2.E: Applications of Derivatives(Exercises)

2.1: Instantaneous Rates of Change: The Derivative

Terms and Concepts

1. T/F: Let $f$ be a position function. The average rate of change on [a,b] is the slope of the line through the points $(a,f(a))$ and $(b,f(b))$.

2. T/F: The definition of the derivative of a function at a point involves taking a limit

3. In your own words, explain the difference between the average rate of change and instantaneous rate of change.

4. In your own words, explain the difference between Definitions 7 and 10.

5. Let $y = f(x)$. Give three different notations equivalent to $“f ′ (x).”$

Problems

In Exercises 6-12, use the definition of the derivative to compute the derivative of the given function.

6. $f(x)=6$

7. $f(x)=2x$

8. $f(t) = 4-3t$

9. $g(x) =x^2$

10. $f(x) = 3x^2-x+4$

11. $r(x) = \frac{1}{x}$

12. $r(s) = \frac{1}{s-2}$

In Exercises 13-19, a function and an x-value $c$ are given.
(Note: these functions are the same as those given in Exercises 6 through 12.)

(a) Find the tangent line to the graph of the function at $c$.
(b) Find the normal line to the graph of the function at $c$.

13. $f(x) = 6,\text{ at }x=-2$.

14. $f(x) = 2x,\text{ at }x=3$.

15. $f(x) = 4-3x,\text{ at }x=7$.

16. $g(x) = x^2,\text{ at }x=2$.

17. $f(x) = 3x^2-x+4,\text{ at }x=-1$.

18. $r(x) = \frac{1}{x},\text{ at }x=-2$.

19. $r(x) = \frac{1}{x-2},\text{ at }x=3$.

In Exercises 20-23, a function $f$ and an x-value $a$ are given. Approximate the equation of the tangent line to the graph of $f$ at $x=a$ by numerically approximating $f'(a)$, using $h=0.1$.

20. $f(x) = x^2+2x+1,\,x=3$

21. $f(x) =\frac{10}{x+1},\,x=9$

22. $f(x) = e^x,\,x=2$

23. $f(x) =\cos x,\,x=0$

24. The graph of $f(x)=x^2-1$ is shown.
(a) Use the graph to approximate the slope of the tangent line to $f$ at the following points: (-1,0),(0,-1) and (2,3).
(b) Using the definition, find $f'(x)$.
(c) Find the slope of the tangent line at the points (-1,0), (0,-1) and (2,3).

25. The graph of $f(x)=\frac{1}{x+1}$ is shown.
(a) Use the graph to approximate the slope of the tangent line to $f$ at the following points: (0,1) and (1, 0.5).
(b) Using the definition, find $f'(x)$.
(c) Find the slope of the tangent line at the points (0, 1) and (1, 0.5).

In Exercises 26-29, a graph of a function $f(x)$ is given. Using the graph, sketch $f'(x)$.

26.

27.

28.

29.

30. Using the graph of $g(x)$ below, answer the following questions.
(a) Where is $g(x)>0$?
(b) Where is $g(x)<0$?
(c) Where is $g(x)=0$?
(d) Where is $g'(x)<0$?
(e) Where is $g'(x)>0$?
(f) Where is $g'(x)=0$?

Review

31. Approximate $\lim\limits_{x\to 5}\frac{x^2+2x-35}{x^2-10.5+27.5}$.

32. Use the Bisection Method to approximate, accurate to two decimal places, the root of $g(x)=x^3+x^2+x-1$ on [0.5, 0.6].

33. Give intervals on which each of the following functions are continuous.
(a) $\frac{1}{e^x+1}$
(b) $\frac{1}{e^x-1}$
(c) $\sqrt{5-x}$
(d) $\sqrt{5-x^2}$

34. Use the graph of $f(x)$ provided to answer the following.
(a) $\lim\limits_{x\to-3^-}f(x)=?$
(b) $\lim\limits_{x\to-3^+}f(x)=?$
(c) $\lim\limits_{x\to-3}f(x)=?$
(d) Where is $f$ continuous?

