Stewart 9e Section 2.2: The Derivative as a Function

1 Differentiation: Definition and Fundamental Properties · Level 1

Use the given graph to estimate the value of each derivative. Then sketch the graph of \(f'\).

(a) \(f'(0)\) (b) \(f'(1)\) (c) \(f'(2)\) (d) \(f'(3)\)

Answer for 1(a)

(e) \(f'(4)\) (f) \(f'(5)\) (g) \(f'(6)\) (h) \(f'(7)\)

Answer for 1(e)

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2 Differentiation: Definition and Fundamental Properties · Level 1

Use the given graph to estimate the value of each derivative. Then sketch the graph of \(f'\).

(a) \(f'(-3)\) (b) \(f'(-2)\) (c) \(f'(-1)\) (d) \(f'(0)\)

Answer for 2(a)

(e) \(f'(1)\) (f) \(f'(2)\) (g) \(f'(3)\)

Answer for 2(e)

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3 Analytical Applications of Differentiation · Level 2

Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV. Give reasons for your choices.

4 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

5 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

6 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

7 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

8 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

9 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

10 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

11 Analytical Applications of Differentiation · Level 2

Trace or copy the graph of the given function \(f\). Then use the method of Example 1 to sketch the graph of \(f'\) below it.

12 Contextual Applications of Differentiation · Level 2

Shown is the graph of the population function \(P(t)\) for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative \(P'(t)\). What does the graph of \(P'\) tell us about the yeast population?

13 Integration and Accumulation of Change · Level 2

A rechargeable battery is plugged into a charger. The graph shows \(C(t)\), the percentage of full capacity that the battery reaches as a function of time \(t\) elapsed (in hours).

(a) What is the meaning of the derivative \(C'(t)\)?

Answer for 13(a)

(b) Sketch the graph of \(C'(t)\). What does the graph tell you?

Answer for 13(b)

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14 Analytical Applications of Differentiation · Level 2

The graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy \(F\) is measured in miles per gallon and speed \(v\) is measured in miles per hour.

(a) What is the meaning of the derivative \(F'(v)\)?

Answer for 14(a)

(b) Sketch the graph of \(F'(v)\).

Answer for 14(b)

(c) At what speed should you drive if you want to save on gas?

Answer for 14(c)

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15 Contextual Applications of Differentiation · Level 2

The graph shows how the average surface water temperature \(f\) of Lake Michigan varies over the course of a year (where \(t\) is measured in months with \(t = 0\) corresponding to January 1). Sketch the graph of the derivative function \(f'\). When is \(f'(t)\) largest?

16 Differentiation: Definition and Fundamental Properties · Level 2

Make a careful sketch of the graph of the sine function and below it sketch the graph of its derivative in the same manner as in Example 1. Can you guess what the derivative of the sine function is from its graph?

17 Differentiation: Definition and Fundamental Properties · Level 2

Let \(f(x) = x^2\).

(a) Estimate the values of \(f'(0)\), \(f'\left(\dfrac{1}{2}\right)\), \(f'(1)\), and \(f'(2)\) by zooming in on the graph of \(f\).

Answer for 17(a)

(b) Use symmetry to deduce the values of \(f'\left(-\dfrac{1}{2}\right)\), \(f'(-1)\), and \(f'(-2)\).

Answer for 17(b)

(c) Use the results from parts (a) and (b) to guess a formula for \(f'(x)\).

Answer for 17(c)

(d) Use the definition of derivative to prove that your guess in part (c) is correct.

Answer for 17(d)

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18 Differentiation: Definition and Fundamental Properties · Level 2

Let \(f(x) = x^3\).

(a) Estimate the values of \(f'(0)\), \(f'\left(\dfrac{1}{2}\right)\), \(f'(1)\), \(f'(2)\), and \(f'(3)\) by zooming in on the graph of \(f\).

Answer for 18(a)

(b) Use symmetry to deduce the values of \(f'\left(-\dfrac{1}{2}\right)\), \(f'(-1)\), \(f'(-2)\), and \(f'(-3)\).

Answer for 18(b)

(c) Use the values from parts (a) and (b) to graph \(f'\).

Answer for 18(c)

(d) Guess a formula for \(f'(x)\).

Answer for 18(d)

(e) Use the definition of a derivative to prove that your guess in part (d) is correct.

Answer for 18(e)

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19 Differentiation: Definition and Fundamental Properties · Level 1

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x) = 3 x - 8\)

20 Differentiation: Definition and Fundamental Properties · Level 1

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x) = m x + b\)

21 Differentiation: Definition and Fundamental Properties · Level 2

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(t) = 2.5 t^2 + 6 t\)

22 Differentiation: Definition and Fundamental Properties · Level 2

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x) = 4 + 8 x - 5 x^2\)

23 Differentiation: Definition and Fundamental Properties · Level 2

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(A(p) = 4 p^3 + 3 p\)

24 Differentiation: Definition and Fundamental Properties · Level 2

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(F(t) = t^3 - 5 t + 1\)

25 Differentiation: Definition and Fundamental Properties · Level 3

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x) = \dfrac{1}{x^2 - 4}\)

