Calculus I
Math 2413
Skills Test
MULTIPLE CHOICE. Record your answer on your scantron. DO NOT USE A CALCULATOR.
Solve the problem.
1) Find the surface area S of a rectangular box with length 3 ft, width 4 ft, and height
3 ft.
A) 72 ft2
B) 54 ft2
C) 66 ft2
D) 33 ft2
2) Find the volume V of a right circular cylinder with radius 6 m and height 20 m. Express the answer in teems of
.
A) V = 720 m 3
B) V = 120 m 3
C) V = 180 m 3
D) V = 60 m 3
3) Find the perimeter. Approximate the result to the nearest tenth using 3.14 for
.
6 in.
3 in.
A) 19.7 in.
B) 24.4 in.
C) 27.4 in.
Find the slope-intercept form of the equation of the line with the given properties.
4) Slope = 6; containing the point (-2, -6)
A) y = 6x - 6
B) y = -6x + 6
C) y = -6x - 6
D) 22.7 in.
D) y = 6x + 6
Multiply the polynomials. Express the answer as a single polynomial in standard form.
5) (3x + 5)3
A) 27x3 +125
C) 27x 3 + 135x 2 + 135x + 125
B) 27x 3 + 135x 2 + 225x + 125
D) 9x 2 + 30x + 25
Find the quotient and the remainder.
6) 8x 3 - 28x2 + 10x + 42 divided by 2x - 5
A) 4x 2 - 4x - 5; remainder 0
B) 4x 2 - 4x - 5; remainder 17
D) x2 - 5; remainder -4
C) 4x 2 - 4x - 5; remainder 20
Reduce the rational expression to lowest terms.
4x + 2
7)
2
20x + 18x + 4
A)
4x + 2
2
20x + 18x + 4
B)
1
5x + 2
C)
1
4x + 5
5x + 18
D)
4x
5x + 2
Perform the indicated operations and simplify the result. Leave the answer in factored form.
x
5
8)
2
2
x - 16 x + 5x + 4
A)
x 2 - 4x + 20
(x - 4)(x + 4)
B)
x 2 + 4x + 20
(x - 4)(x + 4)(x + 1)
C)
x2 - 4
(x - 4)(x + 4)(x +1)
D)
x 2 - 4x + 20
(x - 4)(x + 4)(x + 1)
Simplify the expression. Assume that all variables are positive when they appear.
9) 3x 2 + 7 48x 2 + 3 48x 2
A) 41x 92
10)
x3x +
A)
C)
B) 41x 3
C) 10x 92
D) 10x 3
y
7y
x 33x -
7xy - 3xy + y 7
3x + 7y
10xy +
3x + 7y
B)
7y
x 3-
D)
3x -
7xy - 3xy + y 7
3x - 7y
10xy +
3x - 7y
7y
Find the real solutions of the equation by factoring.
11) 20x 3 + 100x 2 + 120x = 0
A) -
1
, -2
3
12) x3 + 2x 2 - 9x - 18 = 0
A) {-3, 3, 2}
B) {0, -3, -2}
C) {0, 3, 2}
D) {-3, -2}
B) {3, -2}
C) {9, -2}
D) {-3, 3, -2}
Solve the equation by factoring.
7
13) 8x - 55 =
x
A) -
1
,7
8
B)
1
1
,55
8
C) -
1
,8
8
D) {-8, 7}
Find the real solutions of the equation.
14) x 2 - 3x + 49 = x + 4
A) {7}
15) x3/4 - 2x 1/4 = 0
A) {0, 16}
3
2
B) {3}
C) {-3}
D) -
B) { 2}
C) {2}
D) {0, 4}
Factor completely.
16) (x + 8)2 - 12(x + 8) + 27
A) (x 2 + 8x - 9)(x 2 + 8x - 3)
C) (x 2 - 8x + 9)(x 2 - 8x + 3)
B) (x + 17)(x + 11)
D) (x - 1)(x + 5)
2
An expression that occurs in calculus is given. Reduce the expression to lowest terms.
(x 2 + 4) · 7 - (7x + 8) · 5x
17)
(x 2 + 4)3
A)
28x 2 + 40x - 28
(x 2 + 4)3
B)
-35x 2 - 40x + 7
(x 2 + 4)2
C)
-28x 2 - 40x + 28
(x 2 + 4)3
Factor the expression. Express your answer so that only positive exponents occur.
18) x9/7 + x 5/7
A) x5/7 (x 4/7 + 1)
B) x5/7 (x 4/7 + x)
C) x5/7 (x 4/7 - 1)
D)
-42x 2 - 40x + 28
(x 2 + 4)3
D) x5/7 (x 4/7 )
An expression that occurs in calculus is given. Write the expression as a single quotient in which only positive
exponents and/or radicals appear.
