Problem Set: Defining the Derivative

For the following exercises (1-10), use the definition of a derivative to find the slope of the secant line between the values [latex]x_1[/latex] and [latex]x_2[/latex] for each function [latex]y=f(x)[/latex].

1. [latex]f(x)=4x+7; \,\,\, x_1=2, \,\,\, x_2=5[/latex]

2. [latex]f(x)=8x-3; \,\,\, x_1=-1, \,\,\, x_2=3[/latex]

3. [latex]f(x)=x^2+2x+1; \,\,\, x_1=3, \,\,\, x_2=3.5[/latex]

4. [latex]f(x)=\text{−}{x}^{2}+x+2;{x}_{1}=0.5,{x}_{2}=1.5[/latex]

5. [latex]f(x)=\dfrac{4}{3x-1}; \,\,\, x_1=1, \,\,\, x_2=3[/latex]

6. [latex]f(x)=\dfrac{x-7}{2x+1}; \,\,\, x_1=0, \,\,\, x_2=2[/latex]

7. [latex]f(x)=\sqrt{x}; \,\,\, x_1=1, \,\,\, x_2=16[/latex]

8. [latex]f(x)=\sqrt{x-9}; \,\,\, x_1=10, \,\,\, x_2=13[/latex]

9. [latex]f(x)=x^{\frac{1}{3}}+1; \,\,\, x_1=0, \,\,\, x_2=8[/latex]

10. [latex]f(x)=6x^{\frac{2}{3}}+2x^{\frac{1}{3}}; \,\,\, x_1=1, \,\,\, x_2=27[/latex]

For the following functions (11-20),

  1. Use [latex]m_{\tan}=\underset{h\to 0}{\lim}\dfrac{f(a+h)-f(a)}{h}[/latex] to find the slope of the tangent line [latex]m_{\tan}=f^{\prime}(a)[/latex], and
  2. find the equation of the tangent line to [latex]f[/latex] at [latex]x=a[/latex].

11. [latex]f(x)=3-4x, \,\,\, a=2[/latex]

12. [latex]f(x)=\dfrac{x}{5}+6, \,\,\, a=-1[/latex]

13. [latex]f(x)=x^2+x, \,\,\, a=1[/latex]

14. [latex]f(x)=1-x-x^2, \,\,\, a=0[/latex]

15. [latex]f(x)=\dfrac{7}{x}, \,\,\, a=3[/latex]

16. [latex]f(x)=\sqrt{x+8}, \,\,\, a=1[/latex]

17. [latex]f(x)=2-3x^2, \,\,\, a=-2[/latex]

18. [latex]f(x)=\dfrac{-3}{x-1}, \,\,\, a=4[/latex]

19. [latex]f(x)=\dfrac{2}{x+3}, \,\,\, a=-4[/latex]

20. [latex]f(x)=\dfrac{3}{x^2}, \,\,\, a=3[/latex]

For the following functions [latex]y=f(x)[/latex] (21-30), find [latex]f^{\prime}(a)[/latex] using [latex]f^{\prime}(a)=\underset{x\to a}{\lim}\dfrac{f(x)-f(a)}{x-a}[/latex].

21. [latex]f(x)=5x+4, \,\,\, a=-1[/latex]

22. [latex]f(x)=-7x+1, \,\,\, a=3[/latex]

23. [latex]f(x)=x^2+9x, \,\,\, a=2[/latex]

24. [latex]f(x)=3x^2-x+2, \,\,\, a=1[/latex]

25. [latex]f(x)=\sqrt{x}, \,\,\, a=4[/latex]

26. [latex]f(x)=\sqrt{x-2}, \,\,\, a=6[/latex]

27. [latex]f(x)=\dfrac{1}{x}, \,\,\, a=2[/latex]

28. [latex]f(x)=\dfrac{1}{x-3}, \,\,\, a=-1[/latex]

29. [latex]f(x)=\dfrac{1}{x^3}, \,\,\, a=1[/latex]

30. [latex]f(x)=\dfrac{1}{\sqrt{x}}, \,\,\, a=4[/latex]

For the following exercises (31-34), given the function [latex]y=f(x)[/latex],

  1. find the slope of the secant line [latex]PQ[/latex] for each point [latex]Q(x,f(x))[/latex] with [latex]x[/latex] value given in the table.
  2. Use the answers from a. to estimate the value of the slope of the tangent line at [latex]P[/latex].
  3. Use the answer from b. to find the equation of the tangent line to [latex]f[/latex] at point [latex]P[/latex].

