

15. a. 16.

17. b. Avg. Speed = 9.259 e. 10 g. between 80 and 90 m

18. 19. f is a cubic, is a parabola, and is a line

22. disp. = , TD = 5.916 or 5.917 23. a. ft c.


1. 2. 3.

5. ; at

6.123 or 6.124t sec 6.123 195.959 ftsecV 492f xx

8 12f x x 4 6f 13

8f 2y

1 1 1y x 1,4 1 3f

212 6 8f x x x2

22

2 2 2 1 2

2

x x xf x

x1, 2f x x

1, 0f t t 25

3 2f x

x

1 12 23 1

2 2y x x 8y x

2 1 2y x 13

0,x 14

54

dvds

4 48 52

, 5 73.5h

3f

7 40 ftsec

V 10 50 ftsec

V

29 3 5y x123

22 3y x 3

21

f tt

2 36 2 3 2 3g x x x x321

21y x 2

51

f xx

2 1 1y x 38 2 3x 5 12 2,8 , , 8 1 48g

5 3.737 4.105 1.2 or 3.737 4.106 1.2y x y xmsec

msec

3 or 1t t f f

4.5 0 220h 3 118 ftsec

V

xy

yy

yx

yy

x2 23 3dy y x x y

dx x x1, 2 , 5dy

dx

226

6. ; at

7. ;

8. 10.

11. 12. 13. 14.

23. 24. 25.


1. 2. 3. a. b. 4.

5. a. b. 6. 7. a.

7. b. c. increasing, no 8. 9.

10. 13. y = 16 14. a. AROC = 0 b.

15. a. 0 b. 0 c. yes 16. a. 1 b. 1 c. yes


1. 2. 3.

4. 5.

6. 7. 8.

9. (using the limit definition) 10. (using the alternate form)

11. 12. 13. 14.

15. a. b. 4 c. 20 16. a. b. c. t = 0,1

16. d. e. f. g. disp. = 32 h. TD = 34

17. a. 85.722 or 85.723 b. 7.687 c.

2

23 2

dy xdx y y

12

2, 2 , dydx

22 31 2y x

yx

1,1 1y

3; 3, 44

xy y

y 22 2x y

yx

125

5 12y x 0,3 , 0, 1 8 ,1 dVdr 4 64

.719 2.780 or 2.781x 1 0 , 2 1m msec sec

V V 22x

18dydt

24dydt

2cm

sec24dA

dt

cm

sec6dC

dt

ft

min

98

dr

dt

1202dA in

dt sec150

3dV in

dt sec

3

50

dy cm

dt min

9

8

da ft

dt sec

21

8

2dA ft

dt sec160

2dA in

dt min5 ydsec

11.180 ftsec

2

33f

3 24 9 1f x x x2 43 31 1

3 3f x x x 4

94

f tt

2121

2 2 1y x x x2 33 1 1g x x x x

2

2

1 4 2

1

x x xf x

x1, 0y x 18 2 3f x x

2f x x 3 4f

4 36y x x 22

x yyx y

2 1 1y x 34

1 1y x

4 26 6v t t t 12 6a t t

0 1t 2, 1t 1 32 2

v

227

18. a. AROC = b. 19. 20. a. yes b. no, sharp turn

21. a. yes b. no, vertical tangent 22. a. yes b. no, sharp turn 23. a. no b. no, jump 24. a. yes b. yes 25. a. no b. no 26. Assignment 4-1 page 63 1. 3 3. 5. 6. 8. 9. max. f = 10,

9. min. f = 10. max. f = 8, min. f = 0 12. max. f = 2, min. f = 0

14. max. f = , min. f = 15. max. f = , min. f = 0 16. max. value = 3,

16. min. value = 0 17. no max. value, min. value = 0 21. c = 2 22.

23. MVT does not apply. f is not differentiable at x = 0 (sharp turn). 24. 27. c = 1 28. 29. c = 2.414 30. b.

Assignment 4-2 page 66 1. a. , b. c. d.

3. a. , b. c. d.

5. a. , b. c. x = 0 d.

7. problem 2 9. a. b. c. d. none

10. a. b. c. none d.

12. a. , b. c. d.

13. a. b. none c. none d. none

14. a. none b. c. none d. none

16. a. b. none c. none d. none

18. yes 20. no 21. 23. a. never 24. 25. max. f = 73, min. f = 29.

Assignment 4-3 page 70 1. a. b. c. 2. a. b. c. none

4. a. b. c. 5. a.

