Analytical Applications of Derivatives: Get Stronger Answer Key

Related Rates

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

Linear Approximations and Differentials

  1.  
  2. [latex]f^{\prime}(a)=0[/latex]
  3.  
  4. The linear approximation exact when [latex]y=f(x)[/latex] is linear or constant.
  5. [latex]L(x)=\frac{1}{2}-\frac{1}{4}(x-2)[/latex]
  6. [latex]L(x)=1[/latex]
  7. [latex]L(x)=0[/latex]
  8. [latex]0.02[/latex]
  9. [latex]1.9996875[/latex]
  10. [latex]0.001593[/latex]
  11. [latex]1[/latex]; error, [latex]~0.00005[/latex]
  12. [latex]0.97[/latex]; error, [latex]~0.0006[/latex]
  13. [latex]3-\frac{1}{600}[/latex]; error, [latex]~4.632\times 10^{-7}[/latex]
  14. [latex]dy=(\cos x-x \sin x) \, dx[/latex]
  15. [latex]dy=(\frac{x^2-2x-2}{(x-1)^2}) \, dx[/latex]
  16. [latex]dy=-\frac{1}{(x+1)^2} \, dx[/latex], [latex]-\frac{1}{16}[/latex]
  17. [latex]dy=\frac{9x^2+12x-2}{2(x+1)^{3/2}} \, dx[/latex], –[latex]0.1[/latex]
  18. [latex]dy=(3x^2+2-\frac{1}{x^2}) \, dx[/latex], [latex]0.2[/latex]
  19. [latex]12x \, dx[/latex]
  20. [latex]4\pi r^2 \, dr[/latex]
  21. [latex]-1.2\pi \, \text{cm}^3[/latex]
  22. [latex]-100 \, \text{ft}^3[/latex]
  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 [latex]x[/latex] value, the answer is no; answers will vary
  5. When [latex]a=0[/latex]
  6. Absolute minimum at [latex]3[/latex]; Absolute maximum at −[latex]2.2[/latex]; local minima at −[latex]2[/latex], [latex]1[/latex]; local maxima at −[latex]1[/latex], [latex]2[/latex]
  7. Absolute minima at −[latex]2[/latex], [latex]2[/latex]; absolute maxima at −[latex]2.5[/latex], [latex]2.5[/latex]; local minimum at [latex]0[/latex]; local maxima at −[latex]1[/latex], [latex]1[/latex]
  8. Answers may vary.
  9. Answers may vary.
  10. [latex]x=1[/latex]
  11. None
  12. [latex]x=0[/latex]
  13. None
  14. [latex]x=-1,1[/latex]
  15. Absolute maximum: [latex]x=4[/latex], [latex]y=\frac{33}{2}[/latex]; absolute minimum: [latex]x=1[/latex], [latex]y=3[/latex]
  16. Absolute minimum: [latex]x=\frac{1}{2}[/latex], [latex]y=4[/latex]
  17. Absolute maximum: [latex]x=2\pi[/latex], [latex]y=2\pi[/latex]; absolute minimum: [latex]x=0[/latex], [latex]y=0[/latex]
  18. Absolute maximum: [latex]x=-3[/latex]; absolute minimum: [latex]-1\le x\le 1[/latex], [latex]y=2[/latex]
  19. Absolute maximum: [latex]x=\frac{\pi}{4}[/latex], [latex]y=\sqrt{2}[/latex]; absolute minimum: [latex]x=\frac{5\pi}{4}[/latex], [latex]y=−\sqrt{2}[/latex]
  20. Absolute minimum: [latex]x=-2[/latex], [latex]y=1[/latex]
  21. Absolute minimum: [latex]x=-3[/latex], [latex]y=-135[/latex]; local maximum: [latex]x=0[/latex], [latex]y=0[/latex]; local minimum: [latex]x=1[/latex], [latex]y=-7[/latex]
  22. Local maximum: [latex]x=1-2\sqrt{2}[/latex], [latex]y=3-4\sqrt{2}[/latex]; local minimum: [latex]x=1+2\sqrt{2}[/latex], [latex]y=3+4\sqrt{2}[/latex]
  23. Absolute maximum: [latex]x=\frac{\sqrt{2}}{2}[/latex], [latex]y=\frac{3}{2}[/latex]; absolute minimum: [latex]x=-\frac{\sqrt{2}}{2}[/latex], [latex]y=-\frac{3}{2}[/latex]
  24. Local maximum: [latex]x=-2[/latex], [latex]y=59[/latex]; local minimum: [latex]x=1[/latex], [latex]y=-130[/latex]
  25. Absolute maximum: [latex]x=0[/latex], [latex]y=1[/latex]; absolute minimum: [latex]x=-2,2[/latex], [latex]y=0[/latex]
  26. Absolute minima: [latex]x=0[/latex], [latex]x=2[/latex], [latex]y=1[/latex]; local maximum at [latex]x=1[/latex], [latex]y=2[/latex]
  27. No maxima/minima if [latex]a[/latex] is odd, minimum at [latex]x=1[/latex] if [latex]a[/latex] is even

The Mean Value Theorem

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

Derivatives and the Shape of a Graph

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