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Math ����: Review Problems for Exam �
Spring ����
1. [L1] Shown below is the graph of a function f (x).
Determine the following.
a.
lim f (x)
f. The equations of all vertical asymptotes.
g. The equations of all horizontal asymptotes.
h. The value(s) of x for which f (x) has a removable discontinuity.
i. The value(s) of x for which f (x) has a jump
discontinuity.
x ! 4+
b. lim f (x)
x! 4
c. lim f (x)
x !2
d. lim f (x)
x !4
e. lim f (x)
x !•
2. [L2] Evaluate the limits, using algebra and/or limit properties as needed.
3x2 5
+ 3x 3
p
5x 4 1
b. lim
x 1
x !1
e. lim (2 sin(x)
c. lim
g. lim
a. lim
x !2 x 2
x2
x2
x
x
6
2
(2 + h)2
3h
h !0
4
x! 2
d. lim
x !p
f.
3 cos(5x))
lim tan(x)
x ! 3p
2
x !3+
h. lim
x !6
6
3
x
x2
3x
5x
6
3. [L2] Evaluate the following limits, using algebra and/or limit properties as needed.
p
Ä
ä
x3 2x + 3
9x6 x
2
a. lim
3x + 5x + 7
b. lim
c. lim
2
x !•
x !• x3 + 1
x! •
5 2x
4. [L3] Consider the following piece-wise defined function.
8
>
if x  2,
<2x + 1
2
g(x) = x
if 2 < x  3
>
:
2x + c if 3 < x.
Determine the following.
a. lim g(x)
x !2
b. Is g(x) continuous at x = 2? Why or why not?
c. Find the value of c that makes g(x) continuous at x = 3.
Math ����: Review Problems for Exam �
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5. [DM1] Use the limit definition of the derivative to find f 0 (a) for each of the following functions at the given
value of a.
a. f (x) = 7x + 4, a =
b. f (x) = 3x2 + 9x
c. f (x) =
4x2 + 2x
12
5, a = 2
3, a = 1
6. [DM1] Complete the following.
a. The graph of a function g(x) is shown to the right.
Determine the values of g0 (0.5), g0 (0), g0 (1), g0 (3), g(0),
and g( 3). If a value does not exist, briefly explain
why not.
b. The graph of a function f (x) is shown to the right.
Determine the following.
i. An estimate of the value of f 0 ( 3).
ii. An estimate of the value of f 0 (2).
iii. A value of x at which f 0 (x) is positive.
iv. A value of x at which f 0 (x) is negative.
7. [DM2] The position in feet of a race car along a straight track after t seconds is modeled by the function
s(t) = 8t2 12 t3 .
a. Below are several time intervals. Where indicated, compute the average velocity of the vehicle over the time
intervals.
i. Average velocity over [3.8, 4] = 39.58
ii. Average velocity over [3.9, 4] =
iii. Average velocity over [4, 4.001] = 40.0019995
iv. Average velocity over [4, 4.01] =
b. Use the answers from above to draw a conclusion about the instantaneous velocity of the vehicle at t = 4
seconds. Include the units.
Math ����: Review Problems for Exam �
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8. [DM2] The graph below shows the graph of the position s(t) of a particle moving in a straight line.
a. Find the average velocity over the interval [0.4, 0.8].
b. Find the average velocity over the interval [0.6, 1.0].
c. Find the instantaneous velocity at t = 0.8.
d. Find the instantaneous velocity at = t = 1.3.
e. Find two other points t where the instantaneous velocity is the same as at t = 1.3.
9. [DM2] Consider the graph of the function f (x) below.
Using this graph, for each of the following pairs of numbers decide which quantity is larger. Make sure that
you can explain your answer. Note: f (3) refers to the y-coordinate associated with x = 3.
a. f (3)
f (4)
b. f (3)
f (2)
f (2)
f (2)
f (2)
2
f (1)
1
f (3)
3
f (1)
1
c.
d. f 0 (1)
f 0 (4)