Test #4 Quick Quizzes
The Quick quizzes are mostly concept quizzes. Doing only the Quick quizzes will not be
enough to learn the material and pass the test. The quizzes are no calculator. The Quick
quizzes are self-assessment quizzes, and they will not be collected or graded. The answer for
all quizzes are at the end of each quiz. Copyright © 2011 Pearson Education, Inc. All rights reserved
Section 4.1 Quick Quiz
1.
A function f has an absolute maximum on an interval provided
(a) f is continuous on the interval.
(b) the interval is closed.
(c) the interval is closed and f is continuous on the interval.
2.
Which of the following statements is true?
(a) The absolute minimum of a function can occur at an endpoint.
(b) The absolute minimum of a function cannot be a local minimum.
(c) A local maximum can occur at an endpoint.
3.
A critical point of the function f occurs at an interior point c
(a) only when f ‘ (c) = 0 .
(b) only when f ‘(c) fails to exist.
(c) when either f '(c) = 0 or f ‘(c) fails to exist.
4.
If f ‘ (c) = 0 , then
(a) f has a local maximum or local minimum at c.
(b) f has an absolute maximum or minimum at c.
(c) c is a candidate for a local extreme point.
5.
If f ‘ (c) fails to exist and c is in the domain of f, then
(a) f has a local maximum or local minimum at c.
(b) c is a candidate for a local extreme point.
(c) f has an absolute maximum or minimum at c.
6.
Which of the following statements is true about the function f ( x) = x 5 ?
(a) f ‘ (c) = 0 .
(b) f has a local minimum at x = 0.
(c) f has a local maximum at x = 0.
7.
A function with an absolute minimum on an interval I
(a) must have a local minimum on I.
(b) needn’t be continuous on I.
(c) must have an absolute maximum on I.
On the interval [−2, 2] the function f ( x) = x 4
(a) has an absolute maximum but no local maxima.
(b) has an absolute maximum at an interior point of the interval.
(c) has a local minimum but no absolute minimum.
9. The function f ( x) = 10 has the property that
(a) it has no absolute maximum and no absolute minimum.
(b) every point is an absolute maximum and minimum.
(c) it has no critical points.
10. On the interval [−5, 5], the function f ( x)= x - 3
8.
(a) has no local maximum.
(b) has no local minimum.
(c) has neither an absolute maximum nor minimum.
Quick Quiz 4.1 Answers:
1c 2a 3c 4c 5b 6a 7b 8a 9b 10a
Section 4.2 Quick Quiz
1.
The function f ( x) = 3x 2 is
2.
(a) increasing on (-∞, O) and decreasing on (O, ∞).
(b) increasing on (-∞, ∞).
(c) decreasing on (-∞, O) and increasing on (O, ∞).
A function f has the property that f '( x) > O on (a, b) and f '( x) < O on (b, c). It follows that
3.
(a) f is continuous on (a, c).
(b) f is increasing on (a, c).
(c) f is decreasing on (b, c).
A function f has the property that f '( x) > O on (a, b), f '( x) < O on (b, c), and f '(b) = O . It follows that
(a) f has a local maximum at b.
(b) f has a local minimum at b.
(c) f is undefined at b.
(b) f is concave down on I.
(c) f is concave up on I.
4.
If f ' is decreasing on an interval I, then
5.
(a) f is decreasing on I.
If f ''(c) = O , then
(a) an inflection point occurs at x = c.
(b) the concavity of f changes at c.
(c) c is a candidate for the location of an inflection point.
6.
If f is concave down on an interval I, then
(a) f is decreasing on I.
(b) f ''(c) < O on I.
(c) f is positive on I.
7.
The function f has the property that f '(c) = O and f ''(c) > O . It follows that
8.
(a) f has a local minimum at x = c.
(b) f has a local maximum at x = c.
(c) f is increasing on an interval containing c.
The graph of a function f that satisfies f '( x) > O and f ''( x) < O on (O, ∞)
(a) increases at an increasing rate.
(b) increases at a decreasing rate.
(c) decreases at a decreasing rate.
9.
Which is true of f ( x) = sin x on (O,π)?
(a) It is increasing and concave down.
(b) It is decreasing and concave down.
(c) It is positive and concave down.
10. The function f ( x) = x8 satisfies f (O) = f '(O) = f ''(O) = O . Which is true?
(a) A local minimum and an inflection
point occur at x = O.
(b) A local minimum occurs at x = O.
(c) An inflection point occurs at x = O.