26 Differentiation: Definition and Fundamental Properties · Level 3

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(F(v) = \dfrac{v}{v + 2}\)

27 Differentiation: Definition and Fundamental Properties · Level 3

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(g(u) = \dfrac{u + 1}{4 u - 1}\)

28 Differentiation: Definition and Fundamental Properties · Level 2

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x) = x^4\)

29 Differentiation: Definition and Fundamental Properties · Level 3

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(f(x) = \dfrac{1}{\sqrt{1 + x}}\)

30 Differentiation: Definition and Fundamental Properties · Level 4

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. \(g(x) = \dfrac{1}{1 + \sqrt{x}}\)

31 Differentiation: Definition and Fundamental Properties · Level 3

(a) Sketch the graph of \(f(x) = 1 + \sqrt{x + 3}\) by starting with the graph of \(y = \sqrt{x}\) and using the transformations of Section 1.3.

Answer for 31(a)

(b) Use the graph from part (a) to sketch the graph of \(f'\).

Answer for 31(b)

(c) Use the definition of a derivative to find \(f'(x)\). What are the domains of \(f\) and \(f'\)?

Answer for 31(c)

(d) Graph \(f'\) and compare with your sketch in part (b).

Answer for 31(d)

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32 Differentiation: Definition and Fundamental Properties · Level 3

(a) If \(f(x) = x + \dfrac{1}{x}\), find \(f'(x)\).

Answer for 32(a)

(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of \(f\) and \(f'\).

Answer for 32(b)

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33 Differentiation: Definition and Fundamental Properties · Level 2

(a) If \(f(x) = x^4 + 2 x\), find \(f'(x)\).

Answer for 33(a)

(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of \(f\) and \(f'\).

Answer for 33(b)

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34 Differentiation: Definition and Fundamental Properties · Level 3

The table gives the number \(N(t)\), measured in thousands, of minimally invasive cosmetic surgery procedures performed in the United States for various years \(t\). \(t = 2000\): \(N(t) = 5500\), \(t = 2002\): \(N(t) = 4897\), \(t = 2004\): \(N(t) = 7470\), \(t = 2006\): \(N(t) = 9138\), \(t = 2008\): \(N(t) = 10897\), \(t = 2010\): \(N(t) = 11561\), \(t = 2012\): \(N(t) = 13035\), \(t = 2014\): \(N(t) = 13945\) Source: American Society of Plastic Surgeons

(a) What is the meaning of \(N'(t)\)? What are its units?

Answer for 34(a)

(b) Construct a table of estimated values for \(N'(t)\).

Answer for 34(b)

(c) Graph \(N\) and \(N'\).

Answer for 34(c)

(d) How would it be possible to get more accurate values for \(N'(t)\)?

Answer for 34(d)

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35 Differentiation: Definition and Fundamental Properties · Level 3

The table gives the height as time passes of a typical pine tree grown for lumber at a managed site. Tree age (years): \(14, 21, 28, 35, 42, 49\) Height (feet): \(41, 54, 64, 72, 78, 83\) Source: Arkansas Forestry Commission If \(H(t)\) is the height of the tree after \(t\) years, construct a table of estimated values for \(H'\) and sketch its graph.

36 Differentiation: Definition and Fundamental Properties · Level 3

Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures. Temperature (degree C): \(15.5, 17.7, 20.0, 22.4, 24.4\) Weight gained (g): \(37.2, 31.0, 19.8, 9.7, 29.8\) If \(W(x)\) is the weight gain at temperature \(x\), construct a table of estimated values for \(W'\) and sketch its graph. What are the units for \(W'(x)\)?

37 Contextual Applications of Differentiation · Level 2

Let \(P\) represent the percentage of a city's electrical power that is produced by solar panels \(t\) years after January 1, 2020.

(a) What does \(\dfrac{d P}{d t}\) represent in this context?

Answer for 37(a)

(b) Interpret the statement \(\dfrac{d P}{d t}\) at \(t = 2\) equals \(3.5\).

Answer for 37(b)

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38 Analytical Applications of Differentiation · Level 2

Suppose \(N\) is the number of people in the United States who travel by car to another state for a vacation in a year when the average price of gasoline is \(p\) dollars per gallon. Do you expect \(\dfrac{d N}{d p}\) to be positive or negative? Explain.

39 Differentiation: Definition and Fundamental Properties · Level 2

The graph of \(f\) is given. State, with reasons, the numbers at which \(f\) is not differentiable.

40 Differentiation: Definition and Fundamental Properties · Level 2

The graph of \(f\) is given. State, with reasons, the numbers at which \(f\) is not differentiable.

41 Differentiation: Definition and Fundamental Properties · Level 2

The graph of \(f\) is given. State, with reasons, the numbers at which \(f\) is not differentiable.

42 Differentiation: Definition and Fundamental Properties · Level 2

The graph of \(f\) is given. State, with reasons, the numbers at which \(f\) is not differentiable.

43 Differentiation: Definition and Fundamental Properties · Level 3

Graph the function \(f(x) = x + \sqrt{|x|}\). Zoom in repeatedly, first toward the point \((-1, 0)\) and then toward the origin. What is different about the behavior of \(f\) in the vicinity of these two points? What do you conclude about the differentiability of \(f\)?

