The University of British Columbia
Science One Mathematics
Midterm #1
27 October 2009
Time: 50 minutes
Full Name:
Student # :
Signature:
This Examination paper consists of 6 pages (including this one). Make sure you have all 6.
Instructions:
No memory aids allowed. No calculators allowed. No communication devices allowed.
Rules:
– Each candidate must be prepared to produce, upon request, a UBC card for identification.
– Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities
in examination questions.
– 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;
- speaking or communicating with other candidates; and
- 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.
marking:
Q1
/ Q3
/
Q2
/ Q4
/
Names of Instructors: Eric, Fok, Mark
TOTAL
/
Science One Mathematics — Midterm #1 — 27 October 2009 — p. 2 of 6
Q1
[5 marks]
Let (a, a2 ) be any point on the parabola y = x2 .
(a) Find the equation of the line, l1 , going through the point (a, a2 ) and perpendicular to the
tangent line to this parabola at (a, a2 ).
(b) Consider the horizontal line l2 given by y = a2 . Show that the distance between the
y-intercepts of l1 and l2 is equal to 1/2 for any non-zero value of a.
Science One Mathematics — Midterm #1 — 27 October 2009 — p. 3 of 6
Q2
[5 marks]
(a) Define what it means for a function f to be continuous at a point c.
( q
(b) Let f (x) =
|x|
x2
if x < c,
if x ≥ c.
For what value(s) of the constant c is f continuous everywhere?
Science One Mathematics — Midterm #1 — 27 October 2009 — p. 4 of 6
Q3
[10 marks]
Let f (x) =
1
.
1 + x2
(a) Identify intervals on which f is increasing, decreasing, concave up and concave down; and
identify any minima, maxima and inflection points of f .
Science One Mathematics — Midterm #1 — 27 October 2009 — p. 5 of 6
(b) Sketch f . Note: each axis must be labeled and must have at least one point indicated to
provide a scale.
(c) For which values of c is the function g(x) = f (x) + cx increasing everywhere on the real
line?
Science One Mathematics — Midterm #1 — 27 October 2009 — p. 6 of 6
Q4
[10 marks]
Consider the function
f (x) =
x2 − 3x + 2
.
x2 (4 − x)
Prove, using the Intermediate Value Theorem, that f (x) takes on the value 1 at least twice.