AP Calculus Testbank
(Chapter 6)
(Mr. Surowski)
Part I. Multiple-Choice Questions
1. Suppose that f is an odd differentiable function. Then
R1
xf 0 (x) dx =
−1
(A) f (1);
(B) f 0 (1)
(C) f (1) − f (−1)
(D) 0
(E) 2.
2. The slope field indicated below most likely depicts the differential equation
dy
=x+y
dx
dy
(B)
=x+5
dx
dy
(C)
=y+5
dx
dy
(D)
= −x + 5
dx
dy
(E)
= −y + 5
dx
(A)
3. You are given differentiable functions f and g. Which of the
following is a consequence of the integration by parts formula?
R
R
(A) f 00 (x)g(x) dx + f (x)g 00 (x) dx = f 0 (x)g 0 (x)
R
R
(B) f 00 (x)g(x) dx + f 0 (x)g 0 (x) dx = f (x)g(x)
R
R
(C) f 00 (x)g(x) dx + f 0 (x)g 0 (x) dx = f 0 (x)g(x)
R
R
(D) f 00 (x)g(x) dx + f 000 (x)g(x) dx = f 0 (x)g(x)
R
R
(E) f 00 (x)g(x) dx + f (x)g 0 (x) dx = f (x)g(x).
4. Given the slope field below, what is the most plausible behavior
dy
of the solution of the IVP
= f (x, y), y(−4) = 4?
dx
(A) lim y = −∞
x→∞
(B) lim y = 0
x→∞
(C) lim y = +∞
x→∞
(D) lim y = does not
x→∞
exist.
(E) lim y = −2
x→∞
dy
1√
5. You are given the differential equation
= − y, where y(0) =
dx
5
25. For which value(s) of x is y = 0?
(A) ±5
(B) 10
(C) 50
(D) ±10
(E) 5
Z
6.
tan6 x sec2 x dx =
tan7 x
+C
7
tan7 x sec3 x
(B)
+
+C
7
3
tan7 x sec3 x
(C)
+C
21
(D) 7 tan7 x + C
2
(E) tan7 x sec x + C
7
(A)
Z
7.
2
dx =
1 − x2
0
π
π
(A)
(B)
6
3
Z
8.
1/2
√
2
7xe3x dx =
1 3x2
e +C
42
6 2
(B) e3x + C
7
7 2
(C) e3x + C
6
2
(D) 7 e3x + C
(A)
2
(E) 42 e3x + C
(C) −
π
3
(D)
2π
3
(E) −
2π
3
Z
9.
ex e3x dx =
1 3x
e +C
3
1
(B) e4x + C
4
1
(C) e5x + C
4
(D) 4 e4x + C
(A)
(E) 5 e5x + C
Z
10.
√
x 5 − x dx =
3
10
(A) − (5 − x) 2 + C
r3
5x2 x3
(B)
−
+C
2
3
r
10 5x2 x3
(C)
−
+C
3
2
3
3
1
2
(D) 10(5 − x) 2 + (5 − x) 2 + C
3
3
5
2
10
(E) − (5 − x) 2 + (5 − x) 2 + C
3
5
dy
x3 + 1
=
and y = 2 when x = 1, then when x = 2, y =
dx
y
r
r
r
r
27
27
27
3
27
(A)
(B)
(C) ±
(D) ±
(E) ±
2
8
8
2
2
11. If
Z
12.
ln x
dx =
3x
(A) 6 ln2 x + C
1
(B) ln(ln x) + C
6
1
(C) ln2 x + C
3
1
(D) ln2 x + C
6
1
(E) ln x + C
3
Z
13.
sin5 (2x) cos(2x) dx =
sin6 2x
+C
12
sin6 2x
(B)
+C
6
sin6 2x
+C
(C)
3
cos5 2x
+C
(D)
3
cos5 2x
(E)
+C
6
(A)
Z
π/2
2
sin(2x)esin
14.
x
dx =
0
(A) e
Z
(B) e − 1
(C) 1 − e
(D) e + 1
π/2
15.
sin5 x cos x dx =
0
(A)
1
6
(B) −
1
6
(C) 0
(D) −6
(E) 6
(E) 1
Z
16.
1
sin−1 x dx =
0
(A) 0
17. If
18.
π+2
2
(C)
π−2
2
√
3 sin x
(B) 8e
3 cos x
(C) 8e
dx
=
9 − x2
(A) sin−1 3x + C
√
(B) ln |x + 9 − x2 | + C
1
(C) sin−1 3x + C
3
x
(D) sin−1 + C
3 √
1
(E) 3 ln |x + 9 − x2 | + C
Z
19.
