AP Calculus Chapter 3 Testbank
(Mr. Surowski)
Part I. Multiple-Choice Questions (5 points each; please circle the correct
answer.)
4
1. If f (x) = 10x 3 + x, then f 0 (8) =
(A) 11
41
(B)
3
83
(C)
3
21
(D)
3
(E) 21
3x2 + x
, then g 0 (x) =
2. If g(x) = 2
3x − x
(A) 1
6x2 + 1
(B) 2
6x − 1
−6
(C)
(3x − 1)2
−2x2
(D) 2
(x − x)2
36x2 − 2x
(E)
(x2 − x)2
p √
3. If f (x) = 1 + x, then f 0 (x) =
−1
(A) √ p √
4 x 1+ x
1
(B) √ p √
2 x 1+ x
1
(C) p √
4 1+ x
1
(D) √ p √
4 x 1+ x
−1
(E) √ p √
2 x 1+ x
4. Find
dy
given that x3 y + xy 3 = −10.
dx
(A) 3x2 + 3xy 2
(B) −(3x2 + 3xy 2 )
3x2 y + y 3
(C)
3xy 2 + x3
3x2 y + y 3
(D) −
3xy 2 + x3
x2 y + y 3
(E) − 2
xy + x3
5. If f (x) = sin2 x, find f 000 (x).
(A) − sin2 x
(B) 2 cos 2x
(C) cos 2x
(D) −4 sin 2x
(E) − sin 2x
6. If f (x) = 3πx , find f 0 (x) =
3πx
(A)
π ln 3
3πx
(B)
ln 3
3πx
(C)
π
(D) π 3πx−1
(E) π ln 3 (3πx )
7. Find the slope of the normal line to the graph of y = x + cos xy at the point
(0, 1).
(A) 1
(B) −1
(C) 0
(D) 2
(E) Undefined
8. If f (x) = 3x2 − x and g(x) = f −1 (x), then g 0 (10) could be
(A) 59
1
(B)
59
(C) 11
1
(D)
11
1
(E)
10
(
ax3 − 6x
9. If the function f (x) is differentiable and f (x) =
bx2 + 4
a=
(A) 0
(B) 1
(C) −14
(D) −24
if x ≤ 1
then
if x > 1,
(E) 26
10. Two particles leave the origin at the same time and move along the y-axis
with their respective positions determined by the functions y1 = cos 2t and
y2 = 4 sin t for 0 < t < 6. For how many values of t do the particles have the
same acceleration?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
11. Find the value(s) of
dy
at y = 1 given that x2 y + y 2 = 5.
dx
3
(A) − only
2
2
(B) − only
3
3
(C) only
2
2
(D) ±
3
3
(E) ±
2
√
12. If f (x) = x2 3x + 1, then f 0 (x) =
−3x2 − 2x
(A) √
3x + 1
9x2 + 2x
(B) √
3x + 1
−9x2 + 4x
(C) √
2 3x + 1
15x2 + 4x
(D) √
2 3x + 1
−9x2 − 4x
(E) √
2 3x + 1
13. What is the instantaneous rate of change at t = −1 of the function f , if
t3 + t
?
f (t) =
4t + 1
12
4
20
4
12
(A)
(B)
(C) −
(D) −
(E) −
9
9
9
9
9
14. What is the equation of the line tangent to the graph of y = sin2 x at x =
π
?
4
1
π
(A) y − = − x −
2
4
1
π
(B) y − = x −
2
4
1
π
(C) y −√ = x −
4
2
1
π
1
x−
(D) y −√ =
4
2 2
1 1
π
(E) y − =
x−
2 2
4
(
3ax2 + 2bx + 1
15. If the function f (x) =
ax4 − 4bx2 − 3x
real values of x, then b =
1
7
11
(B)
(C) −
(A) −
4
4
16
if x ≤ 1
is differentiable for all
if x > 1
(D) 0
(E) −
1
4
16. The position of a particle moving along the x-axis at time t is given by x(t) =
ecos 2t , 0 ≤ t ≤ π. For which of the following values of t will x0 (t) = 0?
