AP Calculus Chapter 4 Testbank
(Mr. Surowski)
Part I. Multiple-Choice Questions
1. Let f (x) = x3 + 3x2 − 45x + 4. Then the local extrema of f are
(A) a local minimum of −179 at x = 5 and a local maximum of
77 at x = −3.
(B) a local minimum of −77 at x = 3 and a local maximum of
179 at x = −5.
(C) a local minimum of −179 at x = −5 and a local maximum
of −77 at x = 3.
(D) a local minimum of −77 at x = 3 and a local maximum of
77 at x = 5.
(E) a local minimum of 77 at x = −5 and a local maximum of
179 at x = 3.
2. Consider the function f (x) = x3 + 3x2 − 45x + 4. Then
I. f is decreasing on (−5, 3) and increasing on (−∞, −5)∪(3, ∞).
II. f has a local minimum at x = 3.
III. f has a local maximum at x = −5.
(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III
3. For what value of x does the function f (x) = x3 − 9x2 − 120x + 6
have a local minimum?
(A) 10
(B) 4
(C) 3
(D) −4
(E) −10
4. Use differentials to approximate the change in the volume of a
sphere when the radius is increased from 10 to 10.02 cm.
(A) 4213.973
(B) 1261.669
(C) 1256.637
(D) 25.233
(E) 25.133
5. The graph of y = x3 − 5x2 + 4x + 2 has a local minimum at
(A) (0.46, 2.87)
(B) (0.46, 0)
(C) (2.94, −4.05)
(D) (4.06, 2.87)
(E) (1.66, −0.59)
6. Find a positive value, c, for x, that satisfies the conclusion of the
Mean Value Theorem for f (x) = 3x2 −5x+1 on the interval [2, 5].
13
11
23
7
(A) 1
(B)
(C)
(D)
(E)
6
6
6
2
7. The graph of y = x3 − 2x2 − 5x + 2 has a local maximum at
(A) (2.120, 0)
(B) (2.120, −8.061)
(C) (−0.786, 0)
(D) (−0.786, 4.209)
(E) (0.666, −1.926)
8. A 20-foot ladder slides down a wall at 5 ft/sec. At what speed
is the bottom sliding out (in ft/sec) when the top is 10 feet from
the floor?
(A) 0.346
(B) 2.887
(C) 0.224
(D) 5.774
(E) 4.472
9. What are the coordinates of the point of inflection on the graph
of y = x3 − 15x2 + 33x + 100?
(A) (9, 0)
(B) (5, −48)
(C) (1, 119)
(D) (9, −89)
(E) (5, 15)
10. The graph of y = x4 + 8x3 − 72x2 + 4 is concave down for
(A) −6 < x < 2
(B) x > 2
(C) x < −6
√
√
(D) x < −3 − 3 5 or x > −3 + 3 5
√
√
(E) −3 − 3 5 < x < −3 + 3 5
11. The function f is given by f (x) = x4 + 4x3 . On which of the
following intervals is f decreasing?
(A) (−3, 0)
(B) (0, ∞)
(C) (−3, ∞)
(D) (−∞, −3)
(E) (−∞, 0)
12. The value of c that satisfies the Mean Value Theorem on the interval [0, 5] for the function f (x) = x3 − 6x is
5
5
5
(B) 0
(C) 1
(D)
(A) −√
(E) √
3
3
3
13. The graph of the function y = x3 + 12x2 + 15x + 3 has a relative
maximum at x =
(A) −10.613
(B) −.248
(C) −7.317
(D) −1.138
(E) −.683
14. The side of a square is increasing at a constant rate of 0.4 cm/sec.
In terms of the perimeter, P , what is the rate of change of the
area of the square, in cm2 /sec?
(A) 0.05P
(B) 0.2P
(C) 0.4P
(D) 6.4P
(E) 51.2P
15. The second derivative of a function is given by f 00 (x) = x sin x −
2. How many points of inflection does f have on the interval
(−10, 10)?
(A) Zero
(B) Two
(C) Four
(D) Six
(E) Eight
Part II. Free-Response Questions
16. Find the extreme values of the function f (x) = x3 + 3x2 − 36x + 4
on the interval [0, 4].
17. Find the extreme values of the function f (x) =
interval [−3, 3].
18. Find the extreme values of the function f (x) =
interval (−∞, ∞).
x2
x
on the
+1
x2
x
on the
+1
√
19. Find the extreme values of the function g(x) = x 2 − x on the
interval [−2, 2].
20. Find all relative extrema of the function h(x) = x2 e1/x .
√
21. Let f (x) = x 4 − x2 .
(a) Determine all relative extrema of f .
(b) Determine all points of inflection of the graph of y = f (x).
22. Determine all relative extrema of the function f (x) = x + cos x.
23. Let f (x) = 2xe−x , x ≥ 0 and determine the extrema of f (x) on
[0, ∞).
24. Let f (x) = 2xe−x , x ≥ 0.
(a) Determine where f is increasing and where f is decreasing.
(b) Determine where f is concave up and where f is concave
down.
(c) Sketch the graph of y = f (x),
y
6
-
x
?
25. Find the points of inflection of the graph y =
2
.
1 + e−x
26. A particle moves along the x-axis so that its position at time t ≥
0 is given by the function x(t) = t3 − 12t2 + 48t, t ≥ 0, where x
is measured in meters and t is measured in seconds.
(a) Determine those values of t
where the particle is not moving.
(b) Determine when the particle is
moving to the right and when
the particle is moving to the left.
(c) Determine when the particle is (d) At what value(s) of t does the
accelerating and when the particle particle have zero acceleration?
is decelerating.