5. b. c. 7. a. none b. c. none

8. rel. max. (0,6) rel. min. (2,2) 9. rel. max. rel. min. (0,5)

1

3

8

91f 16dx cm

dt sec

0,3 , 2, 1

0,4,5x 32

x 0,8x 2x94

3 15 23

32

c

827

c .215, 1.548 or 1.549c 32

sect

, 2 2, 2,2 2, 4 2,4

2,0 2, , 2 , 0,2 2,0 0,4

, 2 0, 2,0 2x

4, ,4 4, 16

,1 1, 1,8

,0 1, 0,1 110

1, 0,1

,

, 3 , 3,3 , 3,

, 1 , 1,

0sec,2sec 4 , 0a b7 .333c

,1 1, 1,0 2,2 , 2 , 2,

2,1 , 4,6 , 8,9 6,8 6,1 ,0 , 2,

0,2 0,2 , 2, 14 ,0 , 0,101

43,

228

10. rel. max. at , rel. min. at 12. 15. concave down ( ) 16. max. f = 8, min. f = 17. c = 3 18. 20.a. 0 b. -1 c. relative maximum Assignment 4-4 page 73 1. rel. max. rel. min. PI

2. no rel. extrema, PI

3. rel. min. PI 4.

6. rel. min. , P.I. , E.B. horiz. asymp. y = 2

7. Do: , rel. max. , rel. min. , V. A. x = 0 (odd), E. B. like

9. Do: , x-int. , rel. max. 11. x = 1 (nonrem.), x = 2 (rem.)

12. x = 0 (nonrem.) 14. 16. 0 18.a. 19. 20. B

24. (A) , (G) 1

Assignment 4-5 page 76 1. 4. 5. 6. 9. f 11. f These f graphs could be shifted vertically. 12. V.A.: , H.A.: , Rel. Max.: , no Rel. Min.

13. Rel. Min.: , no Rel. Max., P.I.:

14. Rel. Max.: , Rel. Min.: 16.

17. abs. min. f = , abs. max. f = 19.

3x 3x.523 .013f

11, 0, 3, 2a b c d

0,5 2,1 1,3

0,1

3, 11 0,16 , 2,0

0,0 12

1,

0x 1, 1 1,1 12y x

3 3x 3,0 0,314

0, 3 , 0,1 8 , 123

f

4, 2x 0y 19

1,

1, 27 2, 16 , 4,0

0, 5 2, 9 43

c

18 14 2, 12, 16a b c

x

yx

y xy

2, 15

1,12

1 3,2 2 x

y

xy

1, 9

229

Assignment 4-6 page 79 1. x-int.: , 2. Do.: ,

y-int.: , Rel. Max.: x-int.: ,

y-int.: ,

Rel. Max.: ,

Rel. Min.: ,

P.I.: 3. Do.: , 4. x-int.: ,

V. A.: (odd), y-int.: , Rel. Max.: ,

x-int.: (odd), Rel. Min.: , P.I.:

Hole: , H. A.:

5. 6. Min.: , Max.: 7. Abs. Max. f = , Abs. Min. f =

8. Abs. Max. g =14, Abs. Min. g = 0 9. MVT does not apply. (V.A. at x = 0) 10. 11. MVT does not apply. (Sharp turn at x = 3)

12. a. 7 b. -12 13. a = 3, b = -5 14. a. b. DNE c. d. 0

15. 16. 17.

5,0 , 1,0 4 4x

0,5 2,9 0,0 odd , 4,0

0,0

8 ,8

8 , 8

0,0

0,1x 0,0 , 3 ,0 all odd

1x 0,0 1,2

1,0 1, 2 0,0

0, 22y

3, 4 3,2 23

6

3 3 4.732c16

32

f f f

x

y

x

y

x

y

x

y

x

y 2,9

x

y

4,0

8 , 8

8 ,8

4,0

230

x

y

18. 19.

or Either or both branches of this graph could be shifted vertically. 20. 21. 22. This graph could be shifted vertically.

23. 24.

25.

26. 27. a. b.

27. c. d. e. f. g.

27. h. T. D. = 6.151 or 6.152 i. j.

Assignment 5-1 page 82 1. The numbers are both 10. 2. The rectangle’s length and width are both 9 cm. 3. The rectangle’s length and width are both 20 ft. 4. x = 30 ft., y = 40 ft.

5. x = 9 ft., y = 6 ft. 6. x = 2 in. 7. 9. 10. 375 mph

11. f. l. Assignment 5-2 page 84 1. The first number is 30 and the second is 10. 2. The first number is 24.5 and the second is .5. 3. The first number is 0 and the second is 25. 4. x = 400 m, y = 800 m 5. x = .606 ft or .607 ft 6. V = 4.104 ft 7. 8. 36

9. length = 20 cm, width = 20 cm, height = 10 cm 10. Rel. Max.: ,

10. no Rel. Min., P. I.: 11. Rel. Max.: , Rel. Min.: ,

11. Concave Down: 14.

2 2 21 12 23 3 2 3 2y x x x 2 2

2 1x yyx

2

22

3 2 6

2

x xyx

2,13 26 6 1v t t t 0 1 cmsec

v

12 6a t t 3 302

cm

seca 1 cmsec .145 1.145t 2cm

.5t sec , .145 sec .5 ,1.145sec sec

32

3, 3243dV in

dt min

0,4,5x 0,2,4,5x

3 18 , 18

1,13

0,2 , 2, 14 1,1 0,0

,0 , 0,2

12 3

g xx

x

y

x

y

231

Assignment 5-3 page 87 1. a. b. c. 2. b.