44 Differentiation: Definition and Fundamental Properties · Level 3

Zoom in toward the points \((1, 0)\), \((0, 1)\), and \((-1, 0)\) on the graph of the function \(g(x) = (x^2 - 1)^{\dfrac{2}{3}}\). What do you notice? Account for what you see in terms of the differentiability of \(g\).

45 Differentiation: Definition and Fundamental Properties · Level 2

The graphs of a function \(f\) and its derivative \(f'\) are shown. Which is bigger, \(f'(-1)\) or \(f''(1)\)?

46 Differentiation: Definition and Fundamental Properties · Level 2

The graphs of a function \(f\) and its derivative \(f'\) are shown. Which is bigger, \(f'(-1)\) or \(f''(1)\)?

47 Analytical Applications of Differentiation · Level 3

The figure shows the graphs of \(f\), \(f'\), and \(f''\). Identify each curve, and explain your choices.

48 Analytical Applications of Differentiation · Level 3

The figure shows graphs of \(f\), \(f'\), \(f''\), and \(f'''\). Identify each curve, and explain your choices.

49 Contextual Applications of Differentiation · Level 3

The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.

50 Contextual Applications of Differentiation · Level 3

The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices.

51 Differentiation: Definition and Fundamental Properties · Level 2

Use the definition of a derivative to find \(f'(x)\) and \(f''(x)\). Then graph \(f\), \(f'\), and \(f''\) on a common screen and check to see if your answers are reasonable. \(f(x) = 3 x^2 + 2 x + 1\)

52 Differentiation: Definition and Fundamental Properties · Level 3

Use the definition of a derivative to find \(f'(x)\) and \(f''(x)\). Then graph \(f\), \(f'\), and \(f''\) on a common screen and check to see if your answers are reasonable. \(f(x) = x^3 - 3 x\)

53 Differentiation: Definition and Fundamental Properties · Level 3

If \(f(x) = 2 x^2 - x^3\), find \(f'(x)\), \(f''(x)\), \(f'''(x)\), and \(f^{(4)}(x)\). Graph \(f\), \(f'\), \(f''\), and \(f'''\) on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?

54 Contextual Applications of Differentiation · Level 3

(a) The graph of a position function of a car is shown, where \(s\) is measured in feet and \(t\) in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at \(t = 10\) seconds?

Answer for 54(a)

(b) Use the acceleration curve from part (a) to estimate the jerk at \(t = 10\) seconds. What are the units for jerk?

Answer for 54(b)

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55 Differentiation: Definition and Fundamental Properties · Level 3

Let \(f(x) = \sqrt[3]{x}\).

(a) If \(a \neq 0\), use Equation 2.1.5 to find \(f'(a)\).

Answer for 55(a)

(b) Show that \(f'(0)\) does not exist.

Answer for 55(b)

(c) Show that \(y = \sqrt[3]{x}\) has a vertical tangent line at \((0, 0)\).

Answer for 55(c)

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56 Differentiation: Definition and Fundamental Properties · Level 3

(a) If \(q(x) = x^{\dfrac{2}{3}}\), show that \(q'(0)\) does not exist.

Answer for 56(a)

(b) If \(a \neq 0\), find \(q'(a)\).

Answer for 56(b)

(c) Show that \(y = x^{\dfrac{2}{3}}\) has a vertical tangent line at \((0, 0)\).

Answer for 56(c)

(d) Illustrate part (c) by graphing \(y = x^{\dfrac{2}{3}}\).

Answer for 56(d)

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57 Differentiation: Definition and Fundamental Properties · Level 2

Show that the function \(f(x) = |x - 6|\) is not differentiable at \(6\). Find a formula for \(f'\) and sketch its graph.

58 Differentiation: Definition and Fundamental Properties · Level 3

Where is the greatest integer function \(f(x) = \lfloor x \rfloor\) not differentiable? Find a formula for \(f'\) and sketch its graph.

59 Differentiation: Definition and Fundamental Properties · Level 3

(a) Sketch the graph of the function \(f(x) = x |x|\).

Answer for 59(a)

(b) For what values of \(x\) is \(f\) differentiable?

Answer for 59(b)

(c) Find a formula for \(f'\).

Answer for 59(c)

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60 Differentiation: Definition and Fundamental Properties · Level 2

(a) Sketch the graph of the function \(g(x) = x + |x|\).

Answer for 60(a)

(b) For what values of \(x\) is \(g\) differentiable?

Answer for 60(b)

(c) Find a formula for \(g'\).

Answer for 60(c)

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61 Differentiation: Definition and Fundamental Properties · Level 4

Derivatives of Even and Odd Functions. Recall that a function \(f\) is called even if \(f(-x) = f(x)\) for all \(x\) in its domain and odd if \(f(-x) = -f(x)\) for all such \(x\). Prove each of the following.

(a) The derivative of an even function is an odd function.

Answer for 61(a)

(b) The derivative of an odd function is an even function.

Answer for 61(b)

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