1. (Section 2.2, Problem 46) 2. (Section 2.2, Problem 64) 3. (Section 2.3, Problem 4) 4. (Section 2.3, Problem 10) 5. (Section 2.3, Problem 12) 6. (Section 2.3, Problem 18) 7. (Section 2.3, Problem 22) 8. (Section 2.3, Problem 30) 9. (Section 2.3, Problem 36) 10. (Section 2.3, Problem 38) 11. (Section 2.3, Problem 66) 12. (Section 2.4, Problem 6) 13. (Section 2.4, Problem 16) 14. (Section 2.4, Problem 22) 15. (Section 2.4, Problem 42) 16. (Section 2.4, Problem 50) 1 In Exercises 38-42, for the given function f , (a) solve f '( x) 0 ; (b) solve f '( x) 0 ; (c) solve f '( x) 0 . 38. f ( x) 4 x 2 2 x 1 . 1 39. f ( x) x3 4 x 2 9 x 2 . 3 1 40. f ( x) x . x 5 41. f ( x) x x 3 5 x 4 . 42. f ( x) 10 x 6 15 x 4 2 . In Exercises 43-47, find f '' c . 43. f ( x) x 5 4 x 2 10 x 2 ; c 1 . 1 ; c 1. 44. f ( x) 5 x3 x 45. f ( x) 8cos x ; c 3 . 46. f ( x) 2 x 3cos x ; c . 47. f ( x) 4 x 2 6sin x ; c 4 . In Exercises 48-63, find the indicated derivative. 48. 49. 50. 51. 52. 53. 54. dy ; y x 3 sin x . dx dy ; y 4 cos 3sin . d d2y 3 7 ; y x4 x2 2 x 1 . 2 dx 4 2 d 2 x 2 5 . dx d 2 x 1 4 x 5 (Hint: Expand first). dx d 3 1 u 3 . du u d 3 d x 4 x 2 1 . dx dx Math 1431 Page 3 of 5 Section 2.2 Exercises 55. d 4 d2 2 x 2 x x . 2 dx dx d4 4 x 5x . dx 4 d6 57. 6 2sin x . dx d 2t 5 1 58. 4 (Hint: simplify first ) . dt t 56. 59. f ''' x ; f ( x) 6 x 2 10sin x . d3y 1 1 ; y x3 x 2 x 3 . 3 6 4 dx 3 2 61. y '' ; y 4 x x . 60. 62. f ( x) 2 x5 6 x 4 2 x 1 , find d2 f . dx 2 63. f ( x) 5 x 4 3 x 2 2 x 10 , find d3 f . dx 3 x 1 64. Find the points where the line tangent to f ( x) x 2 5 is parallel to the line y 4 x 3 . 65. Find the points where the line tangent to f ( x) x 2 6 x is perpendicular to the line y 2x 5 . 66. Determine the values of A and B so that the line y 11x 18 is tangent to the curve f ( x) Ax 2 Bx at x 3 . 67. Find a function whose derivative is f '( x) 3x 2 2 x 5 . 68. Find a function whose derivative is f '( x) 4 x3 cos x . 69. Find a cubic polynomial f such that f (0) 5, f '(0) 2, f ''(0) 4, f '''(0) 3 . 70. Find a quadratic polynomial f such that f (1) 12, f '(1) 4, f ''(1) 2 . Math 1431 Page 4 of 5 Section 2.2 Exercises Section 2.3 - Exercises In Exercises 1-6, given h( x) f g x , find h '( x) . g( x) sin( x) 1. f ( x) x3 3. f ( x) x 2 1 5. f ( x) x 2 2 x 1 g( x) x 2 2 x g( x) x3 1 2. f ( x) sec( x) 4. f ( x) 3 x 6. f ( x) sin( x) g( x) cot( x) g( x) x 2 3 g( x) cos( x) f In Exercises 7-10, given h( x) x , find h '( x) . g 7. f ( x) 4 x 3 9. f ( x) sin x g( x) x3 2 x g( x) x3 8. f ( x) 3 x g( x) x3 1 10. f ( x) 1 sin x g( x) 2 cos( x) In Exercises 11-30, find f '( x) . 11. f ( x) 5 x 7 13. f ( x) 4 x 3 x 2 x 1 15. f ( x) x 2 3 12. f ( x) x 3 x 2 x 4 3 1 4 5 14. f ( x) x 3 2 x 1 sin x 2 16. f ( x) x 1 2 3x 2 1 17. f ( x) x 3 3 18. f ( x) x x 2 5 3 19. f ( x) cos x 2 1 sin 12 x 20. f ( x) sin( x 4 2 x) 21. f ( x) sin 3 4 x 22. f ( x) tan 2 ( x 4 1) 1 23. f ( x) cos sec 8 x x 24. f ( x) x 4 cot(3 x) 25. f ( x) 3 sin(5 x) 26. f ( x) sin 2 x 1 cos 5 x Math 1431 Page 1 of 5 Section 2.3 Exercises 28. f ( x) 1 sin 2 x 27. f ( x) csc 5 x 1 3 x 30. f ( x) sin 4x 1 29. f ( x) cos 2 1 x sin 2 x In Exercises 31-38, find the equations of the tangent and normal lines to the curve f at the given point. 