3. 4. 6. 7. a. A sq. in.

8. 9. 11. a.

11. b. 13. Both numbers are 1. 15. 180,000

16. V. A.: (odd), Rel. Max.: , no Rel. Min., H. A.: y = 1

18. V.A.: x = 2 (even), x-int.: (odd), y-int.: , Rel. Min.: , H.A: y = 0

23.


1. 2. 3. 4. 6.

7. 10. 11. 12.

14. 15. 16. a. b.

17. 19. a. c.

20. c. 21. 22. a.

22. c. e. 10.333 sec 23.

25. Do.: , V.A.: (odd), no hole, x-int.: (even),

y-int.: , E.B.: H.A. y = 1

26. 27. 28.


1. 2. 3. 5.

6. 8. 9.

10. 11.

3 6y x 1.1 2.7f 1.1 2.669f 2 1y x

26 5.1 3.1 29.1f 22.05y 651

12008.01 2f 1.984 or 1.985y 3162 10

3162 10.171

2yd

3x 29

0,

1,0 14

0, 112

4,

1.190 or 1.191c

21

2C

x

544

5t C

3 2

23 2x x x C

324

3y C

13

Cx

19

Cx

3 23 30 100t t t C 3 14 13 2

x x x C122 2t t C

3 22 2 4f x x x x 3 6y tt

322y x C

322 16y x

3 21 3 73 2 6

5g x x x x 23 4 5v t t t 3 22 5 2s t t t t322

37s t t t 10 sect

3 , 0 412, 4

t tv t

t23

2, 0 4

12 24, 4

t ts t

t t1x 1x

0,0

0,01 38 3f x x 1

34 8y x 3 22 6 2y x x

616

3 2x C 9145

5 3t C 323

4 x C5

313

2x C

6 1 v C4

2116

2 4x x C112

114t C

2 23 3

3

310

3 or 52 5

x C x C 324 133 6

2 or 2 1u u u C u C

x

y

500 yd. fence

232

13. 14. 15.

16. 17.

18. 20. a. b. e. at all times f. 8

22. 23. 25. Tangent and secant lines have the same slope.

26. a. b. 27. 30.

Assignment 5-6 page 101 1. 3 2. 4. 5. a. b. 5 6. a. 2 7. 8. 8 9.

10. 12. 3 3 3 35 2 5 2

32 4 13.

14. 4 43 3

8 8 4 15. 16. a. 3 c. 6 17. a. 5 b. 0

18. 2.484 or 2.485 20 a. 1.098 or 1.099 b. 4.098 or 4.099 22. c. f. 18 g. 22 23. 8.656 24. a. Do.: all reals, V.A.: none, 24. a. H.A.: y = 0, x-int.: (0,0) odd, y-int.: (0,0), symm.: origin 24. b. rel. min. at , rel. max. at c. P.I. at

Assignment 5-7 page 105 1. 2. 3.

4. 5. 6. a. 6. b. mile 360 c. mile 150 6. d.

7. 25 3ft

asec

8.

9. seconds 10. 11. 12. 5 seconds

3x Cx

5 3 12 2 26 2

5 34x x x C

5 32 24 4

5 31 1x x C

323

3 2 1 2 1x x C3 54 2

3 51 1x x C

239 71

2 23x x 2 29 3a t t

313

1 5y t t

14

13 1 3 .732c

0,0 , 0,4 2 ,2 2.1 7.9f 2 4f

8 2 4 2 11012

1 1 8 132

23110

13

163

3 37 4

3328

3

1x 1x 0, 3x

12, 2a b .975y

5y 60 240s t t

4 , 3 hours

6 3 ftsec

speed

0,5 0,2 sec, 6,7 sec 0,2 sec, 5,6 sec

x

y

233

13. Do.: 14. x-int:

V. A.: x = 0 (odd) y-int:

x-int.: rel. max:

no y-int. rel. min:

E. B.: like

Rel. Min.: P. I.: no Rel. Max. P. I.:

15. 16. 17. max. point min. point 18. c = 2.877 19. MVT does not apply. (sharp turn at x = 0) 20. 4 by 8 21. units per sec. 22. abs. max. f = , abs. min. f =

23. local maximum at x = 3 24. 70 feet 25. + C

26. 27. 28. 544 4

15 1516

29. 30. 10 31. 6.294 or 6.295 32.

33. 7.128 or 7.129 Assignment 6-1 page 110

1. 4. 6. 8.

9. a. At 8:00 AM water is flowing from the reservoir at the rate of 1000 .

9. b. 22000 gal e. 33000 gal 10. b. approximately 11000 c. Approximately 11,000 babies were born in the Utah city during 1992, 1993, 1994, and 1995. 11. c. 12. c. e. 100 miles

13. a. d. e. f.

15. 3 1

2 3 2 34 414

5 3 5 2f x x x x x x 17. 21. a.

