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

2. 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)

3. 47. Implicit differentiation
Take deriv.
Find dy/dx

4. 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)

5. 49. Definite integral
Power rule

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

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

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

9. 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)

10. 53. alternate

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

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

13. 58. Optimization Max/Min problem

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

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

16. 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)

17. 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)
B) -2 < t < -1 and 1< t < 2
64. Derivative – chain rule

18. 65a) Implicit differentiation Skip (same as #22)

19. 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.

20. 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

21. 69. Formula with u-subst.

22. 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 = y’

23. 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

24. 73. POI PVA –

25. 75. Derivative with trig