27. Find all relative extrema of the function f (x) = e−x sin x, x > 0.
√
28. Find the critical x-values of the function f (x) = x2 3 x − 1.
29. Find the open intervals on which the function f (x) = 41 x3 − 3x is
increasing or decreasing.
30. Use the second derivative test to find the relative extrema of
f (x) = x4 − 4x3 + 3. As part of your work, be sure to identify all points of inflection. Sketch a graph, indicating all points
of inflection.
y 6
10
-
−2
2
−10
?
x
31. The graph below depicts a function f defined on the interval
(−∞, ∞). In terms of the notation given in the graph, determine
(i) the critical x values of f ;
(ii) the open intervals on which f is increasing and decreasing;
(iii) the open intervals on which f is concave up and concave
down; and
(iv) the values of x at which f has points of inflection.
y6
y = f (x)
6
x0 x1 x2
x3 x4
x5 x6
-
x
?
?
32. Find the production level that produces the maximum profit for
hamburgers in a fast-food restaurant whose profit function (P
dollars) in terms of hamburgers (x) is described as
x2
P = 2.44x −
− 5, 000 ,
20, 000
0 ≤ x ≤ 50, 000.
√
33. Suppose that the semicircle y = 1 − x2 is drawn. Now draw
a rectangle whose base is on the x-axis and height is such that
the rectangle is incribed within the semicircle. What is the maximum area of this rectangle?
34. Suppose that on one side of a 2 km-wide river is an electricitygenerating plant. On the opposite side, and 10 km down the
river is a small town that will be consuming the electricity. If it
costs $80/m to lay cable under the river and $40/m to lay cable
over land, find the most strategic method for laying the cable.
Power Plant
•
•
Town
35. Find the minimum distance from the point (2, 0) to a point on
the parabola whose graph is y = 2x2 .
√
36. Write the linearization of the function f (x) =√ 3 1 + x valid near
x = 0. Use this linearization to approximate 3 1.1.
37. Write the linearization of the function g(x) = cos x valid near
x = 0. Use this to approximate cos(.01). (Are you surprised by
this linearization?)
38. Write the linearization of f (x) = ln x valid near x = 1. Use this
approximate ln(1.2).
39. Assume that y = x2 ln x.
(a) Compute dy in terms of x and dx.
(b) Compute dy given that x = 1 and that dx = 0.1.
40. Let V be the volume of a sphere of radius r and let dV be its differential value (in terms of r and dr). For fixed dr, what would
you say about dV :
(a) that it is an increasing function of r,
(b) that it is a decreasing function of r, or
(c) that it doesn’t depend on r?
41. Let Q be a quantity
√ that depends on a measurable quantity, x,
by the rule Q = 1 + x2 . Suppose that we measure x to be 5.2 ±
0.015.
(a) Estimate the resulting range in computed values for Q.
(b) Estimate the resulting relative error in the computed value
for Q.
42. Assume that ball bearings are to be made whose volume must be
within a 1% error. Use differentials to determine the necessary
tolerance of the radius of the given ball bearings.
43. Suppose that a sphere is expanding at a rate given by dr/dt =
5m/sec. If S denotes the surface area of this sphere and V denotes its volume, compute dV /dt and dS/dt when r = 2. Do the
same for r = 4. Does this make sense?
44. Suppose that you are standing exactly 1km from the Pearl Tower
in Shanghai. Now suppose that someone releases a heliumfilled balloon from the top of the Pearl Tower, and that this balloon is rising at a constant rate of 1m/sec. Let D be the distance
between you (standing on the ground) and the balloon. Let h(t)
be the height of the balloon as a function of time, where we assume that h(0) = 400m.
(a) Compute dD/dt when h = 1000m.
(b) Compute dD/dt when h = 2000m.
(c) Compute lim dD/dt. (Does this make sense?)
h→∞
(d) Show that dD/dt is a increasing function of both h and t.
45. Let h be a function defined for all x 6= 0 such that h(4) = −3 and
x2 − 2
0
the derivative of h is given by h (x) =
for all x 6= 0
x
(a) Find all values of x for which the graph of h has a horizontal
tangent, and determine whether h has a local maximum, a
local minimum, or neither at each of these values. Justify
your answers.
(b) On what intervals, if any, is the graph of h concave up? Justify your answer.
(c) Write an equation for the line tangent to the graph of h at
x = 4.
(d) Does the line tangent to the graph of h at x = 4 lie above or
below the graph of h for x > 4. Why?
46. A cubic polynomial function g is defined by
f (x) = x3 + ax2 + bx + c
where a, b, and c are constants. The function has a critical value
at x = −1, and the graph of f has a point of inflection at the
point (−2, 0). Find a, b, and c.
x
f (x)
f 0 (x)
0
0<x<1 1 1<x<2
−1 Negative 0
Positive
2
2<x<3
3
3<x<4
2
Positive
0
Negative
0
Positive
DNE Negative −3 Negative
f 00 (x) −2 Negative 0
Positive
DNE Negative
4
Positive
0
Positive
47. Let f be a 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 .
6
-
48. An airplane is flying at a constant speed at a constant altitude of
3 km in a straight line that will take it directly over an observer
at ground level. At a given instant the observer notes that the
angle θ is π/3 and is increasing at 1/20 radians per second. Find
the speed of the airplane, in kilometers per hour.
3 km
•
Airplane
θ
•
Observer
49. Continue that we have, as in the above problem, an airplane
flying toward the observer at an altitude of 3 km. This time,
1
dθ
=
, and is
however, assume that angular rate of change,
dt
20
constant.
(i) What is the speed of the airplane when it is directly overhead?
(ii) Is the airplane slowing down or speeding up as it approaches
the observer? Justify your conclusion.