21. b. c. 22. c. f. 0

24. 7 52 4 27 5 3

2 2 8

0x 0,0 3,0

0,03 2 ,0 2,4

0,02y x 3,0and

1,3 1,2

3 2 ,0

1,1 and 1,3 1,2 and 3,2 1,3 1,1

24dxdt

23

23

42 39

162 1x

5 32 21 1

10 62 1 2 1t t C 232

92x x Cx

12415

1 105 3

25 32815

4 2116

4v t t t t

2 1x x 2sin 2t422 1x x

62 61 136 36

3 1 2agalhr

ker3.1 29.4 hi smile

D ,.25 5 5 40mi mihr hr

v v

203 2

ft

sec35 ft

sec55 ft

sec52.5 ft

sec

2 1x 2 3 1y x

1 1 0y x 9 12 2y x 277088105

x

y

x

y

62

234


1. a. b.

1. c. 2.

4. 5.

6. = 46.269 or 46.270 7.

9.

10. 7.888 12. 2.000 13. 16.

17. 18. 20. 21.

23. 24. 25. 3.5 27.


1. 2. 12

1 5 2 12 2 28 2 34 2 15 2 25 20

3. 1 12 2

4.12 2 3.21 2 3.94 2 4.69 7.52 =

5. 12

1 1 2 2 2 5 2 10 17 7. 48 9. 24, This is not

accurate because only a small part of the area under the curve is covered by the three rectangles. 10. 2sin(2x) 13. 1.000 14. a. d. 12 15. 8 16.

17. 18. 19.

20. a. reverse , forward , at rest b. Six minutes after the start of the

test drive the car’s velocity was 6000 feet per minute. d. 2000 ft/min/min f. h. 29,000 ft i. 31,000 ft 21. The first number is 20 and the second is 4. Assignment 6-4 page 122

4. 6.356 5. 1.167 or 1.168 6. 8. 2 9.

10. 11. 1.105 12. 1.683 13.

2 1 3 1 3 16 1 0 1 2 3 2 1 3 3 15

1 1 2 3 2 1 3 3 3 18 2 8 12 17 15 12 6 1401 1 12 4 6

2 116

1 1 14 6 8

2 1312

4 3 7 11 15 1 2.5 2 2 2.5 9

3 33 31 1 1 12 2 2 2

3 3 1 0 1 1 3 2 33 1 12 2

3 3 3 2

2410

x 2 2x322

3x

2 23 33 3

3 02 or 2 2x dx x dx

3 2

21y dy 52

35

18cmsec

92

2 5 126.5

8.83

9.374

4 23

62112

2x x C3 12 22

32y y C 6

0,2 3,10 2,3

9,10

13

1 3 9 9 9 323

14

1 16 23 3 3

16 1 319

3 2x

x

y

500 yd. fence

xy

500 yd. fence

235

14. c = 15. 7.211 16. a. 8 b.

17. b. 3 3 3 312

2 0 2 2 2 4 2 6 8 = 1088

18. a.

19. b. 1 50 250 2 210 2 190 2 190 2 170 2 120 2 80 2 60 2 60 2 40 02

=

62,250 square yards 20. a. 300 ft. b.


1. a. b. c.

2. a. 3. a. 4. a.

5. 8 6. 7. 9. .368 10. x = 2.335 or 2.336

12. 13.

15. 16. 17.

18. 21. 3 22.

24. Do.: , , no V.A., Holes: , no x-int., y-int.: ,

E.B.: H.A. at , y-axis sym. 25. 27.

Assignment 6-6 page 129 (Be sure to show sketches with radii labeled!)

1. a. b. 2. b.

3. a. b.

4. b. 5. b. c.

6. a. b. c.

7. a. b. 8. a. 53332 8

5

.751 43

4 2 6 10 28.103 or 28.104

5 ftsec

1 22

0V x dx

21 2

0

1 12 2

V x dx1 2

02V x dx

1 22

0V x x dx

22 2

22 4V y dy

1 22

22V x x dx

92

1 3 1612 8 3

1 4 6 12524

1921

210 1f x x x x

1 23 23 34

32 1 3 2 1g y y y y y

5 3 12 2 22 8

5 38y x x x C

62112

3f x x C 24 3g y y C32

33 6 3h t t t C 1

2

2x24

4redy

x12

2, 0,1

0y 2 8 1y x 9127

3.370c

21 2

11 x dx

21 2 2

11 1 1x dx

22 2 42

2 2 2x dx

2 22 2

02x x dx

2 2 22

05 5 2x x dx

24 2

1

11 1 dyy

24 2

1

13 3 dxx

24 2

1

15 5 dxx

22 2

0y dy

2 22 2

02 2y dy

2 22 2

04 4 y dy

21 2 2

12 1x dx

21 2 2

11 2 2x dx

236

8. b. 9. a. 16.302 or 16.303 b. 28.688 or 28.689 10. b.

11. b. x-int.: d. range: 14. 4 13 2

8 16 0 15.

16. 18. 0 20. c = 1 Assignment 6-7 page 132

1. 2. 3.

4. 5. 5 6.

7. 8. a. b.

8. c. d. e. f. 3 9. a. b.

