Midterm 1
(Mock Test)
Math 12010-003
Answer the questions in the spaces provided. Show all of your work.
Name:
ID: U
Question:
1
2
3
4
5
6
7
Total
Points:
15
10
15
15
30
10
5
100
Score:
1. Find the following limits.
x2 + 4x + 4
(a) 5 points lim 2
x→−2 x + 3x + 2
(b) 5 points
5x3 − 8x + 10
x→+∞ 2x3 + 4x2 − 5
lim
tan 2x
x→0 sin 2x − 1
(c) 5 points lim
2. Find the points of discontinuity of the following functions.
6x2 + 1
(a) 5 points
5x3 − 20x
 x3 −x

4x2 −4x


2
(b) 5 points f (x) =

x3 + 1



2x − 1
if x < −1
at x = −1
if − 1 < x ≤ 0
if x > 0
x2 − 4x
at the point
3. (a) 5 points Find the equation of the tangent line to curve y =
2x − x3
x = 2.
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(b) 5 points Find the points of horizontal tangents on the curve y =
√
x2 − x + 4.
(c) 5 points A particle travels as a function of time according to the formula
s = 100 + 10t + 0.01t3 , where s is in meters and t is in seconds. Find the velocity
and acceleration of the particle at t = 2.
dy
for the following functions.
dx
(a) 5 points y = (8x4 + 3)2 (x3 − 4x)3
4. Find
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(b) 5 points y =
(2x − 5)3
x2 + x
√
3
(c) 5 points y = 3x 4x4 + 3
dy
for the following functions.
dx
2 x
3
(a) 5 points y = cos
1−x
5. Find
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(b) 10 points cos(xy 2 ) = y 2 + x
p
(c) 15 points x2 y 2 − 7x y 3 − 1 = sin(xy 2 − 1) + 2x
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6. 10 points Find
3x
d3 y
, where y =
.
3
dx
1−x
7. 5 points Find f ′′ (2), where f (x) = x sin
π
x
.
8. 10 points (Bonus Problem): Let f be a function such that lim f (x) exists. If
x→3
lim [(xf (x))2 + 2x − 1] = 14, then find the possible value(s) of lim f (x).
x→3
x→3
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