Solutions to Odd-Numbered Exercises

1. The slope of a linear function stays the same. The derivative of a general function varies according to [latex]x[/latex]. Both the slope of a line and the derivative at a point measure the rate of change of the function.

3. Average velocity is 55 miles per hour. The instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of the car over an interval.

5. The average rate of change of the amount of water in the tank is 45 gallons per minute. If [latex]f\left(x\right)[/latex] is the function giving the amount of water in the tank at any time [latex]t[/latex], then the average rate of change of [latex]f\left(x\right)[/latex] between [latex]t=a[/latex] and [latex]t=b[/latex] is [latex]f\left(a\right)+45\left(b-a\right)[/latex].

7. [latex]{f}^{\prime }\left(x\right)=-2[/latex]

9. [latex]{f}^{\prime }\left(x\right)=4x+1[/latex]

11. [latex]{f}^{\prime }\left(x\right)=\frac{1}{{\left(x - 2\right)}^{2}}[/latex]

13. [latex]\frac{-16}{{\left(3+2x\right)}^{2}}[/latex]

15. [latex]{f}^{\prime }\left(x\right)=9{x}^{2}-2x+2[/latex]

17. [latex]{f}^{\prime }\left(x\right)=0[/latex]

19. [latex]-\frac{1}{3}[/latex]

21. undefined

23. [latex]{f}^{\prime }\left(x\right)=-6x - 7[/latex]

25. [latex]{f}^{\prime }\left(x\right)=9{x}^{2}+4x+1[/latex]

27. [latex]y=12x - 15[/latex]

29. [latex]k=-10[/latex] or [latex]k=2[/latex]

31. Discontinuous at [latex]x=-2[/latex] and [latex]x=0[/latex]. Not differentiable at –2, 0, 2.

33. Discontinuous at [latex]x=5[/latex]. Not differentiable at -4, –2, 0, 1, 3, 4, 5.

35. [latex]f\left(0\right)=-2[/latex]

37. [latex]f\left(2\right)=-6[/latex]

39. [latex]{f}^{\prime }\left(-1\right)=9[/latex]

41. [latex]{f}^{\prime }\left(1\right)=-3[/latex]

43. [latex]{f}^{\prime }\left(3\right)=9[/latex]

45. Answers vary. The slope of the tangent line near [latex]x=1[/latex] is 2.

47. At 12:30 p.m., the rate of change of the number of gallons in the tank is –20 gallons per minute. That is, the tank is losing 20 gallons per minute.

49. At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.

51. The height of the projectile after 2 seconds is 96 feet.

53. The height of the projectile at [latex]t=3[/latex] seconds is 96 feet.

55. The height of the projectile is zero at [latex]t=0[/latex] and again at [latex]t=5[/latex]. In other words, the projectile starts on the ground and falls to earth again after 5 seconds.

57. [latex]36\pi[/latex]

59. $50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold.

61. $21 per unit

63. $36

65. [latex]f\text{'}\left(x\right)=10a - 1[/latex]

67. [latex]\frac{4}{{\left(3-x\right)}^{2}}[/latex]