9. c. disp. = d. T. D. = .779 or .780 or .781 10.

11. 12. 13. approx. 3 miles from the trailhead

14. represents the total number of hikers on the trail at a distance between 1

and 5 miles from the trailhead. 15. larger 16. smaller 17. They are equal. 18. A = 19. A = 20. V = 8

21. a. b.

22. 4

1

1 13 3 3 121.158 or 121.159V dxx x

23. 24. 25. 26. 27.

727

23

2

12 2

xdx

0,0 , 4,0 1y 83

23

.178 or .179, 2.488c

4 214

2v v v C3 1 12 2 22

34 2x x x C

63118

2y C1

2 213

4 3x C5 32 2

5 31 1t t C

2 3cos 2 1 3 cos 2 1x x x121

24a t t

322

34 2y t t

16 3 16 2.3 .206 or .207v 2.3 .007 or .008a

.097 or .098

5

1

5 1avgD x dx

D

avgD 30kerhi s

mile avgD 30kerhi s

mile5

1D x dx

2 2 2

11 2 3x x x dx 16

326

22V x dx

26 2

24 4 2V x dx

1 3 1g x x

x

y

x

y

x

y

x

y

x

y

g

1g

237

28. x = any even integer (nonremovable) 29. (nonrem.)

30. t = 3 (nonrem.) 31. (rem.), (nonrem.) 32. 33.

34. 35. 36. 37.

38. tangent: normal: 39.


1. 4 3. 0 5. 7. 9 9. 11. 13. 14. 15.

16. 17. 19. 21. 23.

24. 25. 26. 27. 28.

30. Method 1: Method 2:

31. 32. 33. 34.

35.

36. a. c. e. f. i. 10 sec. j. 0 sec.

37. 2 85 3

32 8 38. 67.020 or 67.021 40.

2, 3x

2x 3x 15

12, 1a b

2, 3a b3

3 212

6y x x 3 22 2

x yyy x

2, 2 , 2,2

3 202 3

y x 2 203 3

y x 113

0,3 , 2,

116

3

8e 15e 4 x 3x 32x

8x 1x

23avg

f 16

A 43

c t tte te

33 441

42 6x x x

3 234

614

2 3x x xx

x

2 8f 79

54

24 1 15

y Cy y

2 2y x

539 10 31f x x x

53 2

m

sec1

2

m

sec9.5 m

sec29.5 m

sec20

4

34

4V x x dx

x

y

(0,1)y = 0

xy

(0,-1)

y = 0x

y

(0,0)

y = -1

x

y

(0,1)

238

Assignment 7-2 page 139 2. a. b. 3.

5. 7. 9.

11. 12. 13.

14. Rel. Min. at , P.I. at 15. 16.

17. 18. 20. 22.

24. 25. a. 26.

27. a. 4 sec. d. .047 and 7.952 or 7.953 sec. 28. 42.666 or 42.667 30. 32. .291 or .292 or 33. a. b. A = 4


1. a. c. d.

1. e. They are reciprocals. 2. a. 3. a

3. c. 4. a. not possible b. c. 5 5. 6.

7. 9. 10. 1 11. 12. 14.

15. 16. 2

22x x

x

e x x ef xe

18. 16

19. 20. 21.

22. 24. a. b.

24. e. g. cm h. 41.435 cm 26. .471 or .472

28. 3 31 14 9

1 1 11 2 2 21 .5 2 1 1.5

30. 4 32.

1 1 0y x 1y e e x 5 15 xf x e

2 25 2 2 ln5t th t t2

22 xy x e2

22

1 2

1

x xx e xeg x

x

10 x yx y

ye eye xe

2 xyxy

x yeyxe

1 2 1y x

0, 1 21,e

122 13ln 2

1 31 12 2

e e

2 4e2

12

ue C631

182ye C

212

bx a

be C

2

2xx e C

1 32 21 1 3

2 2 20 e e e 3 22 3 6g x x x x

4 2 2

2ln 1 2 2V x dx 7

241,2

3 1g x x2

2 313

3 , 1f x x g x x 13

1 3 , 0f g1 13 3

g x x 2 1, 0g x x x

1 , 22 1

f x g x xx

34

14

15

113

111

32

, 2,2

2

3 320 2d ydx

x x

32 28 6 3 6 ln 6x x22

x

x xe

44 234 4

e 7 34 44 421 9

y y C3

2 21 42 3

x x x C

8 1 12

3 2 32 4A 9

22

cm2e

y 1 1t tv t te e

2 02

cm

seca 1

1.086 or 1.087 1

239

Assignment 7-4 page 147 1. false 2. false 3. false 4. false 5. false 6. true 7. false 8. true 9. 11. 13. 15. 17. x = 8 18.

19. 21. 23. 24. 26. 27. 28. 30.

31. 33. 34. 35. 36.

38. 39. 2.726 or 2.727 41. monotonic increasing 42.

44. 45.


1. 3. 5. 2

312

3 1tg tt t

7.

8. 9. 10. 11.

13. 14. 15.

16. 17. 18. c.