2.2: Interpretations of the Derivative

Terms and Concepts

1. What is the instantaneous rate of change of position called?

2. Given a function $y = f(x)$, in your own words describe how to find the units of $f ′ (x)$.

3. What functions have a constant rate of change?

Problems

4. Given $f(5=10\text{ and }f'(5)=2$, approximate $f(6)$.

5. Given $P(100) = −67$ and $P ′ (100) = 5$, approximate $P(110)$.

6. Given $z(25) = 187$ and $z ′ (25) = 17$, approximate $z(20)$.

7. Knowing $f(10) = 25$ and $f ′ (10) = 5$ and the methods described in this section, which approximation is likely to be most accurate: $f(10.1),\, f(11),\text{ or }f(20)$? Explain your reasoning.

8. Given $f(7) = 26\text{ and }f(8) = 22$, approximate $f ′ (7)$.

9. Given $H(0) = 17$ and $H(2) = 29$, approximate $H ′ (2)$.

10. Let $V(x)$ measure the volume, in decibels, measured inside a restaurant with x customers. What are the units of $V ′ (x)$?

11. Let $v(t)$ measure the velocity, in ft/s, of a car moving in a straight line $t$ seconds after starting. What are the units of $v ′ (t)$?

12. The height H, in feet, of a river is recorded $t$ hours after midnight, April 1. What are the units of $H ′ (t)$?

13. $P$ is the profit, in thousands of dollars, of producing and selling $c$ cars.
(a) What are the units of $P'(c)$?
(b) What is likely true of $P(0P$?

14. $T$ is the temperature in degrees Fahrenheit, $h$ hours after midnight on July 4 in Sidney, NE.
(a) What are the units of $T ′ (h)$?
(b) Is $T ′ (8)$ likely greater than or less than 0? Why?
(c) Is $T(8)$ likely greater than or less than 0? Why?

In Exercises 15-18, graphs of the functions $f(x)\text{ and }g(x)$ are given. Identify which function is the derivative of the other.

15.

16.

17.

18.

Review

In Exercises 19-20, use the definition to compute the derivatives of the following functions.

19. $f(x) =5x^2$

20. $f(x) = \sqrt{x}\text{ at }x=9$.

In Exercises 21-22, numerically approximate the value of $f'(x)$ at the indicated x value.

21. $f(x) =\cos x\text{ at }x=\pi$.

22. $f(x) = \sqrt{x}\text{ at }x=9$.

2.3: Basic Differentiation Rules

Terms and Concepts

1. What is the name of the rule which states that $\frac{d}{ dx} ( x^n ) = nx^{n-1}$, where $n > 0$ is an integer?

2. What is $\frac{d}{dx} \left ( \ln x \right )$?

3. Give an example of a function f(x) where $f ′ (x) = f(x)$.

4. Give an example of a function $f(x)\text{ where }f ′ (x) = 0$.

5. The derivative rules introduced in this section explain how to compute the derivative of which of the following functions?

• $f(x) = \frac{3}{x^2}$
• $g(x) = 3x^2-x+17$
• $h(x)=5\ln x$
• $j(x) = \sin x \cos x$
• $k(x)=e^{x^2}$
• $m(x)=\sqrt{x}$

6. Explain in your own words how to find the third derivative of a function $f(x)$.

7. Give an example of a function where $f ′ (x) \ne 0\text{ and }f ′′(x) = 0$.

8. Explain in your own words what the second derivative “means.”

9. If $f(x)$ describes a position function, then $f ′ (x)$ describes what kind of function? What kind of function is $f ′′(x)$?

10. Let $f(x)$ be a function measured in pounds, where x is measured in feet. What are the units of $f ′′(x)$?

Problems

In Exercises 11-25, compute the derivative of the given function.

11. $f(x) = 7x^2-5x+7$

12. $g(x) = 14x^3+7x^2+11x-29$

13. $m(t) = 9t^2-\frac{1}{8}t^3+3t-8$

14. $f(\theta) = 9\sin \theta + 10 \cos \theta$

15. $f(r) =6e^r$

16. $g(t)=10t^4-\cos t+7\sin t$

17. $f(x)=2\ln x - x$

18. $p(s) = \frac{1}{4}s^4+\frac{1}{3}s^3+\frac{1}{2}s^2+s+1$

19. $h(t) = e^t -\sin t -\cos t$

20. $f(x)=\ln (5x^2)$

21. $f(t) = \ln (17) +e^2+\sin \pi /2$

22. $g(t)=(1+3t)^2$

23. $g(x)=(2x-5)^3$

24. $f(x)=(1-x)^3$

25. $f(x) = (2-3x)^2$

26. A property of logarithms is that $\log_a x = \frac{\log_bx}{\log_b a}$, for all bases a, b>0, $\ne1$.
(a) Rewrite this identity when $b=e$, i.e., using $\log e x=\ln x$.
(b) Use part (a) to find the derivative of $y=\log_a x$.
(c) Give the derivative of $y=\log_{10}x$.

