Name (print, with surname first):
Student number:
University of British Columbia
MATH 110: DECEMBER EXAM for SECTION 001
Date: December 5, 2012
Time: 8:30 a.m. to 11:00 a.m.
Number of pages: 14 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
For examiners’ use only
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after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
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be received.
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• Speaking or communicating with other candidates;
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• Having visible at the place of writing any books, papers or
memory aid devices;
• Using or operating electronic devices — electronic devices
must be completely powered down if present at the place of writing.
Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take
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Candidates must follow any additional examination rules or directions communicated by the invigilator.
Question Mark Possible marks
1
4
2
12
3
12
4
4
5
4
6
4
7
4
8
4
9
7
10
5
Total
60
1. Sketch the graph of each of the following functions.
(a) [2 marks] f (x) = 2 cos(x − π)
(b) [2 marks] g(x) = ln(1 − x)
2
2. Determine whether each of the following statements is true or false. If it is true, provide
justification. If it is false, provide a counterexample.
Note that this question continues onto the next page.
(a) [2 marks] If lim f (x) does not exist and lim g(x) does not exist, then lim (f (x) + g(x))
x→0
x→0
x→0
does not exist.
1
2−x
= .
2
x→2 4 − x
4
(b) [2 marks] lim
(c) [2 marks] If lim f (x) = 2, then f (1) = 2.
x→1
3
(d) [2 marks] If f (x) is a polynomial, then its graph crosses the x-axis.
(e) [2 marks] If a function is continuous, then it is differentiable.
(f) [2 marks] The curve y =
√
x has no horizontal tangent lines.
4
3. Calculate each of the following derivatives. You may leave your answers unsimplified.
Note that this question continues onto the next page.
(a) [3 marks] f 0 (9), where f (x) = g(x)x1/2 , g(9) = 2, and g 0 (9) = 4.
(b) [3 marks] f 0 (x), where f (x) = a +
c
d
b
+ 2 + 3 , and a, b, c and d are constants.
x x
x
5
(c) [3 marks] f 0 (x), where f (x) =
e3x
.
3x
(d) [3 marks] f 0 (θ), where f (θ) is the area of the triangle with vertices at the origin, P
and Q, shown below. Note that the circle is of radius 1.
y
1
θ
6
P
Q
x
4. [4 marks] Explain why there exists a number c in the interval (1, 2) satisfying the equation
−x3 + x2 + 2x = ln(x) + 1.
Justify your answer, citing appropriate theorems.
7
5. [4 marks] Let f (x) = 1 + x12 . Calculate f 0 (x) using the limit definition of derivative. No
credit will be given for other methods of differentiation.
8
6. [4 marks] Let
(
f (x) =
ax + b if x ≤ 1
q
.
x − 34 if x > 1
Find constants a and b such that the function f (x) is differentiable on the interval
(−∞, ∞). Justify your answer.
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7. Let f (x) = sin x − cos x.
(a) [2 marks] Find all values of x satisfying the equation f (x) = 0 in the interval [0, π).
Justify your answer.
(b) [2 marks] Find all values of x satisfying the equation f 0 (x) = 0 in the interval [0, π).
Justify your answer.
10
8. [4 marks] Find the equations of both lines tangent to the curve y =
to the line y = −100x.
11
1
x
which are parallel
9. In this question, you will prove by mathematical induction that the following claim holds
for all positive integers n.
Claim: 1 + 2 + 22 + 23 + · · · + 2n = 2n+1 − 1.
(a) [2 marks] According to mathematical induction, two facts must be proven for the
claim to be true for all positive integers n. The first is the following:
Fact 1: The claim holds for n = 1; in other words, 1 = 21 −1.
Carefully write down the second fact that must be proven for the claim to be demonstrated for all positive integers n.
Fact 2:
(b) [2 mark] Explain why the two facts above imply that
1 + 2 + 22 + 23 + 24 + 25 + 26 = 27 − 1.
(c) [3 marks] Prove the statement labelled “Fact 2 ” which you wrote down in part (a).
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10. [5 marks] Tsunamis are large ocean waves produced by underwater seismic activity. In
deep water, they can travel at extremely high speeds.
A tsunami wave spreads outward in a widening circle. Suppose the radius of the circle is
increasing at a rate of 900 km/h. Calculate how fast the area of the circle is increasing
when the radius is equal to 100 km.
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This page may be used for rough work. It will not be marked.
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