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AP Calculus BC
Chapter 4
REVIEW 4.1–4.4
Name __________________________________
Date ___________________ Period __________
NO CALCULATOR IS ALLOWED FOR THIS PORTION OF THE REVIEW.
1.
∫
4
2
⎡d
⎤
2
⎢ (3t + 2t − 1) ⎥ dt =
⎣ dt
⎦
(A) 12
(B) 40
(C) 46
(D) 55
(E) 66
2. The velocity of a particle moving along the x-axis is given by a third-degree polynomial P(t) . The
roots of P(t) are all in the open interval 0 < t < a . Which of the following statements must be true?
I.
The velocity of the particle will be zero at least once and at most three times for 0 < t < a .
II.
In the interval 0 < t < a , the particle moves both left and right.
III.
The total distance traveled by the particle from t = 0 to t = a is given by
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
3. What is the area of the shaded region in the figure below?
y
y = 12 sin x
O
(A) 1
(B) π
(C) 2
(D) π − 1
(E) 2π
x
∫
a
0
P(t) dt .
4.
Find the antiderivatives for the following:
(a)
∫ (12x
(b)
∫
(c)
∫ (cosθ − sinθ + 2) dθ
3
3
3
t2
)
− 8x − 3 dx
dt
(d)
∫
x2 − x
(e)
∫
5
dx
x
x
dx
5. Solve the following differential equations for the given initial conditions.
(a) f ′(x) = 2x 3 − 5
f (2) = −1
(b) f ′′(x) = − cos x
f ′(π ) = 1
(c) f ′(x) = 4x1/ 3
f (π ) = −π
and f (x) contains the point (8, 20)
1
+ 2.
x2
(a) Find the average value of the function on
the interval [1, 4].
6. Given f (x) =
7. Let F(x) =
∫
x
0
(b) Find the value(s) of c guaranteed by the Mean
Value Theorem for Integrals.
f (t) dt where f is the function graphs to the right.
NOTE: The graph of f is made up of straight lines and a semicircle.
(a) Find F(−4)
(b) Find F(2)
(c) Find F ′(3)
(d) On what intervals is F concave up? Justify your answer.
(e) Which is larger: F(−3) or F(3) ? Justify your answer.
(f) What is the maximum value of F on the interval [–4, 4]? Explain.
x
0
0< x<1
1
1< x <2
2
2< x <3
3
3< x <4
f (x)
–1
Negative
0
Positive
2
Positive
0
Negative
f !(x)
4
Positive
0
Positive
DNE
Negative
–3
Negative
f !!(x)
–2
Negative
0
Positive
DNE
Negative
0
Positive
)
8. Let f be the function that is continuous on the interval ⎡⎣0,4 . The function f is twice differentiable
except at x = 2 . The function f and its derivatives have the properties indicated in the table above,
where DNE indicates that the derivatives of f do not exist at x = 2 .
(a) For 0 < x < 4 , find all values of x at which f has a relative extremum. Determine whether f has a
relative maximum or a relative minimum at each of these values. Justify your answer.
(b) On the axes provided, sketch the graph of a function that has all the characteristics of f .
(c) Let g be the function defined by g(x) =
∫
x
1
f (t) dt on the open interval (0, 4). For 0 < x < 4 , find all
values of x at which g has a relative extremum. Determine whether g has a relative maximum or a
relative minimum at each of these values. Justify your answer.
(d) For the function g defined in part (c), find all values of x , for 0 < x < 4 , at which the graph of g has a
point of inflection. Justify your answer.
y
O
x
A GRAPHING CALCULATOR IS REQUIRED FOR SOME OF THE FOLLOWING PROBLEMS.
1. The function y = sin x + cos x is a solution of which differential equation?
I.
II.
III.
dy
= 2sin x
dx
dy
y+
= 2cos x
dx
dy
− y = −2sin x
dx
y+
(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III
2. The average value of the function f (x) = e− x sin x on the closed interval ⎡⎣1, π ⎤⎦ is
(A) 0.129
(B) 0.145
(C) 0.155
(D) 0.276
(E) 0.310
3. The present price of a new car is $14,500. The price of a new car is changing at a rate of 120 + 180 t
dollars per year. How much will a new car cost 5 years from now?
(A) $15,020
(B) $15,300
(C) $16,440
(D) $18,120
(E) $22,600
4. Given the function f (x) = x 2/ 3 on the interval [1, 5].
(a) Sketch the graph of this function on the given interval.
(b) Approximate the area using a right endpoint Riemann sum with 4 subintervals of equal length.
Show the set-up required for this approximation.
(c) Does this approximation overestimate or underestimate the actual area? Explain
(d) Approximate the area using a midpoint Riemann sum with 4 subintervals of equal length. Show the
set-up required for this approximation.
5. Find F ′(x) for F(x) =
1
∫−2 t 2 + 5 dt .
6. Find F ′(x) for F(x) =
∫
2x
∫
x2
7. Find F ′(x) for F(x) =
x
−2
−2
1
dt .
t +5
2
1
dt .
t +5
2
8. The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by
πt
R(t) = 2 + 5sin 425
( )
A pumping station adds sand to the beach at a rate modeled by the function S, given by
15t
S(t) =
1 + 3t
Both R(t) and S(t) have units of cubic yards per hour and t is measured in hours for 0 ≤ t ≤ 6 . At time
t = 0 , the beach contains 2500 cubic yards of sand.
(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of
measure.
(b) Write an expression for Y(t), the total number of cubic yards of sand on the beach at time t.
(c) Find the rate at which the total amount of sand on the beach is changing at time t = 4 .
(d) For 0 ≤ t ≤ 6 , at what time t is the amount of sand on the beach a minimum? What is the minimum
value? Justify your answers.