(D)
π
2
(E) −
π
2
dy
= 3y cos x, and y = 8 when x = 0, then y =
dx
(A) 8e
Z
(B)
x sin(2x) dx =
x2
cos(2x) + C
2
x2
(B) − cos(2x) + C
4
x
1
(C) − cos(2x) + sin(2x) + C
2
4
x
1
(D) − cos(2x) + cos(2x) + C
2
2
1
1
(E) − cos(2x) + sin(2x) + C
2
4
(A) −
3 sin x
+3
3y 2
(D)
cos x + 8
2
3y 2
(E)
sin x + 8
2
z dz
= z 4−
, where z(0) =
20. Given the differential equation
dt
100
50, what is lim z(t)?
t→∞
(A) 400
(B) 200
(C) 100
(D) 50
(E) 4
Part II. Free-Response Questions
1. Compute the following indefinite integrals:
Z
Z
√
5
(a) (1 − x) dx
(b) x2 1 − x dx
Z
x
(d)
x
e sin(e ) dx
Z
x dx
x4 + 1
(g)
Z
dx
√ p
√
x 1− x
(j)
Z
2
xex dx
(m)
Z
(p)
x3
(e)
Z√
Z
(h)
Z
(k)
Z
(n)
ln ex dx
x sec2
x2 dx
dx
√ √
x 1−x
xex dx
Z
(c)
Z
(f)
√
x
dx
x2 − 1
sin2 x − 1 d(sin x)
Z
(i )
sec 2x tan 2x dx
Z
(l)
x ln x dx
Z
(o)
ex sin x dx
p
3
1 − x2 dx
2. Compute the following definite integrals:
Z e2
dx
(a)
e x ln x
Z √π/4
(b)
0
Zπ/2
x dx
1 + x4
x2 sin x dx
(c)
0
Z
π
(d)
−π
√
cos x dx
4 + 3 sin x
3. Suppose that f is an even differentiable function. Compute
R1
−1
xf 0 (x) dx.
4. Below is sketched the slope field for the differential equation
dy
= f (x, y). Sketch a possible solution of the above differential
dx
equation satisfying the initial value y(−5) = −2.
5. Given the slope field indicated below, sketch solutions of the
IVPs
dy
(i)
= f (x, y), y(−1) = 4, x ≥ −1
dx
dy
(ii)
= f (x, y), y(−1) = −1, x ≥ −1
dx
dy
=
6. Sketch the slope field describing the solutions of the ODE
dx
x − y.
y 6
-
x
?
7. Solve the initial value problem
dy
= 2 cos x + sin x, y(π) = 0.
dx
d2 y
= 2 cos x + sin x, y(π) =
8. Solve the initial value problem
2
dx
1, y 0 (π) = 0.
9. Solve the initial value problem
dy
= −2y, y(0) = 100.
dx
d2 y
10. Solve the initial value problem 2 = −2y, y(0) = 1, y 0 (0) = 0.
dx
dy
=
dt
ry, y(0) = y0 , where y0 is the initial amount (of money) and
r × 100 is the annual interest rate.
11. The (continuous) growth of money is modelled by the IVP
(i) Solve the IVP, given that there is initially $10,000 in the bank,
and that the interest rate is 6.2%.
(ii) How much is in the bank after 2 12 years?
(iii) How long will it take for the money to double in value?
12. Suppose that we have the initial value problem
y0 and that we know that y(150) = 21 y0 . Find k.
dy
= −ky, y(0) =
dt
13. (Carbon-14 Dating). The initial value problem of importance in
Carbon-14 dating is
dy
= −ky, y(0) = y0 .
dt
It is typically assumed that the half-life of the radioactive nuclei
of carbon 14 is roughly 5,700 years.
(i) Using the above, show that k = (ln 2)/5700.
(ii) After how many years of radioactive decay will the original
carbon 14 have decayed by 20%?
14. Newton’s Law of Cooling says that the IVP governing the temperature T of a heated object immersed in an environment at a
temperature of Ts is
dT
= −k(T − Ts ), T (0) = T0 ,
dt
(where k is a positive constant depending on the various media).
(i) Show by differentiation that the solution of the above IVP is
given by T (t) = Ts + (T0 − Ts )e−kt .
(ii) Compute lim T (t).
t→∞
(iii) Does your answer to part (ii) make sense?
15. Assume, as in Problem 14 above, that Newton’s Law of Cooling has solution given by T (t) = Ts + (T0 − Ts )e−kt . We apply
this model to a bowl of hot noodle soup, where the soup was
originally at 90◦ C and cooled to 60◦ C after 10 minutes in a room
whose ambient temperature was 20◦ C.
(i) Find the value of k in the above model.
(ii) What is the temperature of the soup after half an hour?