20. Do.: , Rel. Min.: , no P.I. 21.

23. 24. 25.

27. a. 12

4 10 2 0 2 20 2 30 2 20 2 0 10 = 240


1. 2 2. 3. 4. 6.

7. 8. 9.

51

125log 3 63 729 2 1x p 1ex

2x

1 12

lnf x x 1 1xf x e ln ln lna b c 2ln lna b c

2log xy 2lnacb

2,3t ln 3 12

t 2 4 is extraneoust

.482 or .483x 113

13

4 6f 14dh ftdt min

5x

y2

3

3 22

xf x

x x11 lnh x x xx

2 11

yx x 2

ln 1

ln

tf t

t

11 ln 1

yx x

23 1 6xf x e x

1 lny x 2 ln 2 12 1

xy x yx 2

ln ln22 x

y

y y yyy xy

xy y yy yx x

2

2

3 23 8x xd ydx

e e 12

1y x

0x 1,1 212 2 3x xe e x C

22 418

te C 22 2k

A e kV e

2ln515ln 4

4122

e 212

ln 1x C

212

lnx x C 22ln 3 2y y C 2ln 1x x C

xy

(1,0)

x=0

xy

(-1,0)

x=0

x

y

(1,0)

x=0

240

11. 13. 14.

15. 16. 18. 7 20. 21.

22. 23. 23

11

f xx

25.

26. 27. 28. a. Do.: , Ra.: all reals

30. a. .033 or .034 b. .146 c. .367 or .368 d. .259, 2.239 or 2.240

31. b. c. d.

31. e. 32.

Assignment 7-7 page 158 1. a. Do: all reals, points , HA:

b. Do: , points , VA: 2. HA: 3. , VA: 4.

5. Tangent: , Normal: 6. 7. 8.

9. 10. 11. 12. 13.

14. 15. 16.

17. 18. 19. 20.

21. not possible with current techniques 22. 23.

24. 25. 26.

27. 28. -3 29. 30. 4 31. 0 32. 33.

22 39

41u C 4

12 1 ln

Cx

8 16ln 2x x C

1 xf x e 1 12

ln 1g x x2

3logpq r

ln 5 32

x

2x e

11

1 ln 1 lnyg y

y y y

2 1 2lnxy x yx

22

xyyy

2x

3

11xA e dx

3 2 2

11xV e dx

23

11xV e dx

23

11xV e dx 2, 4, 4a b c

0,1 , 1,e 0y

0x 1,0 , ,1e 0x

3y ln 3y x 3x 2y x

2y x 2y x 1, 125

x 1x

0,1x 3x 323 23t tg t t e t e

2 21 2 12 4x xy e x e2

22

2

2 ln 11

x xx

x 2 21

3 ln 3 ln

yx y xy

y x xy x x2 1

3 1y

x x2

26 2

212 log 11 ln 6

dydx

xx

x0y 1cy cx lnxy c c

32

ln 2 1 24 ee

21 16 3

lnt t C 2 21 12 2

ln 1x x C 212

ln y C

22 xe C2

73 e e

113

14

1,e

,1e

x

y

241

34. a. 34. b. c.

d.

e.

35. 36. -42.050 37.

Assignment 8-1 page 160 1. 2.

3. or 4.

5. 6. 7. 8. 10.

12.

12 2

4

12

2y yy e y e yf y

y 13.

14. 17. 18. 19. 12 20. .746 or .747

22. 349.621 23. 25. a. c.

25. d. 26. 27. 28. b.


1. or or 2. a. ,

2. a. c. 12.602 or 12.603 yrs 3. 551 bacteria 4. 429.741 or 429.742 yrs 5. 19,000 yrs 7. 32.618 or 32.619 min

8. 9. a. 1.520 b. 4.036 or 4.037 11. 12.

14. 16. 17. 18.

Assignment 8-3 page 169 1. a. 0 c. 2 e. 3. b. i,ii.

3. d. 4. c. for

2 1e4

2e

2 22

09 9 xe dx

22

0

12 2

xedx

212

1xf x ee

2cm sec

3 32 13 3

3y y x x C 2 or 2x xy e C y e C12

ln 1 lny x C 12

ln 1 ln 2y x C 22y C x

22 24y x13y e x 100 kty e 21

21y x a

b

lnx

xef x e xx

ln 1 xy e C 3ln 2

12

ln 3

.488 or .489x 2.731t 5 1.617a

4.510 or 4.511TD ln 7 32

x 1x 12

.3462 ty e .3472 ty e1 ln 222

ty e 1 $1056.54A 5 $1316.53A

10 $1733.25A

2

1 12 2e

ln 5 1 2ln 2

212

lnu C 25x C4 43 3y x C 21 1

2 2lny x

95

2 2 or y x C y x Cxy e 0,1

x

y

x

y

242

5. a,b. c.

6. c. 7. c. 8. c. 11. a,c. b.

12. 13. 0 15.

17. (sharp turns) 19. (hole), (VA) 20. (vertical tangent) 22. a. c. 0 23. a.