In Exercises 27-32, compute the first four derivatives of the given function.

27. $f(x) = x^6$

28. $g(x) = 2\cos x$

29. $h(t) =t^2-e^t$

30. $p(\theta)=\theta^4-\theta^3$

31. $f(\theta)=\sin \theta - \cos \theta$

32. $f(x) = 1,100$

In Exercises 33-38, find the equations of the tangent and normal lines to the graph of the function at the given point.

33. $f(x) =x^3-x\text{ at }x=1$

34. $f(t) =e^t\text{ at }t=0$

35. $g(x) = \ln x \text{ at } t=0$

36. $f(x) = 4\sin x \text{ at }x=\pi/2$

37. $f(x) = -2\cos x \text{ at }x=\pi/4$

38. $f(x) = 2x+3\text{ at } x=5$

Review

39. Given that $e^0=1$, approximate the value of $e^{0.1}$ using the tangent line to $f(x)=e^x\text{ at }x=0$.

40. Approximate the value of $(3.01)^4$ using the tangent line to $f(x) = x^4\text{ at }x=3$.

2.4: The Product and Quotient Rules

Terms and Concepts

1. T/F: The Product Rule states that $frac{d}{dx}\left ( x^2\sin x\right ) = 2x\cos x$.

2. T/F: The Quotient Rule states that $\frac{d}{dx}\left ( \frac{x^2}{\sin x}\right ) = \frac{\cos x}{2x}$.

3. T/F: The derivatives of the trigonometric functions that start with "c" have minus signs in them.

4. What derivative rule is used to extend the Power Rule to include negative integer exponents?

5. T/F: Regardless of the function, there is always exactly one right way of computing its derivative.

6. In your own words, explain what it means to make your answers "clear."

Problems

In Exercises 7-10:
(a) Use the Product Rule to differentiate the function.
(b) Manipulate the function algebraically and differentiate without the Product Rule.
(c) Show that the answers from (a) and (b) are equivalent.

7. $f(x) =x(x^2+3x)$

8. $g(x) =2x^2(5x^3)$

9. $h(s)=(2s-1)(s+4)$

10. $f(x) = (x^2+5)(3-x^3)$

In Exercises 11-14:
(a) Use the Quotient Rule to differentiate the function.
(b) Manipulate the function algebraically and differentiate without the Quotient Rule.
(c) Shows that the answers from (a) and (b) are equivalent.

11. $f(x) = \frac{x^2+3}{x}$

12. $g(x) = \frac{x^3-2x^2}{2x^2}$

13. $h(s)=\frac{3}{4s^3}$

14. $f(t) = \frac{t^2-1}{t+1}$

In Exercises 15-29, compute the derivative of the given function.

15. $f(x) =x\sin x$

16. $f(t)=\frac{1}{t^2}(\csc t -4)$

17. $g(x) = \frac{x+7}{x-5}$

18. $g(t) = \frac{t^5}{\cos t -2t^2}$

19. $h(x) = \cot x-e^x$

20. $h(t) = 7t^2+6t-2$

21. $f(x) = \frac{x^4+2x^3}{x^2}$

22. $f(x) = (16x^3+24x^2+3x)\frac{7x-1}{16x^3+24x^2+3x}$

23. $f(t) = t^5(\sec t+e^t)$

24. $f(x) = \frac{\sin x}{\cos x+3}$

25. $g(x) = e^2 \left ( \sin (\pi/4)-1\right )$

26. $g(t) = 4t^3e^t -\sin t\cos t$

27. $h(t) = \frac{t^2 \sin t +3}{t^2\cos t +2}$

28. $f(x)=x^2e^x\tan x$

29. $g(x) = 2x\sin x \sec x$

In Exercises 30-33, find the equations of the tangent and normal lines to the graph of $g$ at the indicated point.

