AP Calculus BC
Final Exam Review Packet
Integration
1. ∫(3𝑥 2 − 2𝑥 + 3)𝑑𝑥
4. ∫
1−3𝑦
√2𝑦−3𝑦 2
𝑑𝑦
3𝑥+4
7. ∫ (𝑥+2)(𝑥 2 +3) 𝑑𝑥
2. ∫ √4 − 2𝑡 𝑑𝑡
3. ∫(2 − 3𝑥)5 𝑑𝑥
5. ∫ 𝑒 𝑥 𝑠𝑖𝑛𝑥 𝑑𝑥
6. ∫ 𝑥 3 𝑒 𝑥 𝑑𝑥
𝑑𝑥
8. ∫ 𝑥 2 +3𝑥−10
3 𝑑𝑦
2𝑦−3
9. ∫2
=
Riemann Sums
x
0
1
2 3 4 5 6
7
8
f(x) 15 12.5 6 -3 -5 2 3.5 7.5 10
10. Some values for a continuous function f(x) are given in the table above. Give an
8
approximation for ∫0 𝑓(𝑥)𝑑𝑥 using:
a. the trapezoidal rule with 8 subintervals
b. the left Riemann sum with 8 subintervals
c. the midpoint sum with 4 subintervals
x
0 1
2
3
4
f(x) 0 1.2 4.3 6.5 1
11. Some values for a continuous function f(x) are given in the table above. Give an
4
approximation for ∫0 𝑓(𝑥)𝑑𝑥 using:
a. the trapezoidal rule with 4 subintervals
b. the right Riemann sum with 4 subintervals
c. the midpoint sum with 2 subintervals
Motion
12. A particle moves along a line in such a way that its position at time t is given by
𝑠 = 𝑡 3 − 6𝑡 2 + 9𝑡 + 3. When does the particle change direction?
13. A particle moves along a line with velocity 𝑣 = 3𝑡 2 − 6𝑡.
a. What is the total distance traveled from t = 0 to t = 3?
b. What is the net change in the position of the particle from t = 0 to t = 3?
14. During the worst 4-hr period of a hurricane the wind velocity, in miles per hour, is given
by 𝑣(𝑡) = 5𝑡 − 𝑡 2 + 100, 0 ≤ 𝑡 ≤ 4. What is the average wind velocity during this period?
15. The position of a particle moving along a straight line is given by
𝑠 = 𝑡 3 − 6𝑡 2 + 12𝑡 − 8.
a. When is the distance from the starting point increasing?
b. When is the speed of the particle decreasing?
c. When is the acceleration positive?
16. A particle moves in the xy-plane according to the parametric equations 𝑥(𝑡) = 2𝑡 2 − 𝑡
and 𝑦(𝑡) = 𝑡 3 − 3𝑡. Find the speed of the particle at t = 2.
17.
a. When is the object furthest to the right?
b. During 3 < t < 4 what is the object’s acceleration?
c. When is the object’s acceleration undefined?
d. When is the object’s acceleration positive?
e. When is the object’s speed increasing?
18. At time t, a particle moving in the xy-plane is at position (x(t), y(t)), where x(t) and y(t)
𝑑𝑥
𝑑𝑦
are not explicitly given. For t ≥ 0, = 4𝑡 + 1 and = 𝑠𝑖𝑛(𝑡 2 ). At time t = 0, x(0) = 0 and
𝑑𝑡
𝑑𝑡
y(0) = -4.
a. Find the speed of the particle at time t = 3.
b. Find the acceleration vector of the particle at t = 3.
c. Find the slope of the line tangent to the path of the particle at t = 3.
d. Find the position of the particle at t = 3.
e. Find the total distance traveled by the particle over the time interval [0, 3].
Function Analysis
19. Find the number of local maximum and minimum points of the function whose
derivative, for all x, is given by 𝑓 ′ (𝑥) = 𝑥(𝑥 − 3)2 (𝑥 + 1)4 .
20. For the function 𝑓(𝑥) = 𝑥 4 − 4𝑥 2 ,
a. Find the number of local maximum and minimum points.
b. Find the number of inflection points.
21. Draw a curve f for which
a. both 𝑓 ′ 𝑎𝑛𝑑 𝑓′′ negative.
b. 𝑓′′ positive but 𝑓′ negative.
22. At which point is
a. f’(x) = 0 and f’’(x) > 0?
b. f’(x) < 0 and f’’(x) = 0?
c. f’(x) = 0 and f’’(x) = 0?
23. Answer the following questions based on the graph
(to the left) of f’(x).
a. Where does f have a point(s) of inflection?
b. Where does f have a local minimum?
c. Where is f concave up?
