Math 121 Midterm Exam Version A Spring 2012 Instructor: Lang Don ...

Math 121 Midterm Exam Version A Spring 2012 Instructor: Lang
Don't forget to bubble in your name and ID on the scantron.
Express as a single logarithm and, if possible, simplify.
1) ln 3x2 - 18x + ln 1
3x
A) ln 9x2(x - 6) B) ln (x - 6)
C) ln (x - 18) D) ln 3x2 - 18x + 1
3x
2) If f(x) = x, g(x) = x , and h(x) = 5x+ 20, find h(g(f(x))).
5
A) 5 x + 20 B) x + 4 C) x + 20 D) x + 4
Use the graph to evaluate the limit.
3) lim
x→0 f(x)
4
3
2
1
y
-4 -3 -2 -1 1 2 3 4 x
-1
-2
-3
-4
A) 0 B) 3 C) -3 D) does not exist
4) Find lim
h→0
2 .
3h + 4 + 2
A) 2 B) Does not exist C) 1 D) 1/2
5) Find lim
x→0
1 + x - 1
x
.
A) 1/4 B) Does not exist C) 1/2 D) 0
1
1)
2)
3)
4)
5)

6) Graph the function y = -|x| and determine the symmetry, if any, of the function.
7)
4
3
2
1
y
-4 -3 -2 -1 1 2 3 4
-1
-2
-3
-4
A) Symmetric about the y-axis
4
3
2
1
y
x
-4 -3 -2 -1 1 2 3 4
-1
-2
-3
-4
C) Symmetric about the x-axis
4
3
2
1
y
-4 -3 -2 -1 1 2 3 4
-1
-2
-3
-4
Find lim - f(x), where f(x) =
x → 5
x
x
16 - x2 0 ≤ x < 4
4 4 ≤ x < 5
5 x = 5
B) Symmetric about the y-axis
4
3
2
1
y
-4 -3 -2 -1 1 2 3 4
-1
-2
-3
-4
D) Symmetric about the origin
4
3
2
1
y
-4 -3 -2 -1 1 2 3 4
-1
A) Does not exist B) 5 C) 4 D) 0
2
-2
-3
-4
x
x
6)
7)

8) Find all vertical asymptotes of the function f(x) =
x - 7
49x - x3 A) x = 0, x = -7 B) x = 0, x = 7
C) x = 0, x = -7, x = 7 D) x = -7, x = 7
9) Find lim
x→ - ∞
2x3 + 4x2
x - 6x2 .
A) 2 B) ∞ C) - 2
3
10) Is f continuous at f(3)?
f(x) =
-x2 + 1,
3x,
-4,
-3x + 6
2,
-1 ≤ x < 0
0 < x < 1
x = 1
1 < x < 3
3 < x < 5
A) No B) Yes
11) Find numbers a and b so that f is continuous at every point, where
f(x) =
-7,
ax + b,
21,
x < -4
-4 ≤ x ≤ 3
x > 3
D) -∞
A) a = -7, b = 21 B) a = 4, b = 33 C) a = 4, b = 9 D) Impossible
12) Find lim
x→∞
36x2 + x - 3
(x - 13)(x + 1) .
A) ∞ B) 6 C) 0 D) 36
13) Find an equation of the tangent line to the curve h(t) = t3 - 16t + 3 at the point (4, 3).
A) y = 32t + 3 B) y = 3 C) y = 32t - 125 D) y = 35t - 125
3
8)
9)
10)
11)
12)
13)

14) The graph of y = f(x) in the accompanying figure is made of line segments joined end to end.
Graph the derivative of f.
(-3, 2)
(-5, 0)
A)
C)
y
(0, -1)
6
4
2
(3, 5) (6, 5)
y
x
-6 -4 -2 2 4 6
-2
-4
-6
6
4
2
y
-6 -4 -2 2 4 6
-2
-4
-6
x
x
4
B)
D)
y
6
4
2
y
x
-6 -4 -2 2 4 6
-2
-4
-6
6
4
2
y
-6 -4 -2 2 4 6
-2
-4
-6
x
x
14)

The graph of a function is given. Choose the answer that represents the graph of its derivative.
15)
y
15
10
5
-15 -10 -5 5 10 15 x
-5
A)
C)
-10
-15
15
10
5
y
-15 -10 -5 5 10 15 x
-5
-10
-15
15
10
5
y
-15 -10 -5 5 10 15 x
-5
-10
-15
5
B)
D)
15
10
5
y
-15 -10 -5 5 10 15 x
-5
-10
-15
15
10
5
y
-15 -10 -5 5 10 15 x
-5
-10
-15
15)

The figure shows the graph of a function. At the given value of x, does the function appear to be differentiable,
continuous but not differentiable, or neither continuous nor differentiable?
16) x = -1
16)
4
2
y
-4 -2 2 4
-2
-4
A) Differntiable
B) Not continuous and not differentiable
C) Continuous but not differentiable
D) None of the above
17) Find dy
dx for y = 16x-2 + 2x 3 + 17x
x
A) -32x -3 + 6x 2 B) -32x -3 + 6x 2 + 17
C) -32x -1 + 6x 2 + 17 D) -32x -1 + 6x 2
18) Find dy
dx for y = (3x - 2)(5x3 - x 2 + 1).
A) 15x 3 + 13x 2 - 39x + 3 B) 60x 3 - 39x 2 + 4x + 3
C) 60x 3 - 13x 2 + 39x + 3 D) 45x 3 + 39x 2 - 13x + 3
19) The curve y = ax2 + bx + c passes through the point (2, 28) and is tangent to the line y = 2x at the
point (0,0). Find a, b, and c.
A) a = 2, b = 0, c = 6 B) a = 7, b = 0, c = 0
C) a = 0, b = 6, c = 2 D) a = 6, b = 2, c = 0
20) Find the second derivative y'' of y = 3x 3 - 7x 2 + 3ex.
A) 14x - 18 + 3ex B) 12x - 14 + 3ex C) 18x - 14 + 3ex D) 14x - 12 + 3ex
6
17)
18)
19)
20)