31. [T] [latex]f(x)=x^2+3x+4, \,\,\, P(1,8)[/latex] (Round to 6 decimal places.)

[latex]x[/latex] Slope [latex]m_{PQ}[/latex] [latex]x[/latex] Slope [latex]m_{PQ}[/latex]
1.1 (i) 0.9 (vii)
1.01 (ii) 0.99 (viii)
1.001 (iii) 0.999 (ix)
1.0001 (iv) 0.9999 (x)
1.00001 (v) 0.99999 (xi)
1.000001 (vi) 0.999999 (xii)

32. [T] [latex]f(x)=\dfrac{x+1}{x^2-1}, \,\,\, P(0,-1)[/latex]

[latex]x[/latex] Slope [latex]m_{PQ}[/latex] [latex]x[/latex] Slope [latex]m_{PQ}[/latex]
0.1 (i) -0.1 (vii)
0.01 (ii) -0.01 (viii)
0.001 (iii) -0.001 (ix)
0.0001 (iv) -0.0001 (x)
0.00001 (v) -0.00001 (xi)
0.000001 (vi) -0.000001 (xii)

33. [T] [latex]f(x)=10e^{0.5x}, \,\,\, P(0,10)[/latex] (Round to 4 decimal places.)

[latex]x[/latex] Slope [latex]m_{PQ}[/latex]
-0.1 (i)
-0.01 (ii)
-0.001 (iii)
-0.0001 (iv)
-0.00001 (v)
−0.000001 (vi)

34. [T] [latex]f(x)= \tan (x), \,\,\, P(\pi,0)[/latex]

[latex]x[/latex] Slope [latex]m_{PQ}[/latex]
3.1 (i)
3.14 (ii)
3.141 (iii)
3.1415 (iv)
3.14159 (v)
3.141592 (vi)

For the following position functions [latex]y=s(t)[/latex], an object is moving along a straight line, where [latex]t[/latex] is in seconds and [latex]s[/latex] is in meters. Find

  1. the simplified expression for the average velocity from [latex]t=2[/latex] to [latex]t=2+h[/latex];
  2. the average velocity between [latex]t=2[/latex] and [latex]t=2+h[/latex], where (i) [latex]h=0.1[/latex], (ii) [latex]h=0.01[/latex], (iii) [latex]h=0.001[/latex], and (iv) [latex]h=0.0001[/latex]; and
  3. use the answer from a. to estimate the instantaneous velocity at [latex]t=2[/latex] seconds.

35. [T] [latex]s(t)=\frac{1}{3}t+5[/latex]

36. [T] [latex]s(t)=t^2-2t[/latex]

37. [T] [latex]s(t)=2t^3+3[/latex]

38. [T] [latex]s(t)=\dfrac{16}{t^2}-\dfrac{4}{t}[/latex]

39. Use the following graph to evaluate a. [latex]f^{\prime}(1)[/latex] and b. [latex]f^{\prime}(6)[/latex].

This graph shows two connected line segments: one going from (1, 0) to (4, 6) and the other going from (4, 6) to (8, 8).

40. Use the following graph to evaluate a. [latex]f^{\prime}(-3)[/latex] and b. [latex]f^{\prime}(1.5)[/latex].

This graph shows two connected line segments: one going from (−4, 3) to (1, 3) and the other going from (1, 3) to (1.5, 4).

For the following exercises, use the limit definition of derivative to show that the derivative does not exist at [latex]x=a[/latex] for each of the given functions.

41. [latex]f(x)=x^{\frac{1}{3}}, \, x=0[/latex]

42. [latex]f(x)=x^{\frac{2}{3}}, \, x=0[/latex]

43. [latex]f(x)=\begin{cases} 1 & \text{ if } \, x<1 \\ x & \text{ if } \, x \ge 1 \end{cases}, \, x=1[/latex]

44. [latex]f(x)=\dfrac{|x|}{x}, \, x=0[/latex]

45. [T] The position in feet of a race car along a straight track after [latex]t[/latex] seconds is modeled by the function [latex]s(t)=8t^2-\frac{1}{16}t^3[/latex].

