Name (print):
Student number:
University of British Columbia
DECEMBER EXAM: Science One Mathematics
Date: December 13, 2013
Time: 12:00 noon to 2:30 p.m.
Number of pages: 13 (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
5
4
10
5
5
6
7
7
7
8
7
9
7
Total
60
• 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.
Please note that your answers must be in “calculator-ready” form, but they do not have to
be simplified unless requested. You must provide full solutions with appropriate justification.
This page may be used for rough work. It will not be marked.
2
1. Determine whether each of the following statements is true or false. If it is true, provide
justification. If it is false, provide a counterexample.
(a) [2 marks] If f (0) = 0 and f (3) = 3, then there exists some c in the interval (0, 3)
such that f 0 (c) = 1.
(b) [2 marks] A critical point can be an inflection point.
(c) [2 marks] If the domain of f is (−∞, ∞), then f has no vertical asymptotes.
3
t + sin(2t)
, or explain why the limit does not exist.
t→0 tan(7t)
(b) [3 marks] Let f (x) = g cos πx
, g(0) = 1, g(1) = −2, g 0 (0) = 3 and g 0 (1) = −4.
2
0
Find f (1).
2. (a) [3 marks] Evaluate lim
4
3. [5 marks] Let
f (x) =
arctan(x) if x < c
.
e−x
if x ≥ c
Explain why there exists a number c such that f is continuous. (Hint: use the Intermediate
Value Theorem.)
5
4. Let f (x) =
ln(x)
.
x2
(a) [1 mark] State the domain of the function.
(b) [1 mark] Find all the x- and y-intercepts of the function.
(c) [2 marks] Find all the vertical and horizontal asymptotes of the function.
(d) [2 marks] Find all the local and global extrema of the function. Your answer(s), if
any, should include both x- and y-coordinates.
(e) [2 marks] Find all the inflection points of the function. Your answer(s), if any, should
include both x- and y-coordinates.
(f) [2 marks] Make a large sketch of the graph of the function, indicating all the information found in parts (a) through (e).
(You may continue your solution to this question on the following two pages.)
6
(You may continue your solution on this page.)
7
(You may continue your solution on this page.)
8
5. [5 marks] Explain why curves of the form xy = a (where a is a constant) intersect curves
of the form x2 − y 2 = b (where b is a constant) at right angles.
9
6. (a) [3 marks] Estimate
√
100.2 using an appropriate linear approximation.
(b) √
[1 mark] Is your estimate greater than, less than, or equal to the actual value of
100.2? Justify your answer.
(c) [3 marks] Explain why the absolute value of the error in your estimate is less than
1
.
or equal to 4000
10
7. [7 marks] The rate of change of atmospheric pressure P with respect to altitude h is
proportional to P , provided that the temperature is constant. At 15◦ C the pressure is
101 kPa at sea level (i.e. h = 0 m) and 87 kPa at h = 1000 m. Determine the pressure at
an altitude of 750 m.
11
8. [7 marks] A boat leaves a dock at 2:00 P.M. and travels due south at a speed of 10
km/h. Another boat has been heading due east at 20 km/h and reaches the same dock
at 3:00 P.M. Determine how fast the distance between the boats is changing (increasing
or decreasing, specify which) at 2:30 P.M.
12
9. [7 marks] A billboard is to be made with 100 m2 of printed area, and with margins of
2 m at the top and bottom and 4 m on each side. Find the outside dimensions of the
billboard with the smallest possible total area.
13