Math 1210-001
Review for Exam 1
10 June 2013
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:
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. If lim f (x) = L and lim g(x) = M , then lim f (x) · g(x) = L · M
x→a
A. Always
x→a
B. Sometimes
x→a
C. Never
f (x)
M
=
x→a g(x)
L
2. If lim f (x) = L and lim g(x) = M , then lim
x→a
A. Always
x→a
B. Sometimes
C. Never
3. If r(x) = p(x)
is a rational function (i.e. p(x) and q(x) are both polynomials), and
q(x)
q(a) = 0, then the line y = a is a horizontal asymptote of the graph of r(x).
A. Always
B. Sometimes
C. Never
4. If r(x) = p(x)
is a rational function (i.e. p(x) and q(x) are both polynomials), and
q(x)
lim r(x) = L, then the line y = L is a horizontal asymptote of the graph of r(x).
x→∞
A. Always
B. Sometimes
C. Never
5. Suppose f (x) is a function which is continuous at x = a. Then f (x) is differentiable at a.
A. Always
B. Sometimes
C. Never
6. If n is an even integer, and a is any real number, f (x) =
A. Always
B. Sometimes
A. Always
B. Sometimes
A. Always
B. Sometimes
A. Always
B. Sometimes
√
n
a is differentiable at a.
C. Never
√
√
√
7. If x and y are real numbers, x + y = x + y.
C. Never
√
√
8. If x and y are real numbers, ( x + y)2 = x + y.
C. Never
√
9. If x and y are real numbers, n xn + y n = x + y.
C. Never
10. If the graph of f (x) has a vertical asymptote to x = a, then f is differentiable at a.
A. Always
B. Sometimes
C. Never
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Section 2: Limits and Continuity
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 function approaches infinity, negative infinity, or neither.
11. lim 4x24 + 2x2 − 16
x→0
sin(3x)
x→0
πx
12. lim
sin(x) sec(x) − sin(x)
x→0
x sec(x)
13. lim
x4 − 1
x→∞ −x2
14. lim
3x2 + 2x − 1
x→−∞ (2x + 1)(4x − 1)
15. lim
16. lim
x→∞ x2
1
+ 3x + 4
17. lim x3
x→∞
4
x→2 (x − 2)2
18. lim
x4 − 8x3 + 22x2 − 24x + 9
x→3
(x − 3)2
19. lim
20. If an = n1 , find lim an .
n→∞
21. If bn =
n
,
1−n
find lim bn .
n→∞
22. We know from “Rules for Limits” that lim x2 = 16. State the -δ definition of the limit
x→4
in this scenario. If you want to guarantee that x2 is no more than 1 unit away from 16,
how close must x be to 4?
√
23. We know from “Rules for Limits” that lim x = 1. State the -δ definition of the limit
x→1
√
in this scenario. If you want to guarantee that x is no more than 0.5 units away from
1, how close must x be to 1?
24. Define f (x) as follows:
(
x + 2 if x ≥ 2
f (x) = x2
if x < 2
3−x
Determine all points of discontinuity of f . Classify each discontinuity as removable or
non removable.
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4
3
2
−24x+9
25. Let g(x) = x −8xx+22x
Determine all points of discontinuity of g. Classify each
2 −4x+3
discontinuity as removable or non removable.
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 =
2
x−1
for x 6= 1, compute
dy
.
dx
26.
27. If f (x) = 2x3 + 3x, compute f 0 (0) and f 0 (3).
27.
28. Compute Dx [x2 + x4 ]
28.
29. Compute Du
1
x2
29.
30. Find the equation of the tangent line to the curve y =
1
at the point (1, 1).
x2
30.
31. Find the equation of the tangent line to the curve f (x) = x3 at the point (2, 8).
31.
32. Find the equation of the tangent line to the curve y = −3x2 + 9 at the point (1, 6).
32.
33. An object is traveling along the y axis, with its distance from the origin given by s =
−16t2 + 32t cm. Find its instantaneous velocity after 1 second.
33.
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.
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