30. $g(s)=e^s (s^2+2)\text{ at }(0,2)$.

31. $g(t) = t\sin t\text{ at }\left (\frac{3\pi}{2},\frac{-3\pi}{2}\right )$

32. $g(x) = \frac{x^2}{x-1}\text{ at }(2,4)$.

33. $g(\theta)=\frac{\cos \theta -8\theta}{\theta +1}\text{ at }(0,-5)$

In Exercises 34-37, find the x-values where the graph of the function has a horizontal tangent line.

34. $f(x) = 6x^2-18x-24$

35. $f(x)=x\sin x\text{ on }[-1,1]$

36. $f(x)=\frac{x}{x+1}$

37. $f(x)=\frac{x^2}{x+1}$

In Exercises 38-41, find the requested derivative.

38. $f(x) = x\sin x;\text{ find }f''(x)$.

39. $f(x) = x\sin x;\text{ find }f^{(4)}(x)$.

40. $f(x) = \csc x;\text{ find }f''(x)$.

41. $f(x) = (x^3-5x+2)(x^2+x-7);\text{ find }f^{(8)}(x)$.

In Exercises 42-45, use the graph of $f(x)$ to sketch $f'(x)$.

42.

43.

44.

45.

2.5: The Chain Rule

Terms and Concepts

1. T/F: The Chain Rule describes how to evaluate the derivative of a composition of functions.

2. T/F: The Generalized Power Rule states that $\frac{d}{dx}\left ( g(x)^n \right ) = n\left ( g(x)\right )^{n-1}$.

3. T/F: $\frac{d}{dx}\left ( \ln (x^2)\right )=\frac{1}{x^2}$.

4. T/F: $\frac{d}{dx} (3^x) \approx 1.1 \cdot 3^x$.

5. T/F: $\frac{dx}{dy} = \frac{dx}{dt}\cdot\frac{dt}{dy}$

6. T/F: Taking the derivative of $f(x)=x^2\sin (5x)$ requires the use of both the Product and Chain Rules.

Problems

In Exercises 7-28, compute the derivatives of the given function.

7. $f(x) = \left ( 4x^3-x \right ) ^{10}$

8. $f(t) = \left ( 3t-2 \right ) ^5$

9. $g(\theta) = \left ( \sin \theta +\cos \theta \right ) ^3$

10. $h(t) = e^{3t^2+y-1}$

11. $f(x) = \left ( x+\frac{1}{x}\right )^4$

12. $f(x) = \cos (3x)$

13. $g(x) =\tan (5x)$

14. $h(t) = \sin^4 (2t)$

15. $p(t) = \cos^3 (t^2+3t+1)$

16. $f(x) = \ln (\cos x)$

17. $f(x) = \ln (x^2)$

18. $f(x) = 2\ln (x)$

19. $g(r) = 4^r$

20. $g(t) = 5^{\cos t}$

21. $g(t) = 15^2$

22. $m(w) = \dfrac{3^w}{2^w}$

23. $h(t) = \dfrac{2^t+3}{3^t+2}$

24. $m(w) = \dfrac{3^w+1}{2^w}$

25. $f(x) = \dfrac{3^{x^2}+x}{2^{x^2}}$

26. $f(x) =x^2\sin (5x)$

27. $g(t) = \cos (t^2+3t)\sin (5t-7)$

28. $g(t) = \cos (\frac{1}{t})e^{5t^2}$

In Exercises 29-32, find the equation of tangent and normal lines to the graph of the function at the given point. Note: the functions here are the same as in Exercises 7 through 10.

29. $f(x) = \left ( 4x^3 -x\right )^{10}\text{ at }x=0$

30. $f(t) = \left ( 3t-2\right )^5\text{ at }t=1$

31. $g(\theta) = \left ( \sin \theta +\cos \theta \right )^3\text{ at }\theta=\pi/2$

32. $h(t) = e^{3t^2+t-1}\text{ at }t=-1$

33. Compute $\frac{d}{dx}\left ( \ln (kx)\right )$ two ways:
(a) Using the Chain rule, and
(b) by first using the logarithm rule $\ln (ab)=\ln a +\ln b$, then taking the derivative.