π
I. t = 0
II. t =
III. t = π
2
(A) I only
(B) II only
(C) I and III only
(D) I and II only
(E) I, II, and III
17. If f (x) = (3x)3x , then f 0 (x) =
(A) (3x)3x (3 ln(3x) + 3)
(B) (3x)3x (3 ln(3x) + 3x)
(C) (9x)3x (ln(3x) + 1)
(D) (3x)3x−1 (3x)
(E) (3x)3x−1 (9x)
18. If f (x) = 2x2 + 4, which of the following will calculate the derivative of f (x)?
[2(x + ∆x)2 + 4] − (2x2 + 4)
(A)
∆x
(2x2 + 4 + ∆x) − (2x2 + 4)
(B) lim
∆x→0
∆x
2
[2(x + ∆x) + 4] − (2x2 + 4)
(C) lim
∆x→0
∆x
(2x2 + 4 + ∆x) − (2x2 + 4)
(D)
∆x
(E) None of the above.
1
, which of the following will calculate the derivative of g(x)?
x+1
1
1
1
(A)
−
∆x x + ∆x + 1 x + 1
1
1
1
(B) lim
−
∆x→0 ∆x
x + ∆x + 1 x + 1
1
1
1
(C) lim
lim
−
∆x→0 ∆x ∆x→0 x + ∆x + 1
x+1
1
1
(D) lim
−
∆x→0 x + ∆x + 1
x+1
(E) None of the above.
19. If g(x) =
The next two questions pertain to the function f , whose graph is given below:
20. For the function f ,
10 y
I. f 0 (−3) > 0
y=f(x)
II. f 0 (0) < 0
III. f is differentiable
on the interval
(0, 1)
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I, II, and III
5
x
-6
-4
-2
2
-5
-10
21. For the function f
I. f 0 (x) > 0 on the interval (−5, −4)
II. f 0 (x) is constant on the interval (4, 6)
III. f 0 is not defined at all points of the interval (1, 5)
(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III
4
6
22. Given the graph of the rational function f below, give a sketch of the graph
of y = f 0 (x) on the same coordinate axes. (Note: the graph of y = f (x)
has a vertical asymptote at x = 1.)
8
y
6
4
2
-4
-2
2
-2
x
4
-4
23. The following graph represents a function g defined on the interval [−4, 4] and
differentiable on (−4, 4). On the same coordinate axes, graph y = g 0 (x) over
the interval (−3, 3).
y=g(x)
4
y
3
2
1
-4
-2
-1
-2
-3
-4
2
4
x
24. The following graph is that of y = h0 (x). On the same coordinate axes, give a
sketch of y = h(x), assuming that h(0) = 1.
4
y
3
2
y=h'(x)
1
-4
-2
2
4
x
-1
-2
25. The following graph is that of y = h0 (x). On the same coordinate axes, give a
sketch of y = h(x), assuming that h(0) = 0.
y
y=h'(x)
x
26. Using the definition of the derivative of a function, find f 0 (x), where f (x) =
x − x4 . Then find f 0 (1).
√
dy
where y = x.
27. Using the definition of the derivative of a function, find
dx
dy
.
Then find
dx x=4
28. Let f (x) = 4x3 − 21x2 − 24x + 23.
(a) Compute f 0 (x).
(b) Find all values of x satisfying f 0 (x) = 0.
1
29. Let f (x) = x + .
x
(a) Compute f 0 (x).
(b) Find all values of x satisfying f 0 (x) = 0.
30. Let y =
x
.
1 + x2
(a) Compute
dy
dx
(b) Compute all values of x for which
31. Let s(x) =
dy
= 0.
dx
sin x
and compute lim s0 (x).
x→∞
x
32. The graph below depicts the velocity v = s0 (t) of a particle moving along a
straight line, where on this straight line positive direction is to the right.
v (velocity)
4
2
t (time)
2
4
6
8
10
-2
-4
(a) Would you say that at time t = 1 the particle is moving to the left,
moving to the right, or not moving at all? Please explain.
