SECTION 14.4 The Tangent Problem; The Derivative 967 In Problems 9–20, find the slope of the tangent line to the graph of f at the given point. Graph f and the tangent line. 9. ( ) ( ) = + f x x3 5 at 1, 8 10. ( ) ( ) = − + − f x x2 1 at 1, 3 11. ( ) ( ) = + − f x x 2 at 1, 3 2 12. ( ) ( ) = − f x x 3 at 1, 2 2 13. ( ) ( ) = f x x3 at 2, 12 2 14. ( ) ( ) = − − − f x x4 at 2, 16 2 15. ( ) ( ) = + f x x x 2 at 1, 3 2 16. ( ) ( ) = − f x x x 3 at 0, 0 2 17. ( ) ( ) = − + − f x x x2 3 at 1, 6 2 18. ( ) ( ) = − + − − f x x x 2 3 at 1, 4 2 19. ( ) ( ) = + f x x x at 2, 10 3 20. ( ) ( ) = − f x x x at 1, 0 3 2 Skill Building In Problems 21–32, find the derivative of each function at the given number. 21. ( ) = − + f x x4 5 at 3 22. ( ) = − + f x x 4 3 at 1 23. ( ) = − f x x 3 at 0 2 24. ( ) = + − f x x2 1at 1 2 25. ( ) = + f x x x 2 3 at 1 2 26. ( ) = − f x x x 3 4 at 2 2 27. ( ) = + − f x x x4 at 1 3 28. ( ) = − f x x x 2 at 2 3 2 29. ( ) = + − f x x x x2 at 1 3 2 30. ( ) = − + − f x x x x 2 at 1 3 2 31. ( ) = f x x sin at 0 32. ( ) = f x x cos at 0 In Problems 33–42, use a graphing utility to approximate the derivative of each function at the given number. 33. ( ) = − + − f x x x 3 6 2at 2 3 2 34. ( ) = − + − f x x x 5 6 10 at 5 4 2 35. ( ) = − + + + f x x x x 1 5 7 at 8 3 2 36. ( ) = − + + + − − f x x x x x 5 9 3 5 6 at 3 4 3 2 37. π ( ) = f x x x sin at 3 38. π ( ) = f x x x sin at 4 39. π ( ) = f x x x sin at 3 2 40. π ( ) = f x x x sin at 4 2 41. ( ) = f x e x sin at 2 x 42. ( ) = − f x e x sin at 2 x 43. Instantaneous Rate of Change The volume V of a right circular cylinder of height 3 feet and radius r feet is π ( ) = = V V r r 3 .2 Find the instantaneous rate of change of the volume with respect to the radius r at = r 3. 44. Instantaneous Rate of Change The surface area S of a sphere of radius r feet is π ( ) = = S S r r 4 .2 Find the instantaneous rate of change of the surface area with respect to the radius r at = r 2. 45. Instantaneous Rate of Change The volume V of a sphere of radius r feet is π ( ) = = V V r r 4 3 .3 Find the instantaneous rate of change of the volume with respect to the radius r at = r 2. 46. Instantaneous Rate of Change The volume V of a cube of side x meters is ( ) = = V V x x .3 Find the instantaneous rate of change of the volume with respect to the side x at = x 3. 47. Instantaneous Velocity of a Ball In physics it is shown that the height s of a ball thrown straight up with an initial velocity of 96 ft/sec from ground level is ( ) = = − + s s t t t 16 96 2 where t is the elapsed time that the ball is in the air. (a) When does the ball strike the ground? That is, how long is the ball in the air? (b) What is the average velocity of the ball from = t 0 to = t 2? (c) What is the instantaneous velocity of the ball at time t? (d) What is the instantaneous velocity of the ball at = t 2? (e) When is the instantaneous velocity of the ball equal to zero? (f) How high is the ball when its instantaneous velocity equals zero? (g) What is the instantaneous velocity of the ball when it strikes the ground? 48. Instantaneous Velocity of a Ball In physics it is shown that the height s of a ball thrown straight down with an initial velocity of 48 ft/sec from a rooftop 160 feet high is ( ) = = − − + s s t t t 16 48 160 2 where t is the elapsed time that the ball is in the air. (a) When does the ball strike the ground? That is, how long is the ball in the air? (b) What is the average velocity of the ball from = t 0 to = t 1? (c) What is the instantaneous velocity of the ball at time t? (d) What is the instantaneous velocity of the ball at = t 1? (e) What is the instantaneous velocity of the ball when it strikes the ground? 49. Velocity on the Moon An astronaut throws a ball down into a crater on the moon. The height s (in feet) of the ball from the bottom of the crater after t seconds is given in the following table: 0 Time, t (in seconds) Height, s (in feet) 1000 1 987 2 969 3 945 4 917 5 883 6 843 7 800 8 749 Applications and Extensions (continued)
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