24. Do: , , VA: (odd), Hole: ,

x-int: (odd), no y-int., HA: , y-axis symm.

Assignment 8-4 page 173 1. 0 2. 4 3. or DNE 4. 6. 8. 4 10. 11. 0 13. 1 15. 0

16. 1.000 19. 20. 21.

22. 23. 24. 25. 5.49%

26. 260,456 people 27. 118.920 or 118.921 g 28. a. 4 c. 1 e. 0

31. 32. 1.248 or 1.249 33. 1.540 Assignment 8-5 page 178

1. a. b. d. f. 2 2. a. c. 3. a. c. 4. a. c.

5. a. c. 6. a. c. 7. a. c. 5.445 8. a. c. 1

9. a. c. 10. a. Quadrant 3 11. a. c. 12. a.

21 34 4

y x

2y x 3 3 4y x2

1xy e

1 318

1f x x

12a

34

3x0x 1x0x 2

130

4

0, 2x 2 1 12 2red

x xy

x x2x 1

20,

1,0 2y

12

1 23

2 5

2

xef x C 21t t

x t Ce

433 3

4 4ln 2

1 13ln 3 ln 3

212

ln 4 5t t C 2ln 2y y C

21 222

x xy e

12

32

13

73

583

529

52

32

23

12

1 270 54

132 12

12

23

8 2 2 or

3 33 43 3,x

x

y

x

y

f

1f

243

14. 16. a. 17.

18. 19. 20.

21. 12

1 2 ln1 2 4 ln 2 2 6 ln3 8 ln 4

23. 24.

Assignment 8-6 page 181 1. 2. 3. 4. B 5. D

6. a. 0 b. 2 c. 3 d. e. 7,8. 9. a. b. 8.141 yrs

9. c. 27.045 yrs (early in 2017) 10. gives

the total amount of coal (in tons) produced by the mine from the beginning of 1990 through 1995. 11. a. 9,386,000 tons b. 9,420,000 tons 12. 9,397,000 tons 13. 0 14. 0 15. 16. 17. -2 18. 0 19. 20. DNE

21. DNE or 22. 0 23. 5 24. 25.

26. 27. 28.

29. 3.064 30. 36.375 31. 55.629 or 55.630 32. 40.643 or 40.644 33. a. 4.909 or 4.910 b. 1.928 34. 388.130 35. a. max at x = -1, min at x = 3, PI at x = 1

35. b. max at x = -1, min at x = 1, PI at x = 0, 1 1,2 2 36. Length 150, width 75

212 337. 2 3 2x y

yy x

38. a. ln( 5)xe C b. 2(ln 3)

2x

C c. 2

2ln 12x

x x C

39. abs max y = 5, abs min y = -14.238 40. 119

Assignment 9-1 page 186 1. a. b. d. .324 or .325 2. 3. in. 5. 7. 8. a. b. 4 c. 0 d. 1 (upward)

9. a. 1 b. none c. (right) d. 0 10. a. b. 112 c. (left ) d. 0

13. 16. 17.

18. a. (others are possible)

3 5 74 4 4 4

, , , 32

1 1 , 2f f 1ln 2

1.442 or 1.443c

2.613 or 2.614 121 ln .9 1.9f

12.515

22 sin 2x x 1 12 2

.859e

3 21 14 4

2y x x 2 16y x3

4 1xy e

1 3.0852,000,000 tR t e

6

0R t dt

3 2 4

1213

5 7 11, , ,

6 6 6 6

2 4 5, , ,

3 3 3 32 23sin 9 2sin 4x x

4 14

e

.309 4.179 or 4.180 .915x 3.410x.936x .646 or .647, 3.788x

18

23 3 3

1 32 2,x ,3x

12

4sin 0 0y xx

y

x

y

244

20. 21. a.

21. b. c. , , is undef.

22.

23. 24. 25.

Assignment 9-2 page 189 1. 2. 3.

5. 6.

7. 9.

10. 12.

13. 14.

15. 17. 18.

19.

20. Tangent:

Normal:

21. 22.

23. 24. 26. 3

27. 0

28. a. b. 29.

31. 32.

34. amp. , per. , ph. sh. (right) 35.

36. disc. at 37. 39. b.

7 7 ,

253sin tan 1

2sin , tan 1 , sec 2

1 1 22 3 3

sin , tan , sec sin 1 cos 0 tan

7 34 7

cos , tan 4,3 3x 0 ,5

3 3, ,

3sin 2y t 1 2cos 2f x x 2 24 cos 1 sin 1g x x x x2csc

cotxf xx

2 22csc 2 cot 2 2 sec tany x x x x x

3 2 2

6

2 sec tan sec 3y y y y y y yf y

ycos 2h x x x

22sin 2 cos

sinf sin cos

2 cos sint tP tt t

2 22

2 2sec ln 22

xf x x xx x

3tan 3g x x

sin cos 2 2sin cosx xh x e x x 12 6

3y x 2 1y x

42 2y x

.986 .272 .821 or .986 .273 .821y x y x

.986 3.664 .821 y x22 3 sin3 1

sin sinx y xxy

y x y x

2 22

2 2 secyy

x y y1

53

450 3 5 3534 34

sin , cos , tan

3 12 2

sin , cos , tan 3 20x34

23 3

4 20, , , ...x 1

22 tan 0 1y x 316

ddt

radianssec

x

y

x

y , 24

5 , 24

245

Assignment 9-3 page 193 1. 2. 3. 4. 5.