Quick Quiz 4.2 Answers:
le
2e
3a
4b
5e
6b
7a
8b
9e
10b
Section 4.3 Quick Quiz
1.
The function f ( x) = x 2 cos x is
2.
Which window is most appropriate for displaying the graph of f ( x) = 4 - x 2 ?
3.
(a) [0, 4] x [-2, 2]
(b) [-10, 10] x [-2, 1]
(c) [-2, 2] x [0, 3]
To exploit symmetry in graphing f ( x) = x 5 - 3x 3 + x , you really need to know the shape of the graph for
(a) even.
(b) odd.
(b) x ≥ 0.
(a) -10 < x < 10.
4.
(b) decreasing.
7.
8.
9.
(c) concave up.
If an even function is decreasing on the interval (0, ∞), then on the interval (-∞, 0) it is
(a) increasing.
6.
(c) x > 5.
If an odd function is increasing on the interval (0, ∞), then on the interval (-∞, 0) it is
(a) increasing.
5.
(c) neither.
(b) decreasing.
(c) positive.
Suppose you graph a function in the default window of your calculator. How can you be sure you have not
missed important features of the graph?
(a) Zoom in and out several times.
(b) Graph the function on a different calculator.
(c) Locate critical points, inflection points, and intervals of increase, decrease, and concavity.
x -2
has
The function f ( x) =
x+2
(a) a zero, a vertical asymptote, and a horizontal asymptote.
(b) a vertical, but not a horizontal, asymptote.
(c) a horizontal, but not a vertical, asymptote.
x-2
The function f ( x) = 2
has
x +2
(a) a zero, a vertical asymptote, and a horizontal
asymptote. (b) a vertical, but not a horizontal,
asymptote.
(c) a horizontal, but not a vertical, asymptote.
Polynomials
(a) may have zeros, but never
asymptotes. (b) may have vertical
asymptotes.
(c) may have horizontal asymptotes.
10. Suppose you determine that f has critical points at x = 2, 4, and 6, and nowhere else. Furthermore, f '
changes sign at the critical points. Then
(a) you must apply the second derivative test to locate local extreme
points. (b) local extreme values occur at x = 2, 4, and 6.
(c) the zeros of f occur at the critical points.
Copyright © 2011 Pearson Education, Inc. All rights reserved.
Quick Quiz 4.3 Answers:
la
2e
3b
4a
5a
6e
7a
8e
9a
10b
Section 4.4 Quick Quiz
1.
The product of a pair of real numbers that have a sum of 40 is P(x) = x (40 - x), where x is one
of the numbers. This product has a maximum when
(a) x = 20.
(b) x = 0.
(c) x = 40.
2.
The sum of a pair of positive real numbers that have a product of 9 is S(x) = x + 9/x, where x is one
of the numbers. This sum has a minimum when
(a) x = 9.
(b) x = 3.
(c) x = 6.
3.
Your task is to maximize Q = xy, subject to the condition that R = x + 2y - 40 = 0. In this formulation,
Q is called the
(a) constraint.
(b) derivative.
(c) objective function.
4.
Your task is to maximize Q = xy, subject to the condition that R = x + 2y - 40 = 0. In this formulation,
R is called the
(a) constraint.
(b) derivative.
(c) objective function.
5.
Suppose you want to maximize the volume V = xyz subject to the conditions that x = y and xy + xz +
yz =
100. The volume expressed in terms of x is
(a) V = 6x3.
6.
(b) V = x(50 - 2x2).
To find the two positive real numbers, x and y, with a sum of 10, that maximize the quantity 2x2 + y2,
one possible objective function is
(a) f(x) = x2 + 2(10 - x)2.
7.
(c) V = x(100 - x2)/2.
(b) f(x) = 3x2 - 20x + 100.
(c) g(y) = 100 - 40y + 3y2.
What is the method presented in this chapter to solve optimization problems?
(a) Find the objective function, eliminate extra variables using the constraints, and find the
extreme points.
(b) Make a table of values of the objective function for all values of the variables and find
the minimum or maximum value.
(c) Find the extreme values of the constraint and substitute into the objective function.
8.
Of all closed curves in the plane (such as rectangles, triangles, circles) with a perimeter of 10, the one
with the greatest area is
(a) an equilateral triangle.
9.
(b) a square.
(c) a circle.
If you cannot find the derivative of the objective function in an optimization problem, the best
way to proceed is
(a) guess and check.
(b) graph the objective function and approximate the extreme
values. (c) find a new objective function.
