Math 115 – Exam 3 Version 1
April 28, 2014
Name:
9 questions for 100 points possible
1. No hats or dark sunglasses. All hats are to be removed.
2. All book bags are to be closed and placed in a way that makes them inaccessible. Do not reach into
your bag for anything during the exam. If you need extra pencils, pull them out now.
3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will
be construed as cheating and you will be asked to leave. This includes using it as a time-piece.
4. No music systems – IPODs, MP3 players, etc. – or calculators; same rules as with cell phones.
5. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are
done with the exam. If you have to use the facilities, do so now. You will not be permitted to leave
the room and return during the exam.
6. Every exam is worth a total of 100 points. Check to see that you have all of the pages. Including the
cover sheet, each exam has 9 pages.
7. Be sure to print your proper name clearly above.
8. If you finish early, quietly and respectfully get up and hand in your exam. You need to show your
student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will
be accepted without ID.
9. When time is up, you will be instructed to put down your writing utensil, close the exam and remain
seated. Anyone seen continuing to write after this announcement will have their exam marked and lose
all points on the page they are writing on. We will come and collect the exams from you. Have your
ID ready.
10. Good luck. You have 50 minutes to complete the exam.
1. (a) [3 points] Given that sin θ =
4
5
and θ lies in Quadrant I, find cos θ
(b) [3 points] What is the exact value of θ in radians when θ sweeps out an arc of length of 9π from a
circle of radius r = 6
(c) [3 points] Suppose θ =
−4π
3 .
In which quadrant will this angle lie?
(d) [3 points] Suppose an angle θ intersects the unit circle at the point
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√ 3
1
2, 2 .
What is tan θ
2. For the following questions let f (x) = (x − 3)2
(a) [3 points] Find a domain on which f (x) is 1 − 1.
(b) [3 points] Find the inverse function, f −1 (x), for f (x) over the domain.
3. For the following questions let g(x) be a 1 − 1 function so that g(−1) = 1, g(0) = 2, g(1) = 3, g(2) = 0,
and g(3) = 5
(a) [3 points] g −1 (2) =
(b) [3 points] g −1 (g(1)) =
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4. Determine whether each statement is TRUE or FALSE. You do not have to show work.
(a) [2 points] If a graph passes the Horizontal Line Test, then it is the graph of a function.
(b) [2 points] Let f (x) be an invertible function. If the graph of y = f (x) has a horizontal asymptote
at y = 3, then the graph of y = f −1 (x) has a vertical asymptote at x = 3.
(c) [2 points] The graph of a rational function can never cross its horizontal asymptote.
(d) [2 points] Given f (x) is a rational function with lim f (x) = 0, then f (2) = 0.
x→2
(e) [2 points] The point (1, 0) is on the graph of y = logb (x) for all b > 0.
(f) [2 points] The graph of y = bx has no x-intercepts for all b > 0.
(g) [2 points] −1 ≤ tan θ ≤ 1 for all angles θ.
(h) [2 points] cos2 θ + sin2 θ = 1 for all θ.
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5. Suppose the f (x) is a rational function with domain (−∞, 0)
following limit statements are true for f (x)
S
S
(0, 3) (3, ∞). Also suppose that the
lim f (x) = lim f (x) = ∞
x→∞
x→−∞
4
3
lim f (x) = −∞
lim f (x) = −
x→0
lim f (x) = ∞
x→3−
x→3+
(a) [2 points] What are the horizontal asymptotes of the graph of y = f (x)? Express your answer as
an equation. If there are no horizontal asymptotes, then write NONE.
(b) [2 points] What are the vertical asymptotes of the graph of y = f (x)? Express your answer as an
equation. If there are no vertical asymptotes, then write NONE.
(c) [2 points] What are the holes(s) (removable discontinuities) of the graph of y = f (x). Express your
answer(s) as ordererd pair(s). If there are no holes, then write NONE.
(d) [2 points] TRUE or FALSE f (x) is continuous on the interval (3, ∞).
(e) [2 points] Does the graph of y = f (x) have a y-intercept? Why or why not?
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6. [14 points] Let f (x) = (x−2)(x+2)
(x−2)(x+3) . Graph the equation y = f (x). Label all intercepts, asympotes, and
removable discontinuities if any exist.
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7. [12 points] Match each graph below to one of these functions. Assume all graphs are drawn to the same
scale.
D. y = log(−x)
A. y = ex + 6
B. y = −ex + 6
C.y = log x1
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8. For each of the following functions, find the domain and the end behavior.
(a) [4 points] f (x) = x(x−2)
(x+4)
S
S
A. D : (−∞, 0) (0, 2) (2, ∞) and lim f (x) = +∞, lim f (x) = −∞, lim f (x) = 0,
x→∞
x→−∞
x→0
lim f (x) = 0
x→2
S
B. D : (−∞, −4) (−4, ∞) and lim f (x) = +∞, lim f (x) = −∞, lim f (x) = −∞,
x→∞
x→−∞
x→−4−
lim f (x) = +∞
x→−4+
S
C. D : (−∞, −4) (−4, ∞) and lim f (x) = +∞, lim f (x) = −∞, lim − f (x) = +∞,
x→∞
x→−∞
x→−4
lim f (x) = −∞
x→−4+
S
S
D. D : (−∞, 0) (0, 2) (2, ∞) and lim f (x) = +∞, lim f (x) = +∞, lim f (x) = +∞,
x→∞
x→−∞
lim f (x) = +∞
x→2
(b) [4 points] g(x) = e(4−x)
A. D : (−∞, ∞) and lim g(x) = +∞, lim g(x) = 0
x→−∞
x→+∞
B. D : (−∞, 4) and lim g(x) = 0, lim g(x) = +∞
x→−∞
x→4
C. D : (−∞, 4] and lim g(x) = −∞, lim g(x) = +∞
x→−∞
x→4
D. D : (−∞, ∞) and lim g(x) = −∞, lim g(x) = +∞
x→−∞
x→+∞
(c) [4 points] h(x) = ln(x − 2)
A. D : (−∞, ∞) and lim h(x) = +∞, lim h(x) = e
x→−∞
x→+∞
B. D : (−∞, 2) and lim h(x) = −∞, lim− h(x) = +∞
x→−∞
x→2
C. D : (2, ∞) and lim h(x) = −∞, lim h(x) = +∞
x→2+
x→∞
D. D : (−∞, ∞) and lim h(x) = 0, lim h(x) = +∞
x→−∞
x→+∞
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x→0
9. For each question, simplify the given expression or solve for x
(a) [4 points] 2(x+3) = 42x
A.
B.
C.
D.
x = −3
x=1
x=3
No Solution
(b) [4 points] log7 (x − 2) + log7 (x + 2) = log7 (3x)
A.
B.
C.
D.
1
2
3
4
(c) [4 points]
A.
B.
C.
D.
4 log5 (3)+4 log5 (2)
log5 (6)
1
2
3
4
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Question
1
2
3
4
5
6
7
8
9
Total
Grade
Page 10
Possible
12
6
6
16
10
14
12
12
12
100