34. Compute $\frac{d}{dx}\left ( \ln (x^k)\right )$ two ways:
(a) Using the Chain Rule, and
(b) by first using the logarithm rule $\ln (a^p)=p\ln a$, then taking the derivative.

Review

35. The “wind chill factor” is a measurement of how cold it “feels” during cold, windy weather. Let $W(w)$ be the wind chill factor, in degrees Fahrenheit, when it is 25 F outside with a wind of $w$ mph.
(a) What are the units of $W' (w)$?
(b) What would you expect the sign of $W'(10)$ to be?

36. Find the derivatives of the following functions.
(a) $f(x) =x^2e^x\cot x$
(b) $g(x) = 2^x3^x4^x$

2.6: Implicit Differentiation

Terms and Concepts

1. In your own words, explain the difference between implicit functions and explicit functions.

2. Implicit differentiation is based on what other differentiation rule?

3. T/F: Implicit differentiation can be used to find the derivative of $y=\sqrt{x}$.

4. T/F: Implicit differentiation can be used to find the derivative of $y=x^{3/4}$.

Problems

In Exercises 5-12, compute the derivative of the given function.

5. $f(x) =\sqrt{x}+\frac{1}{\sqrt{x}}$

6. $f(x) = \sqrt[3]{x}+x^{2/3}$

7. $f(x) =\sqrt{1-t^2}$

8. $g(t) = \sqrt{t}\sin t$

9. $h(x) =x^{1.5}$

10. $f(x)=x^\pi +x^{1.9}+\pi^{1.9}$

11. $g(x) = \frac{x+7}{\sqrt{x}}$

12. $f(t) = \sqrt[5]{t}\left (\sec t +e^t \right )$

In Exercises 13-25, find $\frac{dy}{dx}$ using implicit differentiation.

13. $x^4+y^2+y=7$

14. $x^{2/5}+y^{2/5} = 1$

15. $\cos (x) +\sin (y)=1$

16. $frac{x}{y}=10$

17. $\frac{y}{x} =10$

18. $x^2e^2+2^y=5$

19. $x^2\tan y=50$

20. $\left ( 3x^2+2y^3\right )^4=2$

21. $\left (y^2+2y-x\right )^2 =200$

22. $\frac{x^2+y}{x+y^2}=17$

23. $\frac{\sin (x)+y}{\cos (y) +x}=1$

24. $\ln (x^2+y^2 )=e$

25. $\ln \left (x^2+xy+y^2\right )=1$

26. Show that $\frac{dy}{dx}$ is the same for each of the following implicitly defined functions.
(a) $xy=1$
(b) $x^2y^2=1$
(c) $\sin (xy)=1$
(d) $\ln (xy)=1$

In Exercises 27-31, find the equation of the tangent line to the graph of the implicitly defined function at the indicated points. As a visual aid, each function is graphed.

27. $x^{2/5}+y^{2/5} =1$
(a) At (1,0)
(b) At (0.1, 0.281) (which does not exactly lie on the curve, but is very close).

28. $x^4+y^4=1$
(a) At (1,0).
(b) At $(\sqrt{0.6},\sqrt{0.8})$.
(c) At (0,1).

29. $(x^2+y^2-4)^3=108y^2$
(a) At (0,4).
(b) At $(2,-\sqrt[4]{108})$

30. $(x^2+y^2+x)^2=x^2+y^2$
(a) At (0,1).
(b) At $\left ( -\frac{3}{4},\frac{3\sqrt{3}}{4}\right )$.

31. $(x-2)^2+(y-3)^2=9$
(a) At $\left ( \frac{7}{2},\frac{6+3\sqrt{3}}{2}\right )$.
(b) At $\left ( \frac{4+3\sqrt{3}}{2},\frac{3}{2}\right )$.

In Exercises 32-35, an implicitly defined function is given. Find $\frac{d^y}{dx^2}$. Note: these are the same problems used in Exercises 13-16.

32. $x^4+y^2+y=7$

33. $x^{2/5}+y^{2/5}=1$

34. $\cos x +\sin y =1$

35. $\frac{x}{y} =10$

In Exercises 36-41, use logarithmic differentiation to find $\frac{dy}{dx}$, then find the equation of the tangent line at the indicated x-value.