24. On what interval(s) is/are 𝑓(𝑥) = 𝑥 4 − 2𝑥 2 + 1 concave up?
25. On what interval(s) is/are 𝑓(𝑥) = 𝑥 3 − 2𝑥 2 + 𝑥 + 6 decreasing?
Differential Equations
𝑑𝑦
26. Find y if 𝑑𝑥 = 2
27. Find y if
28. If
𝑑𝑦
𝑑𝑥
𝑑𝑦
𝑑𝑥
𝑦
√𝑥
𝑎𝑛𝑑 𝑦 = 1 𝑤ℎ𝑒𝑛 𝑥 = 4.
= 𝑒 𝑦 and y = 0 when x = 1.
2𝑥
= 4𝑥+𝑦 𝑎𝑛𝑑 𝑦(1) = 1, then what is the approximate value of y when x = 2 using
Euler’s method with two steps?
29. The population of Nowhereville from 1920 to 2000 can be modeled by the logistic
100000
equation 𝑃(𝑡) = 1+52.3𝑒 −0.45𝑡 .
a. What was the population when it was growing the fastest?
b. When is the population growing the fastest?
c. Find lim 𝑃(𝑡).
𝑡→∞
30. The growth rate of Somewhereville from 1950 to 2010 is modeled by the DEQ
𝑑𝑃
𝑑𝑡
= .0065𝑃(7500 − 𝑃).
a. What is the carrying capacity for Somewhereville?
b. What is the population when it is growing the fastest?
c. Find lim 𝑃(𝑡).
𝑡→∞
Derivatives
31. 𝑦 = 𝑥 5 𝑡𝑎𝑛𝑥
32. 𝑦 = √𝑥 2 + 2𝑥 − 1
33. 𝑦 = 𝑙𝑛(𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥)
34. 𝑦 = sin (𝑥)
1
2−𝑥
35. 𝑦 = 3𝑥+1
36. 𝑦 = 3𝑥 2⁄3 − 4𝑥 1⁄2 − 2
37. Find
𝑑𝑦
𝑑𝑥
𝑖𝑓 𝑥 = 𝑒 𝑡 𝑐𝑜𝑠𝑡 𝑎𝑛𝑑 𝑦 = 𝑒 𝑡 𝑠𝑖𝑛𝑡.
38. Find
𝑑𝑦
𝑑𝑥
𝑖𝑓 𝑥 = 3𝑡 + 1 𝑎𝑛𝑑 𝑦 = 4𝑡 − 5.
39. Find
𝑑2 𝑦
𝑑𝑥 2
𝑖𝑓 𝑥 = 𝑡 2 − 1 𝑎𝑛𝑑 𝑦 = 2𝑡 4 − 𝑡 3 .
𝑑𝑟
40. Find 𝑑𝜃 𝑖𝑓 𝑟 = 𝑠𝑒𝑐𝜃𝑡𝑎𝑛2𝜃
41. What is the slope of the curve r = 3 – 2sinat (3, π)?
42. What is the arc length of the graph of y = cosx from [0, π]?
Limits & Continuity
43. If {
𝑓(𝑥) =
𝑥 2 −𝑥
2𝑥
𝑓(0) = 𝑘
for x ≠ 0, and if 𝑓 is continuous at x = 0, then k =
3𝑥(𝑥−1)
𝑓(𝑥) = 𝑥 2 −3𝑥+2 𝑓𝑜𝑟 𝑥 ≠ 1, 2
44. Suppose {
, then f(x) is continuous
𝑓(1) = −3
𝑓(2) = 4
45. Find
a. f(1)
b. lim− 𝑓(𝑥)
𝑥→1
c. lim+ 𝑓(𝑥)
𝑥→1
46. Find
a. lim 𝑓(𝑥)
𝑥→−∞
b. lim 𝑓(𝑥)
𝑥→3
c. lim 𝑓(𝑥)
𝑥→2
d. lim 𝑓(𝑥)
𝑥→−1
Sequences & Series
47. For each series, determine whether it converges or diverges. Give the test used.
1+3𝑛2 +𝑛3
2 𝑛
4
a. ∑∞
𝑛=1 4𝑛3 −5𝑛+2
b. ∑∞
𝑛=1 (7)
c. ∑∞
𝑛=1 𝑛3
d. ∑∞
𝑛=1 5𝑛
e. ∑∞
𝑛=1
f. ∑∞
𝑛=1 (5𝑛−1)
𝑛2
cos 𝑛𝜋
√𝑛
2𝑛
𝑛
48. Write a fourth degree Taylor series expansion about x = 0 for 𝑓(𝑥) = √𝑥 + 1.
49. Write the third degree McLaurin series for 𝑓(𝑥) = 𝑒 −𝑥/2 .
3 𝑛
50. ∑∞
𝑛=1 (5) =
51. Find the radius of convergence for
a. ∑∞
𝑛=1
(𝑛−1)!
2𝑛
𝑛 𝑛
b. ∑∞
𝑛=0 2 𝑥
52. Find the interval of convergence for the following series. Remember to check the
endpoints.
a. ∑∞
𝑛=1
(𝑛−1)!
2𝑛
𝑛 𝑛
b. ∑∞
𝑛=0 2 𝑥