Name (print):
Student number:
University of British Columbia
MATH 110: MIDTERM TEST 2 for SECTION 001
Date: February 13, 2013
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
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 must conduct themselves honestly and in accordance
with established rules for an examination. Should dishonest behaviour be observed, pleas of accident or forgetfulness shall not
be received.
Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be subject to disciplinary action:
• Speaking or communicating with other candidates;
• Purposely exposing written papers to the view of other candidates or imaging devices;
• Purposely viewing the written papers of other candidates;
• 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
any examination material from the examination room without
permission of the invigilator.
Candidates must follow any additional examination rules or directions communicated by the invigilator.
For examiners’ use only
Question Mark Possible marks
1
8
2
5
3
4
4
15
5
8
Total
40
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] A function with a horizontal asymptote never intersects it.
x
= ∞.
x→∞ ln x
(b) [2 marks] lim
(c) [2 marks] If f has a vertical asymptote at x = 2, then f (2) is undefined.
(d) [2 marks] If f (−3) = f (1) = 2, then there exists a point c in the interval (−3, 1)
such that f 0 (c) = 0.
2
2. Let f (x) = sin x − cos x.
(a) [2 marks] Find all values of x in the interval [0, π] satisfying the equation f 0 (x) = 0.
Justify your answer.
(b) [3 marks] Where does f (x) attain its global maximum and global minimum on the
interval [0, π]? Justify your answer.
3
3. [4 marks] Come up with an algebraic expression for a differentiable function f which
has domain (−∞, ∞) and satisfies f (3) = 0, f 0 (3) < 0 and f 00 (3) > 0. Then sketch the
function below.
4
4. Let f (x) = e1/x .
(a) [1 mark] State the domain of f .
(b) [1 mark] Find all the x- and y-intercepts of f , if any exist.
(c) [2 marks] Find the horizontal asymptotes of f , if any exist.
(d) [2 marks] Find the vertical asymptotes of f , if any exist.
5
(e) [3 marks] Determine where f is increasing and where it is decreasing. Does f have
any local extrema?
(f) [3 marks] Determine where f is concave up and where it is concave down. Does f
have any inflection points?
6
(g) [3 marks] Make a large sketch of the graph of f below, including all you have found
in the previous parts of the question.
7
5. [8 marks] This sheet of paper measures 8.5 inches by 11 inches. Suppose you cut out a
square of side length x from each of the four corners, and then fold up the sides to form
an open-topped box of height x, width 8.5 − 2x and depth 11 − 2x inches. What choice
of x results in a box with the largest possible volume?
Note that you must justify your answer, but you do not have to simplify it. Hint: find
the function V (x) describing the volume of the box. What are its roots? What does its
graph look like?
If necessary, you may continue your answer on the following page.
8
This page may be used for extra work on question 5. It will be marked.
9
This page may be used for rough work. It will not be marked.
10