1426 Exam 1 Fall '19 Version A
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MATH 1426
Exam #1 – Version A
Fall 2019
INSTRUCTIONS FOR PART I (80 points):
Write your answers to problems 1-20 on the Scantron SC882-E, marking only one answer per question. You may write on this exam, however, your score in Part I will be determined solely by your answers on the Scantron.
1. Consider the function f
(
t
)
=−
t
2
+
3
t
. Find the average rate of change of f
over the interval [
4
,
4
+
h
]
.
[A] h
+
5
[B] h
−
5
[C] −
h
+
5
[D] −
h
−
5
[E] 5
h
2. The graph of the function y
=
f
(
x
)
is given below. Which of the following statements (I,II,III,IV,V) are true?
I. lim
x→
1
+
¿
f
(
x
)
=
2
¿
¿
II. lim
x →
1
f
(
x
)
=
3
III. lim
x→
3
−
¿
f
(
x
)
=
4
¿
¿
IV. lim
x →
3
f
(
x
)
=
2
V. lim
x →
1
f
(
x
)
does not exist
[A] I only [B] I and III only [C] III and IV only [D] I and V only [E] II only
3. Consider the function f
(
x
)
=
{
x
2
+
1
if x ≤
1
5
if x
>
1
. Which of the following statements (I,II,III) are true?
I. lim
x→
1
+
¿
f
(
x
)
=
2
¿
¿
II. lim
x→
1
−
¿
f
(
x
)
=
2
¿
¿
III. lim
x →
1
f
(
x
)
does not exist
1
MATH 1426
Exam #1 – Version A
Fall 2019
[A] I only [B] II only [C] III only [D] I and II only [E] II and III only
4. Let lim
x→
−
8
f
(
x
)
=
5
and lim
x→
−
8
g
(
x
)
=−
1
. Find lim
x→
−
8
−
3
f
(
x
)
−
2
g
(
x
)
10
+
g
(
x
)
.
[A] −
13
9
[B] −
17
9
[C] −
8
[D] −
7
2
[E] the limit does not exist
5. Given x
3
≤f
(
x
)
≤
|
x
|
for x
on the interval [
−
1,1
]
, find lim
x →
0
f
(
x
)
, if it exists.
[A] 1 [B] 0 [C] −
1
[D] the limit does not exist [E] 1
2
6. Determine the following limit. lim
x→
2
−
¿
x
2
−
4
x
+
3
(
x
−
2
)
2
¿
¿
[A] 0 [B] −
1
[C] −
∞
[D] ∞
[E] 1
7. Determine all
the vertical asymptotes of the function f
.
f
(
x
)
=
x
3
+
2
x
2
+
x
+
2
(
x
+
2
) (
x
+
3
)
x
2
[A] x
=
0
[B] x
=
0
, x
=−
2
[C] x
=
0
, x
=−
3
[D] x
=−
2
, x
=−
3
[E] x
=
0
, x
=−
2
, x
=−
3
2
MATH 1426
Exam #1 – Version A
Fall 2019
8. Select the ONE
statement (A,B,C,D,E) that correctly describes the slant and
horizontal asymptote(s) of the function f .
f
(
x
)
=
2
x
3
+
x
2
+
11
x
−
4
x
2
−
4
[A] f
(
x
)
has no slant asymptote and no horizontal asymptote.
[B] f
(
x
)
has a slant asymptote at y
=
2
x
+
1
and no horizontal asymptote.
[C] f
(
x
)
has a slant asymptote at y
=
19
x
and no horizontal asymptote
[D] f
(
x
)
has a horizontal asymptote at y
=
2
and no slant asymptote.
[E] f
(
x
)
has two horizontal asymptotes at y
=
2
∧
y
=−
2
and no slant asymptote.
9. Determine the limit.
lim
x→
−
∞
18
x
4
−
4
x
2
+
9
√
36
x
8
+
x
6
−
3
x
4
+
5
[A] 0 [B] 18 [C] 1
2
[D] −
1
2
[E] 3
10. On which of the following intervals is the function f
continuous?
f
(
x
)
=
e
x
(
x
2
+
7
x
+
10
)
x
2
−
25
[A] (
−
∞,
−
5
)
,
(
−
5,5
)
,
(
5
,∞
)
[B] (
−
∞,∞
)
[C] (
−
∞,
5
)
,
(
5
,∞
)
[D] (
−
∞,
0
)
,
(
0
,∞
)
[E] (
−
∞,
0
)
,
(
0,5
)
,
(
5
,∞
)
3
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MATH 1426
Exam #1 – Version A
Fall 2019
11. Use the graph of y
=
f
(
x
)
in the figure to determine which of the following statements (I,II,III,IV) are true?
I. lim
x →
1
f
(
x
)
=
∞
II. f
is continuous on the interval (
1,2
)
.
III. f
is continuous at x
=
1.
IV. f
is continuous on the interval [
2
,
5
)
.