(iii) After how many minutes will the soup cool to 30◦ C?
16. Assume, as in Problem 14 above, that Newton’s Law of Cooling has solution given by T (t) = Ts + (T0 − Ts )e−kt . We apply
this model to a bowl of hot noodle soup, where the soup was
originally at 90◦ C and cooled to 60◦ C after 10 minutes in a room,
and then cooled to 50◦ C after another 10 minutes. Determine the
ambient temperature of the room.1
17. (Mr. S on roller skates.) Believe it or not, Mr. S is pretty good
at roller skating! One of his favorite activities is skating as fast
as he can and then stop, seeing how far he can “coast.” Once he
starts coasting, the differential equation that governs his motion
dv
is m
= −kv, where v is his velocity in km/hr, where m is
dt
Mr. S’s mass (in kg), and where k > 0 is a constant representing
friction. Assume that k = 60, 000, that m = 80 kg, and that at the
instant he starts to coast, Mr. S is moving at 30 km/hr.
(a) Determine Mr. S’s velocity, in km/hr, as a function of time.
(b) Determine how far Mr. S will coast (in meters).
1
This problem is quite a bit more difficult than Problem 15, above.
18. The Logistic Differential Equation is given by
dP
k
= P (M − P ) ,
dt
M
where k is a positive constant.
(i) Show by differentiation that the solution of the above differential equation is given by
P =
M
.
1 + Ae−kt
(ii) Compute lim P .
t→∞
19. Suppose that the spread of measles in a given school is predicted
by the logistic function
P (t) =
200
,
1 + 199e−t
where t is the number of days after a student comes into contact
with an infected student.
(i) Write down the corresponding logistic differential equation.
(ii) Compute P (0) and explain what this means.
(iii) After how many days will half of the students have become
infected?
(iv) After how many days will 90% of the students have become
infected?
20. Sketch the slope field describing the solutions of the logistic difdP
ferential equation
= P (5 − P ), P ≥ 0.
dt
P 6
t
-
21. Consider the differential equation
3−x
dy
=
.
dx
y
(a) Let y = f (x) be the particular solution to the given differential equation for 1 < x < 5 such that the line y = −2 is
tangent to the graph of f . Find the x-coordinate of the point
of tangency, and determine whether f has a local maximum,
local minimum, or neither at this point. Justify your answer.
(b) Let y = g(x) be the particular solution to the given differential equation for −2 < x < 8, with the initial condition
g(6) = −4. Find y = g(x).
dy
= x2 (y − 1).
dx
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
y
22. Consider the differential equation
6
•
3•
•
•
2•
•
•
1•
•
•
−1
•
•
1
-
x
(b) While the slope field in part (a) is drawn at only twelve
points, it is defined at every point in the xy-plane. Describe
all points in the xy-plane for which the slopes are positive.
(c) Find the particular solution y = f (x) to the given differential
equation with the initial condition f (0) = 3.
dy
3x2
23. Consider the differential equation
= 2y .
dx
e
(a) Find a solution y = f (x) to the differential equation satisfy1
ing f (0) = .
2
(b) Find the domain and range of the function f found in part
(a).
24. The
f is differentiable for all real numbers. The point
function
1
3,
is on the graph of y = f (x), and the slope at each point
4
dy
= y 2 (6 − 2x).
(x, y) on the graph is given by
dx
d2 y
1
(a) Find 2 and evaluate it at the point 3,
.
dx
4
dy
(b) Find y = f (x) by solving the differential equation
=
dx
1
y 2 (6 − 2x) with the initial condition f (3) = .
4
p
25. Let f be the function satisfying f 0 (x) = x f (x) for all real numbers x, where f (3) = 25.
(a) Find f 00 (3).
(b) Write an expression for y = f (x) by solving the differential
dy
√
= x y with the initial condition f (3) = 25.
equation
dx
26. Consider the differential equation
2x
dy
=− .
dx
y
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
y
6
•
2•
•
•
1•
•
-
−1
x
1
•
−1 •
•
•
−2 •
•
?
(b) Let y = f (x) be the particular solution to the differential
equation with the initial condition f (1) = −1. Write an
equation for the line tangent to the graph of f at (1, −1) and
use it to approximate f (1.1).
(c) Find the particular solution y = f (x) to the given differential
equation with the initial condition f (1) = −1.
27. You are given the initial value problem
dy
= x − y, y(0) = 0.
dx
(i) Use Euler’s method to approximate y(0.2) (use dx = 0.1).
(ii) Show by differentiation that the solution of this IVP is y(x) =
e−x + x − 1.
(iii) Compare the relative error in the estimation of y(0.2). (Use
the definition
exact value − approximate value .)
Relative Error = exact value