31. f ( x) 2x ; x sin x 2 32. f ( x) 4 x 1 x 1 3 ; x 1 33. f ( x) ( x 3 3x 1)( x 2) ; x 1 34. f ( x) ( x 1)( x 2 2) ; x 0 x 1 ; x2 35. f ( x) x 1 x 36. f ( x) 2 ; x2 x 1 37. f ( x) tan 2 x ; x 38. f ( x) sec 4 x ; x ? 8 12 ? In Exercises 39-40, given h( x) f g x , find h '( x) . g( x) x 2 x 39. f ( x) x 3 5 x 40. f ( x) x 4 g( x) x 2 2 x In Exercises 41-50, find f ' c . x2 41. f ( x) ; c0 x2 42. f ( x) (8 x 15)60 ; c 2 5 1 43. f ( x) x 2 2 x Math 1431 3 ; c 1 ? Page 2 of 5 Section 2.3 Exercises b) Find g h f ' 2 . c) Let H ( x) fgh x . Find H '(2) . 65. Given f x 1 x 2 ; 3 a) Solve f ' x 0 . b) Solve f ' x 0 . c) Solve f ' x 0 . 66. Given f x 1 x 2 ; 2 a) Solve f ' x 0 . b) Solve f ' x 0 . c) Solve f ' x 0 . 67. Find a formula for the nth derivative: y a bx , n a positive integer, a, b are constants. n 68. Find a formula for the nth derivative: y x . x 1 69. Give an example of two functions to show that the differentiability of f g does not imply that both f and g are differentiable. 70. Give an example of two functions to show that the differentiability of f g does not imply that both f and g are differentiable. Math 1431 Page 5 of 5 Section 2.3 Exercises Section 2.4 - Exercises dy in terms of x and y . dx In Exercises 1-18, use implicit differentiation to find 1. x 2 y 2 36 2. x 2 xy y 2 1 1 2 1 2 3. x 2 y 2 16 4. x y 9 5. x3 y 3 8 6. x3 xy y 2 4 7. x 2 y y 2 x 2 8. x3 y 3 y x 9. xy x 2 y 10. x3 3 x 2 y 2 xy 2 12 11. 2sin x cos y 1 12. sin x 2 cos 2 y 1 13. (sin x cos y ) 2 2 14. sin x x(1 tan y ) 15. cot y x y 16. y sin( xy ) 17. x sec 1 y 18. x 2 y 2 4 x 6 y 9 0 In Exercises 19-32, find the tangent line to the graph of the given equation at the indicated point. 19. x 2 y 2 9 21. x 2 xy y 2 7 1, 2 23. x3 y 3 6 xy 0 25. xy 1 y 2 ( x2 y 2 ) 2 x2 Math 1431 4 8 , 3 3 1, 1 27. x 2 y 2 9 x 2 4 y 2 0 29. 20. x3 y 2 xy 3 6 3, 1 2, 1 22. ( x 2 y 2 ) 2 4 x 2 y 1, 1 24. ( y 2) 2 4( x 3) 4, 0 26. 7 x 2 6 3xy 13 y 2 16 0 4, 2 3 1, 1 28. ( x 1) 2 ( y 2) 2 20 30. ( x 2 4) y 8 Page 1 of 2 3, 1 3, 4 2, 1 Section 2.4 Exercises 31. (4 x) y 2 x 3 2, 2 32. 3( x 2 y 2 ) 2 100( x 2 y 2 ) 4, 2 In Exercises 33-40, find y ' . 33. sin( x 2 y 2 ) 4 xy 34. xy 4 35. x y 0 x2 4 36. y 2 x 4 37. ( x y )3 x3 y 3 38. x 3 y 3 5 39. x3 y 3 4 xy 1 40. tan( x y ) x 2 3 In Exercises 41-46, find 2 2 2 d2y . dx 2 41. x 2 y 2 36 42. x 2 y 2 2 x 3 43. x 2 y 2 16 44. 1 xy x y 45. y 2 x 3 46. y 2 4 x d2y at the point 3, 2 . dx 2 d2y 1 2 2 48. Given y x 1 , find at the point 2, . 2 dx 2 47. Given y 2 x 2 1 , find 49. Find all points at which the graph of x 2 xy y 2 1 has a horizontal or vertical tangent line. 50. Find all points at which the graph of 4 x 2 y 2 8 x 4 y 4 0 has a horizontal or vertical tangent line. 51. Find all points at which the tangent line to graph of x 2 4 y y 2 4 is parallel to the line 2 y 2 x 1. 52. Find all points at which the normal line to graph of x 2 y 2 10 is parallel to the line 2y x 3. Math 1431 Page 2 of 2 Section 2.4 Exercises