3 review exercises - Review Exercises and Problems for...

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Review Exercises and Problems for Chapter 3 Find derivatives for the functions in Exercises 1–66. Assume a , b , c , and k are constants. 1. 2. 3. 4. 5. h ( θ ) = θ ( θ 1/2 θ 2 ) 6. f ( θ ) = ln (cos θ ) 7. f ( y ) = ln (ln (2 y 3 )) 8. g ( x ) = x k + k x 9. y = e π + π e 10. z = sin 3 θ 11. f ( t ) = cos 2 (3 t + 5) 12. M ( α ) = tan 2 (2 + 3 α ) 13. s ( θ ) = sin 2 (3 θ π ) 14. h ( t ) = ln ( e t t ) 15. 16. 17. 18. 19. s ( x ) = arctan(2 x ) 20. 21. m ( n ) = sin ( e n ) 22. k ( α ) = e tan (sin α ) 23. 24. f ( r ) = ( tan 2 + tanr ) e 25. h ( x ) = xe tan x 26. y = e 2 x sin 2 (3 x ) 27. g ( x ) = tan 1 (3 x 2 + 1) 28. y = 2 sin x cos x 29. h ( x ) = ln e ax 30. k ( x ) = ln e ax + ln b 31. f ( θ ) = e k θ 1 32. f ( t ) = e 4 kt sin t 33. H ( t ) = ( at 2 + b ) e ct 34. 35. f ( x ) = a 5 x 36. 37. 38. 39. 40.
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41. 42. 43. 44. 45. 46. 47. 48. 49. 50. y = 3 x 4 4 x 3 6 x + 2 51. 52. f ( z ) = (ln 3) z 2 + (ln 4) e z 53. 54. f ( x ) = (3 x 2 + π )( e x 4) 55. f ( θ ) = θ 2 sin θ + 2 θ cos θ 2 sin θ 56. 57. r ( θ ) = sin ((3 θ π ) 2 ) 58. y = ( x 2 + 5) 3 (3 x 3 2) 2 59. N ( θ ) = tan(arctan( k θ )) 60. h ( t ) = e kt (sin at + cos bt ) 61. f ( x ) = (2 4 x 3 x 2 )(6 x e 3 π ) 62. f ( t ) = (sin (2 t ) cos (3 t )) 4 63. 64. f ( x ) = (4 x 2 + 2 x 3 )(6 4 x + x 7 ) 65. 66. For Exercises 67–68, assume that y is a differentiable function of x and find dy / dx . 67. x 3 + y 3 4 x 2 y = 0 68. sin ( ay ) + cos ( bx ) = xy 69. Find the slope of the curve x 2 + 3 y 2 = 7 at (2, 1). 70. Assume y is a differentiable function of x and that y + sin y + x 2 = 9. Find dy / dx at the point x = 3, y = 0. 71. Find the equations for the lines tangent to the graph of xy + y 2 = 4 where x = 3. In Problems 78–80, use Figure 3.46 to evaluate the derivatives. Figure 3.46
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78. ( f 1 ) (5) 79. ( f 1 ) (10) 80. ( f 1 ) (15) For Problems 72–77, use Figure 3.45 . Figure 3.45 72. Let h ( x ) = t ( x ) s ( x ) and p ( x ) = t ( x )/ s ( x ). Estimate: (a) h (1) (c) h (0) (e) p (0) 73. Let r ( x ) = s ( t ( x )). Estimate r (0). 74. Let h ( x ) = s ( s ( x )). Estimate: (a) h (1) (b) h (2) 75. Estimate all values of x for which the tangent line to y = s ( s ( x )) is horizontal. 76. Let h ( x ) = x 2 t ( x ) and p ( x ) = t ( x 2 ). Estimate: (a) h ( 1) ′ − (c) p ( 1) ′ − 77. Find an approximate equation for the tangent line to r ( x ) = s ( t ( x )) at x = 1.
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