Calculus AB 1st Semester Review Sheet
1) Find for the following functions:
a) b)
c) d)
e) f)
g)
2) The graph of the function is shown below. For what values in the interval [-2, 4] is not
differentiable?
3) Given: G(x) = and H(x) is a continuous function with values in the table below.
X / -1 / 0 / 1 / 2 / 3 / 4H(x) / 20 / 22 / 26 / 35 / 49 / 66
a) Find the average rate of change of both G and H over the interval [ -1, 4]
b) Find the instantaneous rate of change of G at x = 3
c) Approximate the instantaneous rate of change of H at x = 3
d) Write the equation of the tangent line to G(x) at x = 1
4) If
a) Graph on the grid provided.
b)
c)
d) Is continuous at x = 2? E) Is differentiable at x = 2?
5) Given:
a) Using the difference quotient, find . Show all work.
Multiple Choice. You must show work to support your answers.
6) A) -5 B) C) 0 D) 5 E) 1
7) Let
A) 6409 B) 30720 C) 30729 D) 25609 E) 30756
8) If y is a differentiable function of x, then the slope of the curve of at the point
where y = 1 is
A) B) C) D) E) 0
9) A) 0 B) C) DNE D) -1 E) 1
10) If
A) B) C) D) E)
11) If
A) B) 4(3x2 + 16) C) D) e)
12) The maximum value of the function on the closed interval [1,4] is
A) 1 B) 0 C) 3 D) 6 E) none of these
13) If and y is a differentiable function of x, then
A) B) C) D) E)
14) If is continuous at the point where x = a, which of the following statements may be false?
A) B) C)
D) is defined E)
15) Which of the following functions is not everywhere continuous?
A) y = |x| B) C) D) E)
16) The equation of the line tangent to the curve of at the point where the curve crosses
the y axis is
A) y = 8x – 4 B) y = – 4x C) y = – 4 D) y = 4x E) y = 4x – 8
17) On the graph of below, are both positive on which interval?
18)
A) is not continuous at x = 1 B) is continuous at x = 1, but does not exist
C) exists and equals 1 D) = 2 E)
19) The curve is concave up when
A) x > 3 B) 1 < x < 3 C) x > 1 D) x < 1 E) x < 3
20) The curve has
A) two horizontal asymptotes
B) two horizontal asymptotes and one vertical asymptote
C) two vertical but no horizontal asymptotes
D) one horizontal and one vertical asymptote
E) one horizontal and two vertical asymptotes
21) A function equals for all x except x = 1. In order for the function to be continuous at
x = 1, the value of must be
A) 0 B) 1 C) 2 D) E) none of these
22) Let and let be continuous at x = 5. Then c =
A) B) 0 C) D) 1 E) 6
23) The y intercept of the line tangent to the graph of at x = 1 is
A) 1.0 B) 1.25 C) 1.50 D) 1.75 E) 2.0
24) Let Which of the following statements is (are) true?
I. is defined at x = 6 II. III. is continuous at x = 6
A) I only B) II only C) I and II only D) I, II, and III E) II and III only
25) The number, c, satisfying the Mean Value Theorem for on the interval [1, 1.5] is,
correct to 3 decimal places,
A) 0.995 B) 1.058 C) 1.239 D) 1.253 E) 1.399
26) If exists on the closed interval [a,b], then it follows that
A) is constant on [a,b]
B) there exists a number c in (a, b) such that = 0
C) the function has a maximum on the open interval (a, b)
D) the function has a minimum on the open interval (a, b)
E) the Mean Value Theorem applies
27) The height of a rectangular box is 10 inches. Its length increases at the rate of 2 in/sec; its
width decreases at the rate of 4 in/sec. When the length if 8 inches and the width is 6 inches,
the volume of the box is changing, in cubic inches per second, at the rate of
A) 200 B) 80 C) – 80 D) – 200 E) – 20
28) If , where L is a finite number, then it follows that
A) exists B) is continuous at x = c C)
D) is defined E) none of the preceding is necessarily true
29) A 26-foot ladder leans against a building so that its foot moves away from the building at the rate
of 3 ft/sec. When the foot of the ladder is 10 feet from the building, the top is moving down at the
rate of r ft/sec, where r is
A) B) C) D) E)
30) Find
A) B) C) D) E) none of these
31) The function is increasing on the following intervals
A) B) C) D) (0, 4) E)
32)
A) B) C) 1 D) 3 E) DNE
33) If x and y are real numbers, the domain of the function
A) all x except x = 2 or x = -2 B) all x except x = 4
C) |x| 1 D) x > 1 or x < -1 E) all Real numbers
34) Find
A) B) C) D) E) none of these
35)
A) B) 0 C) D) E) DNE
36) The only function that does not satisfy the Mean Value Theorem on the interval specified is
A) B) C)
D) E)
37)
A) B) 1 C) 1 D) E) DNE
38) Which statement is true?
A) If is continuous at x = c, then exists
B) If = 0, then has a local maximum or minimum at
C) If = 0, then has an inflection point at
D) If is differentiable at x = c, then is continuous at x = c.
E) If is continuous on (a, b), then attains a maximum value on (a, b)
39) The y-intercept of the line tangent to the graph of at x = 1 is
A) 2.44 B) .15 C) .15 D) .64 E) .85
40) Find any critical numbers of the function
A) 0 B) C) D) Both A & B E) Both A & C
41) Locate the absolute extrema of the function on the interval [0, 3]
A) Absolute Maximum: (3, 18); Absolute Minimum: (1, 2)
B) Absolute Maximum: (1, 2); Absolute Minimum (1, -2)
C) Absolute Maximum: (3, 18); No Absolute Minimum
D) No Absolute Maximum; Absolute Minimum ( 0, 0)
E) No absolute maximum or absolute minimum
42) Find the dimensions of the rectangle of maximum area bounded by the x and y axis and the
graph of
A) length 3: width 2.5 B) length 4: width 2 C) length 1: width 3.5
D) length 2: width 3 E) None of these
43) Determine the x value(s), if any, at which the function has a horizontal tangent.
A) x = 0 B) x = 0, x = 8 C) x = 0, x = 8 D) x = 8 E) No horizontal tangents
44) Given the function , for what values of x is
45) The area of a circular region is increasing by a rate of 96 square meters per second. When the
area of the region is 64square meters, how fast, in meters per second, is the radius of the
region increasing?
46) Find the equation of the tangent line to the graph at the point where x = 3.
47) The volume of a cylindrical tin can with a top and bottom is to be 16cubic inches. If a minimum
amount of tin is to be used to construct the can, what must be the height, in inches, of the can?
48) The position equation for the movement of a particle is given by s = where s is measured
in feet and t is measured in seconds. Find the acceleration at two seconds.
49) The radius of a circle is increasing at the rate of 5 inches per minute. At what rate is the area
increasing when the radius is exactly 10 inches?
50) If then there exists a number c in the interval that satisfies the
conclusion of the Mean Value Theorem. Which of the following could be c?
A) B) C) D) E)