[A] I and II only [B] I, III and IV only [C] III and IV only [D] II and IV only [E] I and IV only
12. Find the value of the constant c
that makes the function f
continuous on the interval (
−
∞,∞
)
.
f
(
x
)
=
{
2
x
2
+
x if x≤
1
2
cx
+
5
if x
>
1
[A] 2
3
[B] 1 [C] 1
2
[D] −
1
2
[E] −
1
4
MATH 1426
Exam #1 – Version A
Fall 2019
13. Suppose a function f
is defined on the interval [
a,b
]
. The difference quotient f
(
b
)
−
f
(
a
)
b
−
a
represents which ONE
of the following statements?
[A] The slope of the tangent line to the graph f
at the point (
a ,f
(
a
)
)
.
[B] The derivative of the function f
at point a
.
[C] The average rate of change of f
on the interval [
a,b
]
.
[D] The instantaneous rate of change of f
on the interval [
a,b
]
.
[E] The slope of the tangent line at the point (
a ,f
(
a
)
)
.
14. The limit below represents the slope of a curve y
=
f
(
x
)
at the point (
a ,f
(
a
)
)
. Determine the function f
and the constant a
.
lim
h→
0
2
√
9
+
h
−
6
h
[A] f
(
x
)
=
2
√
x
+
9
,a
=
9
[B] f
(
x
)
=
2
√
x,a
=
0
[C] f
(
x
)
=
2
√
x,a
=
9
[D] f
(
x
)
=
2
√
x,a
=
6
[E] f
(
x
)
=
2
√
x
+
3
,a
=
3
15. Determine an equation of the tangent line to the function f
(
x
)
=
10
+
8
x
2
at the point (
0,10
)
.
[A] y
=
10
[B] y
=
x
[C] y
=−
2
x
+
10
[D] y
=
0
[E] y
=
x
+
10
16. Consider the graph of f
given below. Which ONE
of the following choices correctly identifies the values of f
'
(
x
)
in order from greatest to least?
5
MATH 1426
Exam #1 – Version A
Fall 2019
[A] f
'
(
B
)
≥f
'
(
D
)
≥f
'
(
C
)
≥f
'
(
A
)
≥f
'
(
E
)
[B] f
'
(
A
)
≥f
'
(
C
)
≥f
'
(
D
)
≥f
'
(
B
)
≥f
'
(
E
)
[C] f
'
(
C
)
≥f
'
(
D
)
≥f
'
(
B
)
≥f
'
(
E
)
≥f
'
(
A
)
[D] f
'
(
A
)
≥f
'
(
B
)
≥f
'
(
E
)
≥f
'
(
D
)
≥f
'
(
C
)
[E] f
'
(
A
)
≥f
'
(
B
)
≥f
'
(
C
)
≥f
'
(
E
)
≥ f
'
(
D
)
17. Match the graph of f
with the graph of its derivative f
'
.
[A] [B] 6
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MATH 1426
Exam #1 – Version A
Fall 2019
[C] [D] [E] 18. Consider the graph of the function g
below. Which of the following statements (I,II,III,IV,V) are true? Note
: g
'
is NOT pictured below.
I. g
'
(
2
)
=
0
II. g
'
(
x
)
≥
0
ontheinterval
2
<
x
<
4.
III. g
'
(
x
)
is constant on the interval −
4
<
x
←
2
.
IV. g
'
(
x
)
is increasing in value on the interval 0
<
x
<
2
.
V. g
'
(
x
)
≥
0
ontheinterval
0
<
x
<
2.
[A] I, IV and V only [B] II, III, IV and V only [C] II, IV and V only [D] III and V only [E] I, III and V only
7
MATH 1426
Exam #1 – Version A
Fall 2019
19. If g
'
(
0
)
=
7
and f
(
x
)
=
2
g
(
x
)
−
4
e
x
−
x
2
+
3
x
, what is f
'
(
0
)
?
[A] 13 [B] 11 [C] 14 [D] 17
−
e
[E] 17
−
4
e
20. What is the derivative of h
(
t
)
=
t
34
−
√
t
+
3
e
t
−
3
?
[A] t
33
−
1
2
t
+
3
e
t
[B] 34
t
33
−
1
2
√
t
+
3
e
t
[C] 34
t
33
−
1
2
√
t
+
3
e
t
[D] 34
t
33
−
1
2
t
+
e
t
[E] t
33
−
1
2
√
t
+
3
e
t
INSTRUCTIONS FOR PART II (20 points):
For these questions, you must justify your solutions by showing all your steps
. Write legibly and carefully. Partial credit will be awarded for those parts of your solution that are correct. Only the work and solution written on the exam itself will be graded. Proper mathematical notation is required.
Please put a around your final answer for each question.
21a. Use an algebraic method to find the limit. [5 points]
lim
x →
0
√
2
+
x
−
√
2
x
8
BOX
MATH 1426
Exam #1 – Version A
Fall 2019
21b. Determine the x
-value(s) of any point(s) on the graph of f
(
x
)
=
1
3
x
3
+
2
x
2
+
x
at which the tangent line has slope −
2
. [5 points]
22. Consider the function f
(
x
)
=
4
x
2
−
5
x
+
1
.
(i) Use the limit definition of the derivative
to find f
'
(
x
)
. [7 points]
9
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MATH 1426
Exam #1 – Version A
Fall 2019
(ii) Evaluate f
'
(
0
)
. [1 point] (iii) Write an equation for the tangent line when x
=
0
. [2 points]
END OF EXAM – Version A
10