Related Rates

1. $8$
2. $\frac{13}{\sqrt{10}}$
3. $2\sqrt{3}$ ft/sec
4. The distance is decreasing at $390$ mi/h.
5. The distance between them shrinks at a rate of $\frac{1320}{13}\approx 101.5$ mph.
6.  
7. $\frac{9}{2}$ ft/sec
8.  
9. It grows at a rate $\frac{4}{9}$ ft/sec
10.  
11. The distance is increasing at $\frac{135\sqrt{26}}{26}$ ft/sec
12. $-\frac{5}{6}$ m/sec
13. $240\pi \, \text{m}^2$/sec
14. $\frac{1}{2\sqrt{\pi}}$ cm
15. The area is increasing at a rate $\frac{(3\sqrt{3})}{8} \, \text{ft}^{2} / \text{sec}$.
16. The depth of the water decreases at $\frac{128}{125\pi}$ ft/min.
17. The volume is decreasing at a rate of $\frac{(25\pi )}{16}{\text{ft}}^{3}\text{/min}.$
18. The water flows out at rate $\frac{2\pi}{5} \, \text{m}^3$/min.
19. $\frac{3}{2}$ m/sec
20. The angle decreases at $\frac{400}{1681}$ rad/sec.
21. $100\pi$ mi/min
22. The angle is changing at a rate of $\frac{11}{25}$ rad/sec.
23. The distance is increasing at a rate of $62.50$ ft/sec.
24. The distance is decreasing at a rate of $11.99$ ft/sec.

Linear Approximations and Differentials

1.  
2. $f^{\prime}(a)=0$
3.  
4. The linear approximation exact when $y=f(x)$ is linear or constant.
5. $L(x)=\frac{1}{2}-\frac{1}{4}(x-2)$
6. $L(x)=1$
7. $L(x)=0$
8. $0.02$
9. $1.9996875$
10. $0.001593$
11. $1$; error, $~0.00005$
12. $0.97$; error, $~0.0006$
13. $3-\frac{1}{600}$; error, $~4.632\times 10^{-7}$
14. $dy=(\cos x-x \sin x) \, dx$
15. $dy=(\frac{x^2-2x-2}{(x-1)^2}) \, dx$
16. $dy=-\frac{1}{(x+1)^2} \, dx$, $-\frac{1}{16}$
17. $dy=\frac{9x^2+12x-2}{2(x+1)^{3/2}} \, dx$, –$0.1$
18. $dy=(3x^2+2-\frac{1}{x^2}) \, dx$, $0.2$
19. $12x \, dx$
20. $4\pi r^2 \, dr$
21. $-1.2\pi \, \text{cm}^3$
22. $-100 \, \text{ft}^3$
23.  
24.  
25.  

Maxima and Minima

1. Answers may vary
2. Answers will vary
3. No; answers will vary
4. Since the absolute maximum is the function (output) value rather than the $x$ value, the answer is no; answers will vary
5. When $a=0$
6. Absolute minimum at $3$; Absolute maximum at −$2.2$; local minima at −$2$, $1$; local maxima at −$1$, $2$
7. Absolute minima at −$2$, $2$; absolute maxima at −$2.5$, $2.5$; local minimum at $0$; local maxima at −$1$, $1$
8. Answers may vary.
9. Answers may vary.
10. $x=1$
11. None
12. $x=0$
13. None
14. $x=-1,1$
15. Absolute maximum: $x=4$, $y=\frac{33}{2}$; absolute minimum: $x=1$, $y=3$
16. Absolute minimum: $x=\frac{1}{2}$, $y=4$
17. Absolute maximum: $x=2\pi$, $y=2\pi$; absolute minimum: $x=0$, $y=0$
18. Absolute maximum: $x=-3$; absolute minimum: $-1\le x\le 1$, $y=2$
19. Absolute maximum: $x=\frac{\pi}{4}$, $y=\sqrt{2}$; absolute minimum: $x=\frac{5\pi}{4}$, $y=−\sqrt{2}$
20. Absolute minimum: $x=-2$, $y=1$
21. Absolute minimum: $x=-3$, $y=-135$; local maximum: $x=0$, $y=0$; local minimum: $x=1$, $y=-7$
22. Local maximum: $x=1-2\sqrt{2}$, $y=3-4\sqrt{2}$; local minimum: $x=1+2\sqrt{2}$, $y=3+4\sqrt{2}$
23. Absolute maximum: $x=\frac{\sqrt{2}}{2}$, $y=\frac{3}{2}$; absolute minimum: $x=-\frac{\sqrt{2}}{2}$, $y=-\frac{3}{2}$
24. Local maximum: $x=-2$, $y=59$; local minimum: $x=1$, $y=-130$
25. Absolute maximum: $x=0$, $y=1$; absolute minimum: $x=-2,2$, $y=0$
26. Absolute minima: $x=0$, $x=2$, $y=1$; local maximum at $x=1$, $y=2$
27. No maxima/minima if $a$ is odd, minimum at $x=1$ if $a$ is even