(b) Would you say that at time t = 3 the particle is moving to the left,
moving to the right, or not moving at all? Please explain.
(c) Would you say that at time t = 4 the particle is moving to the left,
moving to the right, or not moving at all? Please explain.
(d) Find (estimate) two values of t at which time the particle is not accelerating.
(e) Find (estimate) a value of t at which time the particle is moving to the
left, but with zero acceleration.
(f) According to this graph, at how many distinct times is the particle at
rest?
(g) For which values of t is the particle not only at rest, but is not accelerating
(i.e., has no forces acting on it)?
(h) According to this graph, at how many distinct times is the particle not
accelerating?
33. Using logarithmic differentiation compute f 0 (x) where f (x) = xx , x > 0.
34. Let P (t) =
1
, where k is a real number.
1 + e−kt
(a) Show that
dP
= kP (1 − P ).
dt
d2 P
(b) Show that 2 = 0 when P = 1/2.
dt
35. Let f and g be differentiable functions and assume that f (1) = 2, f 0 (1) =
1, g(1) = −1, g 0 (1) = 0. Compute h0 (1), given that h(x) = x2 f (x)g(x).
36. In your text1 it was given as a exercise that the dollar cost of producing x
washing machines is c(x) = 2000 + 100x − 0.1x2 . Why is this an absolutely
rediculous cost model? (What is lim c(x)? Is this reasonable?)
x→∞
37. The volume V is a sphere of radius r is given by the formula V = (4/3)πr3 .
Suppose that you know that the radius r is an increasing function of t, and
dr
dV
= 2. Compute
when r = 3.
that when r = 3,
dt
dt
d
38. Compute
dx
cos x
, simplifying as much as possible.
1 + sin x
39. Note that the point (1, 2) is on the curve defined by y 3 − xy 2 − x2 y − 2 = 0.
(a) Compute
dy
at the point (1, 2).
dx
(b) Find an equation of the straight line tangent to the above curve at the
point (1, 2).
(c) Find an equation of the straight line normal to the above curve at the
point (1, 2).
1
Exercise 10, page 130.
40. Suppose that y = e−x cos x. Show that y 00 + 2y 0 + 2y = 0
41. Find f 0 (x), given that f (x) = √
x
and simplify your result as much as
x2 + 9
possible.
√
42. Compute f 0 (1), given that f (x) = sin π x2 + 3 .
43. Let x = t2 − 1, y =
(a) Compute
t−1
give x and y parametrically in terms of t.
t+1
dy
in terms of t.
dx
(b) Find all values of t for which
dy
fails to exist.
dx
(c) Find all values of t for which
dy
= 0.
dx
dy
.
t→∞
t→∞
t→∞ dx
(e) Suppose that the graph of y = f (x), where f is a differentiable function,
has a horizontal asymptote (say with lim y = c, for some real number c.
x→∞
dy
Would you expect that lim
= 0.
x→∞ dx
(d) Compute lim x, lim y, lim
sin x2
44. Let f (x) =
.
x
(a) Compute lim f (x).
x→∞
(b) Compute lim f 0 (x). (If this limit does not exist, say so.)
x→∞
(c) Compute lim |f 0 (x)|. (If this limit does not exist, say so.)
x→∞
45. Let x = t2 + t and let y = cos t.
(a) Find dy/dx as a function of t.
d dy
(b) Find
as a function of t.
dt dx
d
(c) Find
dx
dy
dx
as a function of t.
46. Given the equation y 2 + x2 = xy, compute both dy/dx and d2 y/dx2 .
47. Compute dy/dx, given that
(a) y = e2x cos x
(b) y = eln x
√
(c) y = ln( 1 + x2 )
48. Consider the curves defined by the equations y = f (x)= − 21 x2 + 4 and y =
g(x) = ln x. Show that at the point of intersection of these two curves, the
tangent lines are perpendicular. (Hint: what is f 0 (x)g 0 (x)? What does this
mean?)
√
x x2 + 1
49. Let y = √
and compute dy/dx using logarithmic differentiation.
3
x+2