Name (print):
ID number:
Section (circle):
001
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University of British Columbia
MIDTERM TEST 1 for MATH 110
Date: October 20, 2010
Time: 6:00 p.m. to 7:30 p.m.
Number of pages: 10 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
For examiners’ use only
Rules governing formal examinations:
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination.
Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action:
• Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid
devices, other than those authorized by the examiners;
Question Mark Possible marks
1
6
2
6
3
7
4
9
5
6
6
8
7
8
8
3 (bonus)
Total
50
• Speaking or communicating with other candidates;
• Purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness
shall not be received.
Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take
any examination material from the examination room without
permission of the invigilator.
Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
1. (a) [2 marks] State Pythagoras’ theorem. Include a picture, and refer to the picture in
your statement of the theorem.
Parts (b) and (c) refer to the picture to the right, which
depicts a rectangular box with dimensions 3 × 4 × 12.
A
1. (b) [2 marks] What is the length of the diagonal of the
base (i.e. the line segment BD)?
12
B
3
4
D
C
1. (c) [2 marks] What is the length of the diagonal of the box (i.e. the line segment AD)?
2
2. (a) [2 marks] For what values of x do the curve y = 4x2 + 6x − 4 and the line y = 2x − 1
intersect?
2. (b) [2 marks] Write down the equation of a line that intersects the curve y = 2(x+1)2 +3
exactly once.
2. (c) [2 marks] Prove that any two lines with different slopes intersect exactly once.
3
3. Find the domain of each of the following functions.
3. (a) [2 marks] f (x) =
2x2
3. (b) [3 marks] f (x) = √
3. (c) [2 marks] f (x) = √
1
− 2x − 12
2x2
1
− 2x − 12
1
−x3
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4. (a) [3 marks] Let
2
x2 + ax + b
=1+ 2
hold for all x. Find a and b.
2
x +x+1
x +x+1
4. (b) [3 marks] Let f (x) = cx and g(x) = x4 . Suppose the equation (f ◦ g)(x) = (g ◦ f )(x)
is true for all x. Find c.
x3 − dx + d
= 4. Find d.
x→∞
dx3 − d
4. (c) [3 marks] Let lim
5
5. For parts (a) through (d), evaluate the limit, or explain why it does not exist.
49 − 1
x→−7 7 + 1
5. (a) [1 mark] lim
81 − x
√
x→81 9 −
x
5. (b) [2 marks] lim
x5 + 2x3
x→−∞ x4 − 5x2 + 2
5. (c) [3 marks] lim
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6. (a) [2 marks] Define what it means for a function f to be continuous at a point a.
6. (b) [3 marks] Let
f (x) =
x2 − 8
if x ≤ c
.
10x − 33 if x > c
Find c so that f is continuous everywhere.
6. (c) [3 marks] Let
g(x) =
1 if x = 1, 12 , 13 , 14 , . . .
.
0 otherwise
Prove that g(x) is not continuous at 0.
7
7. [8 marks] Sketch the graph of a single function f satisfying all of the following conditions:
• f is defined everywhere but at x = 2;
• f is discontinuous at x = −3, 0 and 2;
• f (−3) = 2, f (0) = 1 and f (4) = −2;
• lim f (x) = 4;
x→−3
• lim− f (x) = ∞ and lim+ f (x) = 3; and
x→0
x→0
• f (x) 6= 0 for all x.
8
8. [3 bonus marks] Prove that the graph of the function
f (x) =
2x4 + x − 1
x−3
crosses the x-axis somewhere in the interval (0, 1).
9
This page may be used for rough work. It will not be marked.
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