The Mean Value Theorem

1.  
2. One example is $f(x)=|x|+3, \, -2 \le x \le 2$
3.  
4. Yes, but the Mean Value Theorem still does not apply
5. $(−\infty,0), \, (0,\infty)$
6. $(−\infty,-2), \, (2,\infty)$
7. 2 points]
8. 5 points
9. $c=\frac{2\sqrt{3}}{3}$
10. $c=\frac{1}{2}, \, 1, \, \frac{3}{2}$
11. $c=1$
12. Not differentiable
13. Not differentiable
14. Yes
15. The Mean Value Theorem does not apply since the function is discontinuous at $x=\frac{1}{4}, \, \frac{3}{4}, \, \frac{5}{4}, \, \frac{7}{4}$.
16. Yes
17. The Mean Value Theorem does not apply; discontinuous at $x=0$.
18. Yes
19. The Mean Value Theorem does not apply; not differentiable at $x=0$.
20. $c=\pm \frac{1}{\pi} \cos^{-1}(\frac{\sqrt{\pi}}{2})$; $c=\pm 0.1533$
21. The Mean Value Theorem does not apply.
22. $\frac{1}{2\sqrt{c+1}}-\frac{2}{c^3}=\frac{521}{2880}$; $c=3.133,5.867$
23. Yes
24. It is constant.

Derivatives and the Shape of a Graph

1.  
2. It is not a local maximum/minimum because $f^{\prime}$ does not change sign
3.  
4. No
5.  
6. False; for example, $y=\sqrt{x}$.
7.  
8. Increasing for $-2 < x < 1$ and $x > 2$; decreasing for $x < 2$ and $-1 < x < 2$
9. Decreasing for $x < 1$; increasing for $x > 1$
10. Decreasing for $-2 < x < -1$ and $1 < x < 2$; increasing for $-1 < x < 1$ and $x < -2$ and $x > 2$
11. 1. Increasing over $-2 < x < 1, \, 0 < x < 1, \, x > 2$; decreasing over $x < -2 \, -1 < x < 0, \, 1 < x < 2$
    2. maxima at $x=-1$ and $x=1$, minima at $x=-2$ and $x=0$ and $x=2$
12. 1. Increasing over $x > 0$, decreasing over $x < 0$
    2. Minimum at $x=0$
13. Concave up on all $x$, no inflection points
14. Concave up on all $x$, no inflection points
15. Concave up for $x < 0$ and $x > 1$, concave down for $0 < x < 1$, inflection points at $x=0$ and $x=1$
16. Answers will vary
17. Answers will vary
18. 1. Concave up for $x > \frac{4}{3}$, concave down for $x < \frac{4}{3}$
    2. Inflection point at $x=\frac{4}{3}$
19. 1. Increasing over $x < 0$ and $x > 4$, decreasing over $0 < x < 4$
    2. Maximum at $x=0$, minimum at $x=4$
    3. Concave up for $x > 2$, concave down for $x < 2$
    4. Infection point at $x=2$
20. 1. Increasing over $x < 0$ and $x > \frac{60}{11}$, decreasing over $0 < x < \frac{60}{11}$
    2. Minimum at $x=\frac{60}{11}$
    3. Concave down for $x < \frac{54}{11}$, concave up for $x > \frac{54}{11}$
    4. Inflection point at $x=\frac{54}{11}$
21. 1. Increasing over $x > -\frac{1}{2}$, decreasing over $x < -\frac{1}{2}$
    2. Minimum at $x=-\frac{1}{2}$
    3. Concave up for all $x$
    4. No inflection points
22. 1. Increases over $-\frac{1}{4} < x < \frac{3}{4}$, decreases over $x > \frac{3}{4}$ and $x < -\frac{1}{4}$
    2. Minimum at $x=-\frac{1}{4}$, maximum at $x=\frac{3}{4}$
    3. Concave up for $-\frac{3}{4} < x < \frac{1}{4}$, concave down for $x < \frac{3}{4}$ and $> \frac{1}{4}$
    4. Inflection points at $x=-\frac{3}{4}, \, x=\frac{1}{4}$
23. 1. Increasing for all $x$
    2. No local minimum or maximum
    3. Concave up for $x > 0$, concave down for $x < 0$
    4. Inflection point at $x=0$
24. 1. Increasing for all $x$ where defined
    2. No local minima or maxima
    3. Concave up for $x < 1$, concave down for $x > 1$
    4. No inflection points in domain
25. 1. Increasing over $-\frac{\pi }{4} < x < \frac{3\pi }{4}$, decreasing over $x > \frac{3\pi }{4}, \, x < -\frac{\pi }{4}$
    2. Minimum at $x=-\frac{\pi }{4}$, maximum at $x=\frac{3\pi }{4}$
    3. Concave up for $-\frac{\pi }{2} < x < \frac{\pi }{2}$, concave down for $x < \frac{\pi }{2}, \, x > \frac{\pi }{2}$
    4. Infection points at $x=\pm \frac{\pi }{2}$
26. 1. Increasing over $x > 4$, decreasing over $0 < x < 4$
    2. Minimum at $x=4$
    3. Concave up for $0 < x < 8\sqrt[3]{2}$, concave down for $x > 8\sqrt[3]{2}$
    4. Inflection point at $x=8\sqrt[3]{2}$
27. $f > 0, \, f^{\prime} > 0, \, f^{\prime \prime} < 0$
28. $f > 0, \, f^{\prime} < 0, \, f^{\prime \prime} < 0$
29. $f > 0, \, f^{\prime} > 0, \, f^{\prime \prime} > 0$
30. True, by the Mean Value Theorem]
31. True, examine derivative