10. The volume of a right circular cylinder with radius r and height h is V = π r2 h; its lateral surface area
(curved surface) is S = 2 π rh. To create a cylinder with a volume of 100 cm3 and a small lateral
surface area, you should
(a) make the radius large and the height small.
small. (c) make the radius equal to the height.
(b) make the height large and the radius
Quick Quiz 4.4 Answers:
la
2b
3e
4a
5e
6b
7a
8e
9b
10a
Section 6.1 Quick Quiz
1.
The difference in the position of an object at two times, s(b) - s(0), is
(a) the distance traveled.
2.
(b) the displacement.
(c) the average velocity.
One can find the position of an object moving along a line given
(a) the speed.
(b) the initial position and the speed.
(c) the initial position and the velocity.
3.
An object that moves along a line with a velocity given by v(t) = 2 - t has a displacement between t = 0 and
t = 4 of
(a) 0.
(b) 4.
(c) 2.
4.
For a general velocity function v of an object moving along a line, ∫a 𝑣(𝑡)𝑑𝑡 is the
b
(a) distance traveled by the object over the time interval [a, b].
(b) the displacement of the object over the time interval [a, b].
(c) the position of the object at t = b.
b
b
5.
For a general velocity function v of an object moving along a line,
b
∫a |𝑣(𝑡)|𝑑𝑡
∫ 𝑣(𝑡)𝑑𝑡
a
(a) distance traveled by the object over the time interval [a, b].
(b) the displacement of the object over the time interval [a, b].
(c) the position of the object at t = b.
6.
If the velocity and the position of an object are v and s, respectively, then in general
b
s(0) + ∫a 𝑣(𝑡)𝑑𝑡
(a) distance traveled by the object over the time interval [0, b].
(b) (b) the displacement of the object over the time interval [0, b].
(c) (c) the position of the object at t = b.
7.
b
Suppose a quantity changes in time at a rate given by Q '(t ) . Then ∫a 𝑄′(𝑡)𝑑𝑡
(a) the value of Q at time t = b.
(b) the net change in Q between t = a and t = b.
(c) the average rate of change of Q between t = a and t = b.
8.
A tank is full at t = 0 at which time water begins flowing out at a rate given by V '(t) = 10(2 - t) gallons per
hour until t = 2 hours when the tank is empty. At t = 1
(a) the flow rate is a maximum.
(b) the tank is half full.
(c) 15 gallons have been drained from the tank.
(a) X > Y.
(b) X < Y.
(c) X = Y.
Quick Quiz 6.I Answers:
1b
2c
3a
4b
Sa
6c
7b
8c
Section 6.2 Quick Quiz
1.
b
If f(x) ≥ g(x) on the interval [a, b], ∫a 𝑓(𝑥) − 𝑔(𝑥)𝑑𝑥 is
(a) the area between the curves y = f(x) and y = g(x) on the interval [a, b].
(b) the average difference between the functions f and g.
(c) the net area of the region bounded by y = f(x) and the x-axis.
2.
The area of the region between the curves y = x and y = 2 - x2 is:
2
1
1
(b) ∫−2(2 − 𝑥 2 − 𝑥)𝑑𝑥
(c) ∫−2(𝑥 2 + 𝑥 − 2)𝑑𝑥
a) ∫−1(2 − 𝑥 2 − 𝑥)𝑑𝑥
3.
The area between the graphs of f and g on the interval [a, b] is
b
(a) ∫a 𝑓(𝑥) − 𝑔(𝑥)𝑑𝑥
b
(b) ∫a 𝑔(𝑥) − 𝑓(𝑥)𝑑𝑥
b
(c) ∫a |𝑓(𝑥) − 𝑔(𝑥)|𝑑𝑥
4.
The area of the region in the first quadrant between the curves y = x and y = x3 is
1
2
1
(b) ∫1 (𝑥 − 𝑥 3 )𝑑𝑥
(c) ∫0 ( 3√𝑦 − 𝑦)𝑑𝑦
(a) ∫0 (𝑥 3 − 𝑥)𝑑𝑥
5.
To find the area of the region bounded by y = 2x + 4 and y = x2 - 4
(a) it is easiest to set up an integral with respect to x.
(b) it is easiest to set up an integral with respect to y.
(c) the same amount of work is needed whether you integrate with respect to x or y.
1
1
6. Let A= ∫0 (𝑥 1/3 − 𝑥)𝑑𝑥 and B= ∫0 (y − 𝑦 3 )𝑑𝑦
(a) A = B.
(b) A > B.
(c) A < B.
