Name:
Student ID:
Section:
Instructor:
Math 112 (Calculus I)
Midterm Exam 1 Winter 2014
RED
Instructions:
• For questions which require a written answer, show all your work. Full credit will be given only
if the necessary work is shown justifying your answer.
• Simplify your answers.
• Calculators are not allowed.
• Should you have need for more space than is allocated to answer a question, use the back of the
page the problem is on and indicate this fact.
• Please do not talk about the test with other students until after the last day to take the exam.
For Instructor use only.
#
MC
Possible Earned
#
Possible Earned
39
18
8
14
8
19
6
15
8
20
6
16
6
21
6
17
6
22
7
Sub
67
Sub
Total
33
100
Winter 2014
Math 112 – Midterm Exam 1
Page 2 of 9
Part I: Multiple Choice Mark the correct answer on the bubble sheet provided.
1. Simplify the following: y =
sin(2x)
.
sin(x)
a) y = 2
b) y = sin(x)
c)
y = sin2 (x)
d) y = cos(x)
e)
f)
None of the Above
2. Let f (x) =
a)
y = 2 cos(x)
3x − c if x < 0
. What value of c makes f (x) continuous?
ex
if x ≥ 0
1
d) −e
b)
−1
c)
e
e)
3
f)
−3
3. Which functions f (x), g(x), and h(x) will give a composition of
h ◦ g ◦ f = sin(1 + |x|)?
a) f (x) = sin x, g(x) = |x|, h(x) = 1 + x
b) f (x) = sin |x|, g(x) = 1 + x, h(x) = x
c)
f (x) = sin(1 + x), g(x) = |x|, h(x) = sin x
d) f (x) = |x|, g(x) = 1 + x, h(x) = sin x
e)
f (x) = 1 + x, g(x) = |x|, h(x) = sin |x|
f) f (x) = 1 + x, g(x) = sin x, h(x) = |x|
g) f (x) = 1 + x, g(x) = |x|, h(x) = sin x
4. Find all the vertical asymptotes of f (x) =
a) x = 3, 1
√
d) x = 2
g)
h)
None of these.
x2 − 4x + 3
.
x2 − 1
x = −1
b)
x=1
c)
e)
x = −3, −1
f) x = −1, 1
None of the above.
3x2 − 17x + 4
.
x→∞ 2x2 + 6x − 13
5. Calculate the following limit: lim
a) −
d)
1
4
13
b)
2
e) −
17
6
c)
∞
f)
3
2
Continue to Next Page
2
Winter 2014
Math 112 – Midterm Exam 1
Page 3 of 9
6. Which of the following functions is differentiable at x = 0?
a) f (x) =
2
x
c) h(x) =
e) v(x) =
b) g(x) = |x|
2
x
d) u(x) =
0
x + 3 if x < 0
x + 5 if x ≥ 0
x+3
2x + 3
if x < 0
if x ≥ 0
f) w(x) =
if x ≤ 0
if x > 0
|x|
x
7. If ln(x2 − 1) − ln(x − 1) = 2, then x =
a)
2
b)
e2
c) e2 − 1
d)
2e − 1
e)
2e
f)
None of the Above
x2 + 3x + 2
.
8. Calculate the following limit: lim 2
x→(−2) x − 3x + 2
a)
1
b)
−1
c)
2
d)
3
e)
0
f)
Undefined
9. The graph of f (x) is first shifted downward by 3 units.
tally by a factor of 4. Which of the following describes
x
a) g(x) = f
−3
4
x
c) g(x) = f
+3
4
e) g(x) = f (4x) − 3
g) g(x) = f (4x + 3)
After this the graph is stretched horizonthe function g(x)?
x
b) g(x) = f
+3
4
x
d) g(x) = f
−3
4
f) g(x) = f (4x) + 3
h)
g(x) = f (4x − 3)
10. Which of the following is an odd function?
a) u(x) =
sin(x) + cos(x)
x2
b) g(x) =
d) v(x) =
sin2 (x) − cos2 (x)
x3
e) h(x) =
x+1
+x+1
c) f (x) =
|x|
x4
x2 + 1
x4 + x2 + 1
f) w(x) =
sin(x)
x
x3
Continue to Next Page
3
Winter 2014
Math 112 – Midterm Exam 1
11. Calculate the following limit: lim−
x→1
a)
0
d) −2
−1
12. Calculate sin
5π
6
π
d) −
4
a)
Page 4 of 9
(x + 1)|x − 1|
.
x2 − 1
b)
−1
c)
1
e)
2
f)
Does Not Exist
1
− .
2
7π
6
π
e) −
3
b)
π
6
5π
f) −
6
c) −
13. The equation of the tangent line to y = f (x) at the point (3, 5) is y = 2x − 1. Which limit must
be correct?
f (x) − 1
f (x) + 1
=3
b) lim
=3
a) lim
x→2
x→2
x−2
x−2
f (x) − 1
f (x) + 1
c) lim
=2
d) lim
=2
x→3
x→3
x−3
x−3
f (x) − 5
f (x) − 3
=2
f) lim
=2
e) lim
x→3
x→3
x−3
x−3
Continue to Next Page
4
Winter 2014
Math 112 – Midterm Exam 1
Page 5 of 9
Part II: Written Response Neatly write the solution to each problem. Complete explanations
are required for full credit.
14. (8 points) Use the definition of the derivative to find the derivative of f (x) =
1
.
x
15. (8 points) Find the equation of the tangent line to g(x) = x2 when x = 2.
5
Winter 2014
Math 112 – Midterm Exam 1
Page 6 of 9
16. (6 points) Simplify. Your answer should contain no trigonometric functions.
−1 1
sin sec
x
17. (6 points) Calculate the following limit.
1
lim
x→4
− 14
x−4
x
6
Winter 2014
Math 112 – Midterm Exam 1
Page 7 of 9
18. (8 points) Calculate the following limit. An explanation is required for full credit.
!
5
lim z 4 sin
z→0
z
.
19. (6 points) Let f (x) = 2x + 3. It is true that lim f (x) = 9.
x→3
Find the largest value of δ such that if |x − 3| < δ, then |f (x) − 9| < 1.
7
Winter 2014
Math 112 – Midterm Exam 1
Page 8 of 9
π
20. (6 points) Show that the equation ex sin(x) = cos(x) has a solution in the interval 0,
.
2
21. (6 points) Suppose a car drives on a straight road starting at noon. The distance, in miles, that
80t2
the car has traveled after t hours is modeled by the equation d(t) =
.
2t + 1
a). What is the physical interpretation of
d(4) − d(0)
?
4
b). What is the physical interpretation of lim
t→2
d(t) − d(2)
?
t−2
8
Winter 2014
Math 112 – Midterm Exam 1
Page 9 of 9
22. (7 points) Below is the graph of f (x).
Find the following limits. Assume all finite answers are integers.
a) lim f (x) =
b) lim f (x) =
c) lim f (x) =
d) lim− f (x) =
x→−∞
x→−3
e) lim+ f (x) =
x→0
x→∞
x→0
f) lim− f (x) =
x→3
g) lim+ f (x) =
x→3
END OF EXAM
9