21) Find the derivative of y = x3
x - 1 .
A) y' = -2x3 + 3x 2
(x - 1) 2
C) y' = -2x3 - 3x 2
(x - 1) 2
22) Find the derivative of y = 6x2e-x.
B) y' = 2x3 + 3x 2
(x - 1) 2
D) y' = 2x3 - 3x 2
(x - 1) 2
A) 6xex(2 - x) B) 6xe-x(2 - x) C) 6xe-x(x + 2) D) 12xe-x(1 - x)
23) The area A = πr2 of a circular oil spill changes with the radius. At what rate does the area change
with respect to the radius when r = 7 ft?
A) 14 ft2/ft B) 7π ft2/ft C) 49π ft2/ft D) 14π ft2/ft
24) Find y'' if y = -2 cos x.
A) y'' = -2 sin x B) y'' = 2 sin x C) y'' = -2 cos x D) y'' = 2 cos x
25) The position of a body moving on a coordinate line is given by s = t2 - 6t + 9, with s in meters and
t in seconds. When, if ever, during the interval 0 ≤ t ≤ 6 does the body change direction?
A) t = 12 sec B) no change in direction
C) t = 3 sec D) t = 6 sec
Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values
of x. Find the derivative with respect to x of the given combination at the given value of x.
x f(x) g(x) f'(x) g'(x)
26) 3 1 9 8 3
26)
4 3 3 5 -4
f(g(x)), x = 4
A) -32 B) 8 C) 24 D) -20
27) Find the derivative of the function y = (4x3 - 2x2 + 8x - 6)ex3
A) (12x5 - 6x4 + 24x3 + 12x2 - 4x + 8)ex3 B) (12x2 - 4x + 8)ex3
C) (12x5 - 6x4 + 24x3 -18x2)ex3 D) (12x5 - 6x4 + 24x3 - 6x2 - 4x + 8)ex3
28) Find dy
dt if y = t7(t3 + 4) 6
A) 126t19(t3 + 4) 5 B) 7t6(t3 + 4) 5 (18t3 + 4)
C) t6(t3 + 4) 5 (25t3 + 28) D) t7(t3 + 4) 5 (25t2 + 28)
29) Find the slope of the tangent line to the curve y4 + x3 = y2 + 12x at the point (0, 1).
A) y = - 2x B) y = 6x + 1 C) y = - 3x - 1 D) y = 3x + 1
7
21)
22)
23)
24)
25)
27)
28)
29)

30) Find dy/dx by implicit differentiation where x4/3 + y4/3 = 1.
A) - x
y
1/3
B) y
x
1/3
C) - y
x
1/3
D) x
y
31) Assume that the profit generated by a product is given by P(x) = 3 x, where x is the number of
units sold. If the profit keeps changing at a rate of $600 per month, then how fast are the sales
changing when the number of units sold is 1600?
A) $8/month B) $160,000/month
C) $8000/month D) $16,000/month
32) Find the derivative dy/dx if y = ln(ln 8x).
A) 1
x
B)
1
ln 8x
33) Use logarithmic differentiation to find dy
dx
A) (x + 10)x ln(x + 10) + x
x + 10
if y = (x + 10)x.
C) 1
8x
B) ln(x + 10) + x
x + 10
C) x + (10)x-1 D) x ln(x + 10)
34) Find dy
dx if y = -sin-1 (3x2 - 4).
A)
C)
3
1 + (3x2 - 4) 2
-6x
1 - (3x2 - 4) 2
B)
D)
6x
1 - (3x2 - 4) 2
6x
1 + (3x2 - 4) 2
35) Use logarithmic differentiation to find dy
dx if y = (x4 + 1) 5 (x - 1)4x5.
A) (x4 + 1) 5 (x - 1)4x5 25
x
B) 20x3
+
x4 + 1
4 + 5
x - 1 x
C) (x4 + 1) 5 (x - 1)4x5 20x3
x4 + 1
+ 4
x - 1
+ 4 + 5
x - 1 x
D) (x4 + 1) 5 (x - 1)4x5(5ln(x4 + 1) + 4ln(x - 1) + 5ln x)
8
D)
1/3
1
x ln 8x
30)
31)
32)
33)
34)
35)

Answer Key
Testname: 121 MIDTERM A
1) B
2) C
3) D
4) D
5) C
6) B
7) C
8) A
9) B
10) A
11) C
12) B
13) C
14) A
15) A
16) Continuous but not differentiable
17) B
18) B
19) D
20) C
21) D
22) B
23) D
24) D
25) C
26) A
27) D
28) C
29) B
30) A
31) D
32) D
33) A
34) C
35) C
9