4
10 June 2013
Review for Exam 1
Math 1210-001
The exam will consist of 15 problems and 1 bonus problem, and will last 60
minutes. Answer the questions in the spaces provided. You may use a
scientific or graphing calculator. You may not use any books, notes, cell
phones etc. You must show all of your work to receive credit. Good luck!
Name:
V\iAU
*
O\LL1S tn
J2
Tips:
• If you get stuck on a problem, don’t panic. Move on and come back to it later.
• If you aren’t sure how to show your work, write a few sentences explaining your thought
process.
• For the problems in Section 1, try to find one example where the statement is true and
one where it is not true. If you can find both, then it is “sometimes” true. If there are
absolutely no examples where it is true, then the statement is “never” true. But, just
because you can only think of many examples where the statement is not true doesn’t
mean there aren’t any. (This is where looking hard for both types of examples comes
in handy.)
• Once you have figured out the answer to the “Sometimes/Always/Never” questions,
think about why it is the correct answer. Try to think of ways to change each statement
so that it falls into each of the three categories.
Section 1: Multiple Choice
Instructions: Decide whether each statement is always, sometimes, or never true. Circle
the letter of your answer.
1. Iflimf(x)=Landlimg(x)=M,thenllmf(x)•g()L.M
Always
B. Sometimes
2. If limf(x)
L and lirn g(x)
A. Always
® Sometimes
x—a
x—a
C. Never
=
M, then urn
x—a
o) (rcc
L
g(x)
C. Never
is a rational function (i.e. p(x) and q(x) are both polynomials), and
q(a) = 0, then the line y = a is a horizontal asymptote of the graph of r(x).
Trv
rC%)
Sometimes C. Never
A. Always
y
iyu1f.
3. If r()
=
4. If r(x)
=
lim r(x)
. p(x) and q(x) are both polynomials), and
. (i.e.
.
. a rational
function
is
L, then the line y = L is a horizontal asymptote of the graph of r(x).
p(x)
® Always
5. Suppose
f(x)
B. Sonietimes
C. Never
is a function which is continuous at
Sometimes
A. Always
C. Never
=
a. Then f(x) is differentiable at a.
.\- Cc)
ft 0.
*O.
/ä is differentiable at a.
.
X)
6. If n is an even integer, and a is any real number, f(x)
Sometimes
A. Always
C. Never
7. If r and y are real numbers, /x + y
A. Always
©
Sometimes
Sometimes
C. Never
Sometimes
rJ:
C. Never
9. If x and y are real numbers, ç/x +
B. Sometimes
Tw
nj FkSe
=
=
+
o)
T+IT)
y.
+ 0
-
+ y.
Tr
yT
x
C. Never
10. If the graph of f(x) has a vertical asymptote to x
A. Always
‘>
+
(/ + /)2
A. Always
<
=
8. If x and y are real numbers.
A. Always
=
Never
Page 2
=
a
a, then
f
+
is differentiable at a.
0
J.A,_
s
)
1+1
Section 2: Limits and Contilluity
Instructions: For the following questions, determine whether each of the following limits
exists. If so, compute the limit precisely (do not approximate). If not, determine whether
the fllnction approaches infinity, negative infinity, or neither.
II. Jim (4x
24 + 2x
2
12. Jim
sin(3x)
x—O
=
—
16)
—.
3
—
7TX
sin(x) sec(x) sin(x)
xsec(x)
—
13. lim
14. lim
15.
2
—x
3x
+
2
2x—1
(2x + 1)(4x 1)
Jim
—
1
=0
x+3+4
2
16.lim
3
17. limx
18. lim
x—2
(x
—
—
19. Jim
x—3
2)2
2 24x +9
3 + 22x
8x
(x 3)2
—
—
—
20. If a
= ,
21. If b
=
find lim a.
---,
=
find lim b.
n—*co
0
—
16. State the c-5 definition of the limit
2
22. We know from “Rules for Limits” that lim x
x—4
2
in this scenario. If you want to guarantee that
how close must x be to 4?
x- —‘
—
1
is no more than 1 unit away from 16,
<4i
23. We know from “Rules for Limits” that lim
=
x—*1
1. State the
E-
definition of the limit
in this scenario. If you want to guarantee that \/ is no more than 0.5 units away from
1,howclosemustxbetol?
24. Define
f(x)
\x\< o 9-S
as follows:
I+2
f(x)=
2
I---3—x
Determine all points of discontinuity of
non removable. No
f.
ifx>2
ifx<2
Classify each discontinuity as removable or
c
0
Page 3
= x48X322x2_24x+9
Determine all points of discontinuity of g. Classify each
25. Let g(x)
discontinuity as removable or non removable. .z. = I, z. 3
rer,-oyaLIe
Section 3: The Derivative
Instructions: Use the definition of the derivative to answer the following questions. Write
your final answer on the line provided.
26. If y
for
=
I
1, compute
a
.
27. If f(x)
=
—a
2x + 3i, compute f’(O) and f’(3).
27.
28. Compute D
[12
+ x]
28.
29. Compute
29.
30. Find the equation of the tangeilt line to the curve y
3
U
at the point (1, 1).
=
-23
30.
31. Find the equation of the tangent line to the curve f(x)
=
at the point (2, 8).
2.
A_j2(7L
DTZ.
l
32. Find the equation of the tangent line to the curve y
=
_3x2
+ 9 at the point (1, 6).
t-tc) (QCL)
32.
33. An object is traveling along the y axis, with its distance from the origin given by s
2 + 32t cm. Find its instantaneous velocity after 1 second.
—16t
33.
=
0 c/s
34. Sylvester throws a ball in the air. Its height is given by h(t) = —4(x 2)2 + 16. Find its
instantaneous velocity after 2 seconds. (Hint: Expand and simplify the function before
applying the definition of the derivative.)
—
34
0
(4- I Se.cuycec.i)
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