Review for Final Day 3 #48 – 75 48. D 49. E 50. A 51. C 52. B 56. A

46. Mean Value Theorem F(x) continuous on [a, b] and differentiable on (a, b) then there exists a number c on (a, b) such that… F’(c) = ( F(b) – F(a) ) (b – a)

47. Implicit differentiation Take deriv. Find dy/dx

48. Increasing function Find f’(x) Set f’(x) =0 and solve to get critical #’s Interval test Increasing when f’(x) > 0 (positive) + – + f’(x) -3 -1

49. Definite integral Power rule

50. Indefinite integral Choose u Find du Substitute u and du into the integral Take the anti-deriv. Substitute back the original variable.

51.Concave down Find y’’ < 0 – + 2 y’’

52. POI Find y’’ Solve y’’ = 0 Interval test Find POI (sign change) – – + -2 0 y’’

53. Equation of tangent line using implicit differentiation Find dy/dx Find slope at given point (x, y) Use Point-slope form to write the equation y – y1 = m ( x – x1)

53. alternate

54. Trig derivative with chain rule Find the deriv. Substitute in the given value 55. Derivative Find the deriv. Substitute in the given value

56.Trig derivative 57. Indefinite integral with u-subst.

58. Optimization Max/Min problem

59. Decreasing and concave up Find y’ < 0 for decreasing and y’’>0 for concave up + – – + 0 2 4 Decreasing (0, 2)(2, 4)

60. increasing + – + -3 -1 D) Increasing (-, -3)  (-1, )

62. Limit approaching infinity (HA) 61. Deriv. Quotient Rule Use quotient rule to find the deriv. Substitute in the value 62. Limit approaching infinity (HA)

64. Derivative – chain rule 63. Particle moving left Find v(t) Set v(t) = 0 Interval test Find where v(t) is negative v(t) + - + - + -2 -1 1 2 B) -2 < t < -1 and 1< t < 2 64. Derivative – chain rule

65a) Implicit differentiation Skip (same as #22)

66. Trig Integral with u-subst. Choose u Find du Substitute in u and du Integrate Substitute back the original variable 66. Trig Integral with u-subst.

67. Integral with u-substitution Choose u Find du Substitute in u and du Integrate Substitute back the original variable 68. Trig Integral with u-subst

69. Formula with u-subst.

70. Relative min. Find y’ Set y’=0 solve for critical #’s Interval test Rel. min. y’ changes from neg. to pos. – – + Relative Min. at x = 1 y’ 0 1

72. Maximum on closed interval 71. Limits and continuity D) f(x) is continuous at x = 5 72. Maximum on closed interval Max. occurs when f’(x) changes from positive to negative f(-1) =-24 f(1) = 4 E f(2) = 3 + - + 1 2

73. POI 74. PVA + – + + -2 -1 0

75. Derivative with trig