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מדריכי לימוד > Precalculus II

Solutions for Derivatives

Solutions to Try Its

1. 3 2. f′(a)=6a+7{f}^{\prime }\left(a\right)=6a+7 3. f′(a)=−15(5a+4)2{f}^{\prime }\left(a\right)=\frac{-15}{{\left(5a+4\right)}^{2}} 4. 32\frac{3}{2} 5. 0 6. −2-2, 0, 0, −3-3 7. a. After zero seconds, she has traveled 0 feet. b. After 10 seconds, she has traveled 150 feet east. c. After 10 seconds, she is moving eastward at a rate of 15 ft/sec. d. After 20 seconds, she is moving westward at a rate of 10 ft/sec. e. After 40 seconds, she is 100 feet westward of her starting point. 8. The graph of ff is continuous on (−∞,1)∪(1,3)∪(3,∞)\left(-\infty ,1\right)\cup \left(1,3\right)\cup \left(3,\infty \right). The graph of ff is discontinuous at x=1x=1 and x=3x=3. The graph of ff is differentiable on (−∞,1)∪(1,3)∪(3,∞)\left(-\infty ,1\right)\cup \left(1,3\right)\cup \left(3,\infty \right). The graph of ff is not differentiable at x=1x=1 and x=3x=3. 9. y=19x−16y=19x - 16 10. –68 ft/sec, it is dropping back to Earth at a rate of 68 ft/s.

Solutions to Odd-Numbered Exercises

1. The slope of a linear function stays the same. The derivative of a general function varies according to xx. 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 f(x)f\left(x\right) is the function giving the amount of water in the tank at any time tt, then the average rate of change of f(x)f\left(x\right) between t=at=a and t=bt=b is f(a)+45(b−a)f\left(a\right)+45\left(b-a\right). 7. f′(x)=−2{f}^{\prime }\left(x\right)=-2 9. f′(x)=4x+1{f}^{\prime }\left(x\right)=4x+1 11. f′(x)=1(x−2)2{f}^{\prime }\left(x\right)=\frac{1}{{\left(x - 2\right)}^{2}} 13. −16(3+2x)2\frac{-16}{{\left(3+2x\right)}^{2}} 15. f′(x)=9x2−2x+2{f}^{\prime }\left(x\right)=9{x}^{2}-2x+2 17. f′(x)=0{f}^{\prime }\left(x\right)=0 19. −13-\frac{1}{3} 21. undefined 23. f′(x)=−6x−7{f}^{\prime }\left(x\right)=-6x - 7 25. f′(x)=9x2+4x+1{f}^{\prime }\left(x\right)=9{x}^{2}+4x+1 27. y=12x−15y=12x - 15 29. k=−10k=-10 or k=2k=2 31. Discontinuous at x=−2x=-2 and x=0x=0. Not differentiable at –2, 0, 2. 33. Discontinuous at x=5x=5. Not differentiable at -4, –2, 0, 1, 3, 4, 5. 35. f(0)=−2f\left(0\right)=-2 37. f(2)=−6f\left(2\right)=-6 39. f′(−1)=9{f}^{\prime }\left(-1\right)=9 41. f′(1)=−3{f}^{\prime }\left(1\right)=-3 43. f′(3)=9{f}^{\prime }\left(3\right)=9 45. Answers vary. The slope of the tangent line near x=1x=1 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 t=3t=3 seconds is 96 feet. 55. The height of the projectile is zero at t=0t=0 and again at t=5t=5. In other words, the projectile starts on the ground and falls to earth again after 5 seconds. 57. 36π36\pi 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. f’(x)=10a−1f\text{'}\left(x\right)=10a - 1 67. 4(3−x)2\frac{4}{{\left(3-x\right)}^{2}}

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