  1. Find the average velocity of the vehicle over the following time intervals to four decimal places:
    1. [4, 4.1]
    2. [4, 4.01]
    3. [4, 4.001]
    4. [4, 4.0001]
  2. Use a. to draw a conclusion about the instantaneous velocity of the vehicle at [latex]t=4[/latex] seconds.

46. [T] The distance in feet that a ball rolls down an incline is modeled by the function [latex]s(t)=14t^2[/latex], where [latex]t[/latex] is seconds after the ball begins rolling.

  1. Find the average velocity of the ball over the following time intervals:
    1. [5, 5.1]
    2. [5, 5.01]
    3. [5, 5.001]
    4. [5, 5.0001]
  2. Use the answers from a. to draw a conclusion about the instantaneous velocity of the ball at [latex]t=5[/latex] seconds.

47. Two vehicles start out traveling side by side along a straight road. Their position functions, shown in the following graph, are given by [latex]s=f(t)[/latex] and [latex]s=g(t)[/latex], where [latex]s[/latex] is measured in feet and [latex]t[/latex] is measured in seconds.

Two functions s = g(t) and s = f(t) are graphed. The first function s = g(t) starts at (0, 0) and arcs upward through roughly (2, 1) to (4, 4). The second function s = f(t) is a straight line passing through (0, 0) and (4, 4).

  1. Which vehicle has traveled farther at [latex]t=2[/latex] seconds?
  2. What is the approximate velocity of each vehicle at [latex]t=3[/latex] seconds?
  3. Which vehicle is traveling faster at [latex]t=4[/latex] seconds?
  4. What is true about the positions of the vehicles at [latex]t=4[/latex] seconds?

48. [T] The total cost [latex]C(x)[/latex], in hundreds of dollars, to produce [latex]x[/latex] jars of mayonnaise is given by [latex]C(x)=0.000003x^3+4x+300[/latex].

  1. Calculate the average cost per jar over the following intervals:
    1. [100, 100.1]
    2. [100, 100.01]
    3. [100, 100.001]
    4. [100, 100.0001]
  2. Use the answers from a. to estimate the average cost to produce 100 jars of mayonnaise.

49. [T] For the function [latex]f(x)=x^3-2x^2-11x+12[/latex], do the following.

  1. Use a graphing calculator to graph [latex]f[/latex] in an appropriate viewing window.
  2. Use the ZOOM feature on the calculator to approximate the two values of [latex]x=a[/latex] for which [latex]m_{\tan}=f^{\prime}(a)=0[/latex].

50. [T] For the function [latex]f(x)=\dfrac{x}{1+x^2}[/latex], do the following.

  1. Use a graphing calculator to graph [latex]f[/latex] in an appropriate viewing window.
  2. Use the ZOOM feature on the calculator to approximate the values of [latex]x=a[/latex] for which [latex]m_{\tan}=f^{\prime}(a)=0[/latex].

51. Suppose that [latex]N(x)[/latex] computes the number of gallons of gas used by a vehicle traveling [latex]x[/latex] miles. Suppose the vehicle gets 30 mpg.

  1. Find a mathematical expression for [latex]N(x)[/latex].
  2. What is [latex]N(100)[/latex]? Explain the physical meaning.
  3. What is [latex]N^{\prime}(100)[/latex]? Explain the physical meaning.

52. [T] For the function [latex]f(x)=x^4-5x^2+4[/latex], do the following.

  1. Use a graphing calculator to graph [latex]f[/latex] in an appropriate viewing window.
  2. Use the [latex]\text{nDeriv}[/latex] function, which numerically finds the derivative, on a graphing calculator to estimate [latex]f^{\prime}(-2), \, f^{\prime}(-0.5), \, f^{\prime}(1.7)[/latex], and [latex]f^{\prime}(2.718)[/latex].

53. [T] For the function [latex]f(x)=\dfrac{x^2}{x^2+1}[/latex], do the following.

  1. Use a graphing calculator to graph [latex]f[/latex] in an appropriate viewing window.
  2. Use the [latex]\text{nDeriv}[/latex] function on a graphing calculator to find [latex]f^{\prime}(-4), \, f^{\prime}(-2), \, f^{\prime}(2)[/latex], and [latex]f^{\prime}(4)[/latex].