Quick Quiz 6.2 Answers:
1a
2b
3c
4c
5a
6a
Section 6.3 Quick Quiz
1.
If a region R in the first quadrant is revolved about the x-axis, the disk or washer method
(a) requires integrating with respect to x.
(b) requires integrating with respect to y.
(c) cannot be applied.
2.
If a region R in the first quadrant is revolved about the y-axis, the disk or washer method
(a) cannot be applied.
(b) requires integrating with respect to x.
(c) requires integrating with respect to y.
3.
The region bounded by the line y = 2 - x and the two axes is revolved about the x-axis. The volume of the
solid generated is
2
2
(a) π ∫0 (2 − 𝑥)𝑑𝑥
4.
(b) π ∫0 y(2 − 𝑦)𝑑𝑦
The region in the first quadrant bounded by y =
the solid generated is
2
x and y = x is revolved about the x-axis. The volume of
1
(a) π ∫0 (𝑥 − 𝑥 2 )𝑑𝑥
2
(c) π ∫0 (2 − 𝑥)2 𝑑𝑥
(b) π ∫0 (𝑥 − 𝑥 2 )𝑑𝑥
1
(c) π ∫0 (𝑦 2 − y)𝑑𝑥
5.
The region in the first quadrant bounded by the curve x = 16 - y2 and the coordinate axes is revolved about
the y-axis. The volume of the solid generated is
4
4
2
(a) π ∫0 (16 − 𝑦 2 )2 𝑑𝑦
(b) π ∫0 (16 − 𝑦 2 )𝑑𝑦
(c) π ∫0 (16 − 𝑦 2 )2 𝑑𝑦
6.
The region bounded by the curves y = sec x and y = 2 is revolved about the x-axis. The volume of the solid
generated is
5π/3
π/3
π/3
(a) (a) π ∫π/3 (2 − sec 𝑥)2 𝑑𝑥
(b) π ∫−π/3(4 − 𝑠𝑒𝑐 2 𝑥)𝑑𝑥 (c) 2π ∫0 (𝑠𝑒𝑐 2 𝑥 − 4)𝑑𝑥
Quick Quiz 6.3 Answers:
1a
2c
3c
4b
5a
6b
Section 6.4 Quick Quiz
1.
If a region R in the first quadrant is revolved about the x-axis, the shell method
(a) requires integrating with respect to x.
(b) requires integrating with respect to y.
(c) cannot be applied.
2.
If a region R in the first quadrant is revolved about the y-axis, the shell method
(a) requires integrating with respect to x.
(b) requires integrating with respect to y.
(c) cannot be applied.
3.
The region in the first quadrant bounded by y =
the solid generated is
1
1
(a) 2π ∫0 (√𝑥 − 𝑥)𝑑𝑥
4.
x and y = x is revolved about the y-axis. The volume of
(b) π ∫0 (𝑦 2 − 𝑦 4 )𝑑𝑦
1
(c) π ∫0 (√𝑦 − 𝑦 2 )𝑑𝑦
The volume of a box with sides of length 2, 3, and 4 given by the general slicing method is
4
4
(a)∫0 (𝑦 2 − 𝑦 4 )𝑑𝑦
2
(b) ∫2 3𝑥 2 𝑑𝑥
(c) ∫0 3𝑥 2 𝑑𝑥
5.
The disk method corresponds to the general slicing method when the cross-sectional areas are given by
(c) A( x) =π f ( x) 2 .
(a) A( x) = x 2 .
(b) A( x) = x f ( x) .
6.
If the region bounded by y = cos x and the x- and y-axes (0 ≤ x ≤ π/2) is revolved around the x-axis, the
volume of the solid generated is most easily found using
(a) the shell method integrating with respect to x.
(b) the shell method integrating with respect to y.
(c) the disk method integrating with respect to x.
7.
The base of a solid is the unit circle in the xy-plane and cross-sections through the solid perpendicular to the
base are equilateral triangles. To find the volume of the solid, you should use
(a) the general slicing method.
8.
(b) the disk method.
(c) the shell method.
The region in the first quadrant bounded by y = 4 - x2 and the coordinate axes is revolved about the x-axis.
The volume of the solid generated is
4
2
(a) 2π ∫0 (y√4 − 𝑦)𝑑𝑦
(b) 2π ∫0 𝑥(4 − 𝑥 2 )𝑑𝑥
4
(c) 2π ∫0 𝑦(4 − 𝑦 2 )𝑑𝑦
Quick Quiz 6.4 Answers:
1b
2a
3b
4a
5c
6c
7a
8a