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.<\/p>\n
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.<\/p>\n
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].<\/p>\n
7. [latex]{f}^{\\prime }\\left(x\\right)=-2[\/latex]<\/p>\n
9. [latex]{f}^{\\prime }\\left(x\\right)=4x+1[\/latex]<\/p>\n
11. [latex]{f}^{\\prime }\\left(x\\right)=\\frac{1}{{\\left(x - 2\\right)}^{2}}[\/latex]<\/p>\n
13. [latex]\\frac{-16}{{\\left(3+2x\\right)}^{2}}[\/latex]<\/p>\n
15. [latex]{f}^{\\prime }\\left(x\\right)=9{x}^{2}-2x+2[\/latex]<\/p>\n
17. [latex]{f}^{\\prime }\\left(x\\right)=0[\/latex]<\/p>\n
19. [latex]-\\frac{1}{3}[\/latex]<\/p>\n
21. undefined<\/p>\n
23. [latex]{f}^{\\prime }\\left(x\\right)=-6x - 7[\/latex]<\/p>\n
25. [latex]{f}^{\\prime }\\left(x\\right)=9{x}^{2}+4x+1[\/latex]<\/p>\n
27. [latex]y=12x - 15[\/latex]<\/p>\n
29. [latex]k=-10[\/latex] or [latex]k=2[\/latex]<\/p>\n
31. Discontinuous at [latex]x=-2[\/latex] and [latex]x=0[\/latex]. Not differentiable at \u20132, 0, 2.<\/p>\n
33. Discontinuous at [latex]x=5[\/latex]. Not differentiable at -4, \u20132, 0, 1, 3, 4, 5.<\/p>\n
35. [latex]f\\left(0\\right)=-2[\/latex]<\/p>\n
37. [latex]f\\left(2\\right)=-6[\/latex]<\/p>\n
39. [latex]{f}^{\\prime }\\left(-1\\right)=9[\/latex]<\/p>\n
41. [latex]{f}^{\\prime }\\left(1\\right)=-3[\/latex]<\/p>\n
43. [latex]{f}^{\\prime }\\left(3\\right)=9[\/latex]<\/p>\n
45. Answers vary. The slope of the tangent line near [latex]x=1[\/latex] is 2.<\/p>\n
47. At 12:30 p.m., the rate of change of the number of gallons in the tank is \u201320 gallons per minute. That is, the tank is losing 20 gallons per minute.<\/p>\n
49. At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.<\/p>\n
51. The height of the projectile after 2 seconds is 96 feet.<\/p>\n
53. The height of the projectile at [latex]t=3[\/latex] seconds is 96 feet.<\/p>\n
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.<\/p>\n
57. [latex]36\\pi[\/latex]<\/p>\n
59. $50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold.<\/p>\n
61. $21 per unit<\/p>\n
63. $36<\/p>\n
65. [latex]f\\text{'}\\left(x\\right)=10a - 1[\/latex]<\/p>\n
67. [latex]\\frac{4}{{\\left(3-x\\right)}^{2}}[\/latex]<\/p>\n\n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t