19) 10x 3/2(x 3 + x 2 ) - 12x 5/2 - 12x 3/2
A) x3/2(x - 1)[10(x 2 - 1) - 2] + 10x 3
C) 10x 3/2(x 3 - 1) + 10x 3 - 2x 3/2(x - 1)
20)
B) 2x 3/2 (x + 1)(5x 2 - 6)
D) 10x 3/2(x + 1)(5x 2 - 6)
(49 - x 2)1/2 + 3x2 (49 - x 2 )-1/2
49 - x 2
A)
3x2
49 - x2
B)
49 - 4x 2
(49 - x 2 )3/2
C)
3x2
(49 - x 2 )3/2
D)
49 + 2x 2
(49 - x 2 )3/2
For the given functions f and g, find the requested composite function value.
x-6
, g(x) = x 2 + 9;
Find (g f)(-2).
21) f(x) =
x
A)
145
16
B)
7
13
C) 13
3
D) 25
Graph the function.
22) f(x) = x
A)
B)
C)
D)
4
23) f(x) = x + 5
2
if x < 1
if x 1
A)
B)
C)
D)
Use the graph of the function f to solve the inequality.
24) f(x) 0
A) (- , -3]
[4, 8]
B) (- , -3]
[4, )
C) (- , -3)
Solve the inequality algebraically. Express the solution in interval notation.
25) (x - 8)2 (x + 9) > 0
A) (
, -9]
B) (-9, )
C) (
5
, -9)
(4, )
D) (- , -3)
(4, 8)
D) (
(9, )
, -9)
Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the
real numbers.
26) f(x) = 3x 3 - 14x 2 + 3x + 20
A) 1,
3
, -4; f(x) = (3x - 5)(x - 1)(x + 4)
5
B) -1,
3
, -4; f(x) = (3x - 5)(x - 4)(x + 1)
5
5
, 4; f(x) = (3x - 5)(x - 4)(x + 1)
3
D) -4,
5
, 1; f(x) = (3x - 5)(x - 1)(x + 4)
3
C) -1,
Write as the sum and/or difference of logarithms. Express powers as factors.
9
(8x) 1 + 2x
,
x>6
27) ln
(x - 6)7
A) 8ln x +
2
ln (1 + 2x) - 7ln (x - 6)
9
C) ln 8 + ln x - 9ln (1 + 2x) - 7ln (x - 6)
Express as a single logarithm.
x2 - 4x - 45
x 2 - 3x - 40
- ln
+ ln (x 2 - 18x + 81),
28) ln
x-8
x+9
A) ln
3(x - 9)(x + 9)
2(x - 8)
B) ln
B) ln 8 + ln x +
1
ln (1 + 2x) - ln 7 - ln (x - 6)
9
D) ln 8 + ln x +
1
ln (1 + 2x) - 7ln (x - 6)
9
x>0
(x - 9)3
(x - 8)2 (x + 9)
C) ln
(x - 9)3 (x + 9)
(x - 8)2
D) ln
3(x - 9)
2(x - 8)(x + 9)
f(x) = sin x, g(x) = cos x, h(x) = tan x, F(x) = csc x, G(x) = sec x, H(x) = cot x. Provide an appropriate response.
29) Find F
A) C)
3
. What point is on the graph of F?
2 3
;
3
2 3
;
3
3
3
,
,-
2 3
3
B) -2;
2 3
3
D) 2;
3
3
, -2
,2
Find the exact value of the expression.
3
30) sin-1
2
A)
B)
4
2
3
C)
3
4
D)
3
Simplify the expression.
cos
+ tan
31)
1 + sin
A) 1
B) cos
+ sin
C) sec
Find the real zeros of the trigonometric function on the interval 0
32) f(x) = 2 cos2 x + 3 sin x - 3
A)
6
, ,
7
6
B)
2
, ,
3 2 3
x<2
C)
6
D) sin2
5
, ,
6 2 6
D)
2
,
6
,
1
6
Solve the system of equations by elimination.
33)
9x + 15y = 15
2x - 5y = -5
A) x = 1, y = 0; (1, 0)
B) x = 0, y = 0; (0, 0)
C) x = 0, y = 1; (0, 1)
Solve the system of equations to find the points at which the two graphs intersect.
34)
y = x2 - 6x + 9
x+ y=5
A) x = 4, y = 1; x = 1, y = 4
or (4, 1), (1, 4)
C) x = -4, y = 9; x = -1, y = 6
or (-4, 9), (-1, 6)
B) x = 4, y = 9; x = 1, y = 4
or (4, 9), (1, 4)
D) x = 3, y = 2 or (3, 2)
7
D) x = 1, y = 1; (1, 1)