36. $y=\left ( 1+x\right )^{1/x},\quad x=1$

37. $y=2x^{x^2},\quad x=1$

38. $y=\frac{x^x}{x+1},\quad x=1$

39. $y=x^{\sin (x)+2},\quad x=1$

40. $y=\frac{x+1}{x+2},\quad x=1$

41. $y=\frac{(x+1)(x+2)}{(x+3)(x+4)},\quad x=1$

2.7: Derivatives of Inverse Functions

Terms and Concepts

1. T/F: Every function has an inverse.

2. In your own words explain what it means for a function to be "one on one."

3. If (1,10) lies on the graph of $y=f(x)$, what can be said about the graph of $y=f^{-1}(x)$?

4. If (1,10) lies on the graph of $y=f(x)\text{ and }f'(1)=5,$ what can be said about $y=f^{-1}(x)$?

Problems

In Exercises 5-8, verify that the given functions are inverses.

5. $f(x) =2x+6\text{ and }g(x)=\frac{1}{2}x-3$

6. $f(x) = x^2+6x+11,\,x\ge 3$ and $g(x) = \sqrt{x-2}-3,\, x\ge 2$

7. $f(x) = \frac{3}{x-5},\,x\ne 5$ and $g(x) = \frac{3+5x}{x},\, x\ne 0$

8. $f(x) = \frac{x+1}{x-1},\, x\ne 1\text{ and }g(x)=f(x)$

In Exercises 9-14, an invertible function $f(x)$ is given along with a point that lies on its graph. Using Theorem 22, evaluate $\left ( f^{-1}\right )^\prime (x)$ at the indicated value.

9. $f(x)=5x+10$
Point = (2,20)
Evaluate $\left ( f^{-1}\right )^\prime (20)$

10. $f(x)=x^2-2x+4,\,x\ge 1$
Point = $(3,7)$
Evaluate $\left ( f^{-1}\right )^\prime (7)$

11. $f(x)=\sin 2x,\,-\pi/4 \le x \le \pi/4$
Point = $(\pi/6,\sqrt{3}/2)$
Evaluate $\left ( f^{-1}\right )^\prime (\sqrt{3}/2)$

12. $f(x)=x^3-6x^2+15x-2$
Point = $(1,8)$
Evaluate $\left ( f^{-1}\right )^\prime (8)$

13. $f(x)=\frac{1}{1+x^2},\,x\ge 0$
Point = $(1,1/2)$
Evaluate $\left ( f^{-1}\right )^\prime (1/2)$

14. $f(x)=6e^{3x}$
Point = $(0,6)$
Evaluate $\left ( f^{-1}\right )^\prime (6)$

In Exercises 15-24, compute the derivative of the given function.

15. $h(t) = \sin^{-1}(2t)$

16. $f(t) = \sec^{-1}(2t)$

17. $g(x) = \tan^{-1}(2x)$

18. $f(x) = x\sin^{-1}(x)$

19. $g(t) = \sin t \cos^{-1}t$

20. $f(t) = \ln te^t$

21. $h(x) = \frac{\sin^{-1}x}{\cos^{-1}x}$

22. $g(x) = \tan^{-1}(\sqrt{x})$

23. $f(x) = \sec^{-1}(1/x)$

24. $f(x) = \sin (\sin^{-1}x)$

In Exercises 25-27, compute the derivative of the given function in two ways:
(a) By simplifying first, then taking the derivative, and
(b) by using the Chain Rule first then simplifying.

Very that the two answers are the same.

25. $f(x) = \sin \left ( \sin^{-1}x\right )$

26. $f(x) =\tan^{-1} (\tan x)$

27. $f(x)=\sin \left (\cos^{-1}x\right )$

In Exercises 28-29, find the equation of the line tangent to the graph of $f$ at the indicated value.

28. $f(x)=\sin^{-1}x\text{ at }x=\frac{\sqrt{2}}{2}$

29. $f(x)=\cos^{-1}(2x)\text{ at }x=\frac{\sqrt{3}}{4}$

Review

30. Find $\frac{dy}{dx}$, where $x^2y-y^2x=1$.

31. Find the equation of the line tangent to the graph of $x^2+y^2+xy=7$ at the point (1,2).

32. Let $f(x) =x^3+x$. Evaluate $\lim\limits_{s\to 0}\frac{f(x+s)-f(x)}{s}$.