1. A56 Review for Final Day 2 #15 – 45

2. 15. B
23. A
31. C
39. D
16. D
24. E
32. B
40. C
17. C
25. C
33. E
41. E
18. A
26. A
34. B
42. C
19. B
27. B
35. C
43. D
20. C
28. A
36. E
44. C
21. E
29. C
37. B
45. E
22. D
30. E
38. D
Calculus A56 Answers for Review for Semester 1 final exam

3. 15. Limit as x approaches a number
Substitute the number in… and get your answer
16. Limit as x approaches 
Use rule from MA OR
Use L’Hospital’s Rule

4. 17. Limit as x approaches a number
Substitute the number in…
Use L’Hospital’s Rule when we get an indeterminate form 0/0 or /
18. Limit as x approaches a number
Substitute the number in…
Use L’Hospital’s Rule when we get an indeterminate form 0/0 or /
Hole in graph does not effect the two sided limit

5. 19. Limit as x approaches a number
Piecewise function…limit from left and limit from right
Jump in graph means the two-sided limit DNE

6. 20. Limit and continuity
Two-sided exists if there is a hole in the graph
3 part continuity test: 1) limit exists, 2) function exists and 3) limit = function value

7. 21. Mean Value Theorem
f(x) is continuous on [a, b] and differentiable on (a, b)… there exists a number c on (a, b) such that…
f’(c) = f(b) – f(a) b – a
Continuous: no jumps, no holes, no vertical asymptotes
Differentiable:
Not differentiable at x=0

8. 22. Implicit differentiation
Find dy/dx
Substitute in the point (x, y) to find slope

9. 23. Implicit differentiation
Find dy/dx
Substitute in the point (x, y) to find slope

10. 24. Equation of tangent line
Find dy/dx
Use product rule:
first(d sec) + sec(d first)
Substitute in the point to find the slope
Write the equation of the tangent line using y – y1 = m(x – x1)

11. 25. Derivative using product rule and chain rule
Product rule: First(deriv second) + second(deriv first)

12. 26. Quotient Rule
Bottom (d top) – top (d bottom) bottom2

13. 27. Chain Rule
28. Trig Derivative

14. 29. Implicit Differentiation

15. 30. Relative minimum and maximum
Find the derivative f’(x)
Set the derivative f’(x) =0
Solve for x to find critical numbers
Interval test to find r. min (neg. to pos.) and r. max (pos. to neg.)
Critical #’s :
y’ – –
- 2
r. min r.max r.min E

16. 31. Related Rates Sphere
Given dV/dt
Given V
Find ds/dt

17. 32. Related Rates Cone Write the ratio of r/h Write given info
Find dV/dt
Substitute the r into the Volume equation
V in terms of h… take the deriv.
Substitute in given info to solve

18. 33. Optimization – Max/Min problem
Define var.
x = first pos. #
y = second pos. #
Write equations
Find deriv.
Set deriv.=0
Solve for critical #’s
Interval test
Answer question
– – –

19. 34. Particle is at rest 35. Definite Integral
Set first derivative (velocity) equal to zero and solve
35. Definite Integral

20. 36. Definite Integral with u-subst
Choose u
Find du
Subst.
Integrate
Evaluate the bounds

21. 37. Definite integral with u-subst.
Choose u
Find du
Subst.
Integrate
Evaluate the bounds

22. 38. Definite integral with u-subst.
Choose u
Find du
Subst.
Integrate
Evaluate the bounds

23. 39. Definite Integral with u-subst.
Choose u
Find du
Subst.
Integrate
Evaluate the bounds

24. 41. Derivative – product rule
40. Integral formula
Use formula to find anti-derivative
Evaluate bounds
41. Derivative – product rule

25. 42. Related Rates Right Triangles
Draw picture
Find x, y and z
Find dx/dt, dy/dt, dz/dt
Use the formula :
x(dx/dt) + y(dy/dt) = z(dz/dt)
y=24 ft z = 26 ft
Find dz/dt = 0
dy/dt
x = 10 ft
dx/dt = 3 ft/sec

26. 43. Definite integral with u-subst.
Choose u
Find du
Subst.
Integrate
Evaluate the bounds

27. 44. PVA initial value
Position, velocity, acceleration

28. 45. Concave up
Find the second deriv. – 3
y’’
Concave up x < 3 E