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Math 131 WIR, copyright Angie Allen
Math 131 Week-in-Review #4 (Exam 1 Review: Sections 1.1-1.3, 1.5, 1.6, and 2.1-2.5)
Note: This collection of questions is intended to be a brief overview of the exam material (with emphasis on sections 2.4
and 2.5). When studying, you should also rework your notes, the previous week-in-reviews for this material, as well as your
suggested and online homework.
1. For the function f whose graph is shown below, find each of the following.
a) State the domain of f .
y
b) Where is f continuous? Remember to indicate left/right hand continuity where applicable.
f(x)
8
7
6
5
4
3
2
c) Show (using the definition of continuity) that f is continuous at
x = −5.
1
8
7 6 5
4 3
2 1
1
−1
−2
−3
−4
−5
−6
−7
d) For what values of c does lim f (x) not exist? For each value of c, use
x→c
limits to describe the way in which the limit does not exist.
e) Which condition in the definition of continuity fails
first at x = 6?
f) Show (using the definition of continuity) that f is continuous from the
left at x = 2.
g) Determine whether f (x) is continuous on the following intervals:
(−∞, 1]
(1, 2]
[2, 6)
h) Which condition in the definition of continuity fails
first at x = 1?
−8
2 3 4
5
6
7 8
x
2
Math 131 WIR, copyright Angie Allen
2. Determine, algebraically, where the function f (x) =
5 ln(x − 3)
√
is continuous.
x+7
3. Classify each of the following as a power function, root function, rational function, polynomial function (state its degree), algebraic function, trigonometric function, exponential function, or logarithmic function.
x2 + 3x
a) f (x) =
x+π
4. Determine, algebraically, if the function f (x) =
√
x3 + x9
b) h(x) =
1 − x2
|x|
is even, odd, or neither.
|x| + 5
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Math 131 WIR, copyright Angie Allen
5. Consider the function f (x) below and answer the following questions:
a) Find the domain of f (x).

10x − 4x2








x+1
2
f (x) =
x −x−2





2


 x −8
23−x
x < −2
−2 ≤ x ≤ 3
x>3
b) Determine where f (x) is continuous. Remember to indicate left/right hand continuity where applicable.
c) Find lim f (x), if it exists.
x→−3
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Math 131 WIR, copyright Angie Allen
6. Find the horizontal asymptotes of the function f (x) =
7. Given f (x) =
p
2 − e3x ,
a) Find the domain of f (x).
b) Find f −1 , if it exists.
3x2 + 4x − 6
, if they exist.
5(x + 2)(x − 2)
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Math 131 WIR, copyright Angie Allen
8. Find the domain of the function f below.
 √

e x







x+2

2
f (x) =
x −x−6



√

5 2


x − πx



log7 (x + 9)
9. If f (x) =
x≥2
−5 ≤ x < 2
x < −5
√
3
, g(x) = 6e2x , and h(x) = x + 5, find f ◦ g ◦ h and state its domain.
x+2
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Math 131 WIR, copyright Angie Allen
10. Solve the following for x:
log7 (log3 (2x + 1) + 45) = log5 25
3x3 − 15x2 − 18x
algebraically, if they exist. If there are
4x4 − 4
vertical asymptotes, use limits to describe the behavior near each vertical asymptote.
11. Find the horizontal and vertical asymptotes of the curve y =
7
Math 131 WIR, copyright Angie Allen
x2 − 9
, if it exists. If it does not exist, use limits to describe the way in which it does not exist.
x→3 |x − 3|
12. Evaluate lim
√
4
2x3 + 3 x3
13. Find lim √
.
7 9
x→−∞
x −x
14. The population of a small city in Texas from 1994 to 2002 is shown in the table below. (Midyear estimates are given.)
Year
P
1994
29, 036
1996
29, 672
1998
32, 300
2000
36, 205
2002
38, 260
Find an appropriate model to fit the data (for interpolation purposes), AND briefly explain your choice.
Math 131 WIR, copyright Angie Allen
8
x2 + 4x + 3
, if it exists. If it does not exist, use limits to descibe the way in which it does not exist.
x→−1 (x + 1)2
15. Find lim
16. If an object is launched into the air with a velocity of 40 feet per second, its height t seconds later is given by
s(t) = −16t 2 + 40t feet.
a) Find the average velocity on the interval [1.5, 2.5].
b) Estimate the instantaneous velocity when t = 1.5 by calculating slopes of nearby secant lines from the left.
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Math 131 WIR, copyright Angie Allen
x+3
.
17. Find lim √
x→∞ 9x2 + 1
18. Find the value of A such that the function f (x) below is continuous for all real numbers x.
f (x) =
 x
 3e

x < −2
Ax2 + x + 1 x ≥ −2
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Math 131 WIR, copyright Angie Allen
e6x − e−3x
.
x→∞ e−5x − e4x
19. Find lim
20. Estimate lim
x→0
exist.
sin x − x
numerically, if it exists. If it does not exist, use limits to describe the way in which it does not
x3
21. Solve the following for x:
a) 27x−5 34x = log8 8
b) log7 (3 − x) = log7 54 − log7 (−x)
Math 131 WIR, copyright Angie Allen
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22. Find lim ax3 − bx2 + c, where a < 0, b > 0, and c > 0.
x→−∞
√
23. Starting with the function f (x) = x, write the function g(x) that results from shifting f (x) to the left 3 units, compressing horizontally by a factor of 2, stretching vertically by a factor of 4, and reflecting about the x-axis.
24. A bacteria population starts with 10 cells and quadruples every 6 hours. Write a function B(h) that gives the number of
bacteria after h hours. Write your answer in exponential form.