Name (print):
ID number:
Section (circle):
001
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University of British Columbia
MIDTERM TEST 1 for MATH 110
Date: October 19, 2011
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
5
5
5
6
6
7
5
8
2 (bonus)
Total
40
• 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 the Pythagorean theorem. You may include a picture.
1. (b) [4 marks] Find the area of a square whose diagonal is of length 2.
2
2. Find the domain and range of each of the following functions.
2. (a) [2 marks] f (x) = (x − 3)2 + 2
2. (b) [4 marks] f (x) =
p
2−
√
x
3
3. Evaluate each of the following limits.
9−4
x→2 3 − 2
3. (a) [1 mark] lim
1 − x2
x→∞
x
3. (b) [3 marks] lim
3. (c) [3 marks] lim (f ◦ f )(x), where f (x) = x3 + 2
x→0
4
4. [5 marks] Let
f (x) =
1
1
1
, 100
, 1000
,...
10 if x = 1, 10
.
0 otherwise
Explain why lim f (x) does not exist.
x→0
5
5. (a) [3 marks] Write down an algebraic expression for the parabola which has x-intercepts
−2 and 3, and which passes through the point (0, 6).
5. (b) [2 marks] Write down an algebraic expression for a function which is defined everywhere but discontinuous at x = 0.
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6. Recall that a function f is said to be rational if it is of the form
f (x) =
p(x)
,
q(x)
where p(x) and q(x) are polynomials.
5. (a) [2 marks] Write down the contrapositive of the following (false) statement:
If f is a rational function, then f is continuous at all real numbers.
5. (b) [4 marks] Disprove the statement in part (a).
7
7. [5 marks] Sketch the graph of a single function f satisfying all of the following conditions:
· f is defined everywhere but at x = −1;
· f is continuous everywhere but at x = −1 and x = 2;
· f (x) 6= 0 for all x
· f (2) = 3
· lim+ f (x) = 4;
x→2
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8. [2 bonus marks] Explain why all polynomials of odd degree must have at least one root.
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This page may be used for rough work. It will not be marked.
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