Unit 4 Answers

4-1 Answers

1. 3

2. (5,−2)

3. x = 0, 4, 5

4. x = 1

5. CN: x = 32

6. CN: x = 0, 8

7. CN: None

8. CN: x = −2

9. abs. max.: 10 abs. min.: −94

10. abs. max.: 8 abs. min.: 0

11. abs. max.: 4 abs. min.: 0

12. abs. max.: 2 abs. min.: 0

13. abs. max.: 2 abs. min.: 0

14. abs. max.: −3 abs. min.: −15

15. abs. max.: 23 abs. min.: 0

16. abs. max.: 3 abs. min.: 0

17. abs. max.: none abs. min.: 0

18. abs. max.: none abs. min.: 0

19. abs. max.: 2 abs. min.: 0

20. MVT Does Not Apply

21. MVT Does Apply; c = 2

22. MVT Does Apply; c = 32

23. MVT Does Not Apply

24. MVT Does Apply; c = 827

25. MVT Does Not Apply

26. MVT Does Not Apply

27. MVT Does Apply; c = 1

28. c = −.215, 1.548 (or 1.549)

29. c = 2.414

30. (a) vavg = −48 ft/sec on [0, 3]

(b) t = 1.5 sec

4-2 Answers

1. (a) (−∞,−2), (2,∞)

(b) (−2, 2)

(c) rel. min. (2,−4)

(d) rel. max. (−2, 4)

2. (a) (−∞,−2), (0,∞)

(b) (−2,−1), (−1, 0)

(c) rel. min. (0, 0)

(d) rel. max. (−2,−4)

3. (a) (−2, 0), (2,∞)

(b) (−∞,−2), (0, 2)

(c) rel. min. (±2, 0)

(d) rel. max. (0, 4)

4. (a) (0,∞)

(b) (−∞, 0)

(c) rel. min. (0, 0)

(d) no rel. max.

5. (a) (−∞,−2), (0,∞)

(b) (−2, 0)

(c) rel. min. x = 0

(d) rel. max. x = −2

6. (a) (−2, 0), (3,∞)

(b) (−∞,−2), (0, 3)

(c) rel. min. x = −2, 3

(d) no rel. max.

7. Problem 2

8. Problem 3

9. (a) (4,∞)

(b) (−∞, 4)

(c) rel. min. (4,−16)

(d) no rel. max.

10. (a) (−∞, 1)

(b) (1,∞)

(c) no rel. min.

(d) rel. max. (1, 8)

11. (a) (−∞,−2), (1,∞)

(b) (−2, 1)

(c) rel. min. (1,−7)

(d) rel. max. (−2, 20)

12. (a) (−∞, 0), (1,∞)

(b) (0, 1)

(c) rel. min.(1, 1

10

)(d) rel. max. (0, 1)

13. (a) (−∞,∞)

(b) never

(c) no rel. min.

(d) no rel. max.

14. (a) never

(b) (∞,−3), (−3, 3), (3,∞)

(c) no rel. min.

(d) no rel. max.

15. (a) (−1,∞)

(b) (−∞,−1)

(c) rel. min. (−1, 0)

(d) no rel. max.

16. (a) (−∞,−1), (−1,∞)

(b) never

(c) no rel. min.

(d) no rel. max.

17. (a)(32 ,∞

)(b)

(−∞, 3

2

)(c) rel. min.

(32 ,−2716

)(d) no rel. max.

18. Yes

19. Yes

20. No

21. 0 ≤ t < 2 sec

22. 70 ft

23. (a) Never

(b) t > 2 sec

24. a = −4, b = 0

25. abs. max.: 73 abs. min.: −7

26. abs. max.: 4 abs. min.: 2

27. CN: all reals

28. MVT can not be used

29. c = −.333

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4-3 Answers

1. (a) (−∞, 1)

(b) (1,∞)

(c) (1, 0)

2. (a) (−2, 2)

(b) (−∞,−2), (2,∞)

(c) none

3. (a) (−∞,−1), (0, 1)

(b) (−1, 0), (1,∞)

(c) (−1,−1), (0, 0), (1, 1)

4. (a) (−2, 1), (4, 6), (8, 9)

(b) (6, 8)

(c) (6, 1)

5. (a) (−∞, 0), (2,∞)

(b) (0, 2)

(c) (0, 2), (2,−14)

6. (a) (1,∞)

(b) (−∞, 1)

(c) (1,−2)

7. (a) never

(b) (−∞, 0), (0,∞)

(c) none

8. rel. max.: (0, 6) rel. min.: (2, 2)

9. rel. max.:(±3, 101

4

)rel. min.: (0, 5)

10. rel. max: at x = −3 rel. min.: at x = 3

11. rel. max: at x = 0 rel. min.: at x = −2, 4

12. Show your work!

13. Show your work!

14. Show your work!

15. Concave Down

16. abs. max.: 8 abs. min.: −1

17. c = 3

18. d = 2, b = 0, a = 1, c = −3

19. (a) Picture!

(b) Picture!

(c) Picture!

(d) Picture!

20. (a)dy

dx= 0

(b) y′′ = −1

(c) rel. max. at (1,−2)

21. g′(4) < 0

22. g′(4) > 0

23. g′(4) > 0

24. (a) t > 0

(b) −3 < t < 0, t > 3

(c) t > 3.644

Page 3

4-4 Answers

1. Graph!

2. Graph!

3. Graph!

4. Graph!

5. Graph!

6. Graph!

7. Graph!

8. Graph!

9. Graph!

10. False

11. Removable: x = 2 Nonremovable: x = 1

12. Removable: none Nonremovable: x = 0

13. Removable: none Nonremovable: multiples of 3

14. 14

15. DNE

16. 0

17. y − 64 = 88.722(x− 5) [or y − 64 = 88.723(x− 5)]

18. (a) (0,−3), (0, 1)

(b) rel. min.: (0,−3) rel. max.: (0, 1)

19. (±√

8,−1)

20. B

21. C

22. A

23. D

24. (a) m = −23

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(b) m = −23

(c) m = 0

(d) m undefined

(e) m ≈ −1

(f) m = 0

(g) m ≈ 1

(h) m ≈ 3

(i) m ≈ −1

(j) m = 0

Page 5

4-5 Answers

1. Make Sure You Understand!

2. Make Sure You Understand!

3. Make Sure You Understand!

4. Make Sure You Understand!

5. Make Sure You Understand!

6. Make Sure You Understand!

7. Make Sure You Understand!

8. Make Sure You Understand!

9. Make Sure You Understand! The Graph of f couldbe shifted vertically.

10. Make Sure You Understand!

11. Make Sure You Understand!

12. Make Sure You Understand!

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13. Make Sure You Understand!

14. rel. max.: (0,−5) rel. min.:(2,−9)

15. Make Sure You Understand!

16. c =√

43

17. abs. max.: 14 abs. min.: −18

18. Let y = −x2 − 6x. Then:

19. c = 16 b = −12 a = 2

20. loc. min. at x = 3

21. loc. max. at x = 3

22. False. [Find a counterexample!]

23. True

24. False. [Find a counterexample!]

25. True

26. False. [Find a counterexample!]

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