Name (print):
Student number:
University of British Columbia
MATH 110: MIDTERM TEST 1 for SECTION 001
Date: October 17, 2012
Time: 6:00 p.m. to 7:30 p.m.
Number of pages: 9 (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 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;
For examiners’ use only
Question Mark Possible marks
1
5
2
5
3
6
4
4
5
6
6
6
7
8
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) [3 marks] Sketch the circle (x − 3)2 + (y − 2)2 = 4 and the line y − 4 = −(x − 5) on
the axes below.
(b) [2 marks] Find the coordinates of the point on the circle closest to the line.
2
2. [5 marks] Find the domain of the function g(x) =
3
x+3
√
.
4 − x2 − 1
3. Sketch the graph of each of the following functions.
(a) [3 marks] f (x) =
x2 − 1
x−1
√
(b) [3 marks] g(x) = − x − 3
4
4. (a) [2 marks] Write down an expression for a function f whose domain is all real numbers,
but whose limit lim f (x) does not exist.
x→0
(b) [1 mark] Write down an expression for a function g which is equal to its own inverse.
(c) [1 mark] Write down an expression for a function h such that h(h(x)) is equal to
h(x).
5
5. 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 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
6
6. Let f (x) =
1
.
x2
(a) [4 marks] Use the limit definition of derivative to find f 0 (x). No credit will be given
for using other methods to calculate the derivative.
(b) [2 marks]Find the equation of the line perpendicular to the curve y = f (x) at the
point 21 , 4 .
7
7. (a) [2 marks] Write down an expression for a function whose graph is a parabola intersecting the x-axis at x = 1 and x = 2.
(b) [6 marks] Find nonzero constants a, b, c and d such that the graph of the function
f (x) = ax3 + bx2 + cx + d has horizontal tangent lines at x = 1 and x = 2, and
intersects the y-axis at y = 3.
8
This page may be used for rough work. It will not be marked.
9