6. 7. 8. 10.

11. 12. 14.

15. 4.403 or 4.404 16. .888 18.

19. 20. 21. 23. 0 24. 2 25. 2 26. 1

27. tantan tantan0 612 412 12

2 2e e e e 29.

30. 31. 32. 34. 2

Assignment 9-4 page 197 1. 2. 3. 5. 7. 9. 11. 0 12.

13. 14. 15. 16. 17.

19. 20. 21.

23. amp. = 3, per. = 2, ph. sh. = (right)

24. 26. 2 2

42sin cos sin 2t t t t t

t 27.

29. 30. b. c. 5 58

12 12 2 11

31. 1.528 or 1.529 32. b. 33. b. 33. d. 3.876 or 3.877 years e. 4217 rabbits g. years Assignment 9-5 page 201

1. 2. 3. 4. 6. 1 lnsin + C4x

7. 11 5tan + C7 7

t 8.

6 9. 11.

12

2 3 6 122ln ln 4 tan 2 C31

3sec 1 C

1sinxC 21

2sin y C 51

5sec x C ln sin 2y C

ln sin ln cosx x C tan 1xe C 13

tan 3x x C.5149 22

04 3 4 3cos 2x x dx

.5149 2

0

12

3cos 2 3x x dx 2 23

1.295 sec sec tanxf x e x x

2 sin cosh 2 224 csc 3g t t t 3cossin

xx

3 423

1.249 .339 or .340 23 3

4x 1x 35cos y310

sin x 26

1 9y

x

21

y

y

eg ye

3

32 1

th tt

2

22 arccos

1x x xx

14

22sin ln ty

t2 21 1

2 2lny x x C

,2 1a b 316

1221

4x

y e 2000cos 2 300sinN t t t2

12 3012

ln 2 4 2 12 2

arctan t C

322

3arctan C 5 5arctanx x C

x

y

246

13.

14. 15. 16.

17. 1 18. 19. 2 20.

21. 22. 24.

25. 26. 27. 28.

29. 30. 31. 32.


1. 2. 3. 4.

6. 7. 8. or

9. 11. 12.

13. 1 22sin + Cx 14.

16. 18.

19. 20. 21.

22. 24. 25. 26. 27. 2.336 or 2.337

28.a. b.

29. a. 0 b. 0 c. 1 d. e. 31. Assignment 9-7 page 207

1. 2. 3.

4. 6

5. 6.

7. 8. 9.

10. 11. 1 3tan + C3x 12.

212 3 3

arctanve C

5 23 32 1

15 33 1 3 1x x C

12 2

12

2

44

1

x

x

eye 2

1arcsin 1

f xx x

13

tan 2x C ln sin te C 13 cot 3x x C32

2 625

32

x

3 12 24

36x x x C 7114 2 1x C

5 395

2t t t C322

3ln y C

C11 tan C ln cos C 21

2ln 1 tan C

2 2 2ln 1x x x C5

3 2215

sin x C 3 213x x xe e e C

21 52 4 4

ln 16 arctan xx C

3 cos 3 sin 3y x x x 2cos sint tg tt

23 ln sec tanh y y y

21txt 4

2 11 1

vy

v2

4 2 24y 1y 3

4 4,x

4

2sin or 2siny x C y x C 2sin 7y x

1 221

22x

y e

12ln 4 4 4 2x x x C 1 tan 22

x x C 313

ln cos t C

23 3

arctan x C 313

2y y ye e e C

2 2ln 1 or 2 ln 1y ye C y e C122 ln x C 23arctan 2ln 9

t t C

ln x xe e C 2 1 2arctan 1x x C

x

y

x

y

247

13. 14. 2

2 1ln arctan 21 4

f x x xx x

15. 22

4 2

23 ln arcsin1 arcsin

xg x xx x

16.

17.I. 17. II. matches with c matches with b

17. III. 17. IV. matches with d matches with a

18.a. I II

III

18.b. IV

19. 20. a. b. 505 or 506 bacteria c. 123,500 bacteria

21. a. 8 ft. by 9 ft. b. 22. a. b. 6 hours

2

21 4 arctan 2

yx x

22 cos tt

t t yeye

2 23 3

2 02x x dx x x dx

2 22 2

2 25 6 5 6x x dx x x dx

2 1 22 2 2

2 2 15 6 5 6 5 6x x dx x x dx x x dx

2

21 5x dx2 1y C t .458k

272 ft 323fthr

x

y

x

y

x

y

x

y

the essentials of calculus - wordpress.com...221 19. a. do: xt 4, ra: yd0 220.a. do: all reals ra:...

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