Name (print):
ID number:
University of British Columbia
MIDTERM TEST: SCIE 001 MATHEMATICS
Date: October 20, 2011
Time: 8:30 a.m. to 9:20 a.m.
Number of pages: 8 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
Rules governing formal examinations:
For examiners’ use only
Each candidate must be prepared to produce, upon request, a
UBC card for identification.
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;
• 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.
Question
Mark
Maximum mark
1
6
2
6
3
6
4
6
5
5
6
6
Total
35
1. Calculate the following limits, or explain why they do not exist (δ − ε proofs are not necessary, but show
steps in the calculation).
x
1. (a) [3 marks] lim q
x→∞
2
(2x + 1)
x2 − 14x + 49
x→7
sin (x − 7)
1. (b) [3 marks] lim
2
2. Find the derivatives of the following functions (after differentiating, simplification is not necessary).
2. (a) [3 marks] f (x) =
p
x+
√
x
2. (b) [3 marks] e3x ln(5x + 1)
3
3. [6 marks]Let f (x) =
x
1
2x
+1
if x ≤ 2
.
if x > 2
3. (a) Sketch the function, making sure to label your axes.
3. (b) Let ε = 34 . Find some δ > 0 such that |f (x) − 2| < ε whenever 0 < |x − 2| < δ. Explain why your
choice of δ is suitable. You may refer to your picture in part (a).
4
4. [6 marks] Let f be a function such that |f (x)| ≤ x2 for all x.
4. (a) Prove that f is continuous at 0 (a δ − ε argument is not necessary).
4. (b) Prove that f is differentiable at 0 (a δ − ε argument is not necessary).
5
5. [5 marks] A function f is said to have a fixed point at c if f (c) = c. For example, the function f (x) = x2
has fixed points at 0 and 1. Let f be any continuous function with domain [0, 1] and range [0, 1]. Prove
that f has a fixed point. (Hint: consider the function g(x) = f (x) − x.)
6
6. [6 marks] Find all nonnegative values of the constant E ≥ 0 such that there is a function Ψ(x) which
is: not identically zero, differentiable on the interval [0, 1], twice differentiable on the interval (0, 1), and
which satisfies the second-order ordinary differential equation
Ψ00 + EΨ = 0,
0 < x < 1,
and the two boundary conditions
Ψ0 (0) = 0,
Ψ0 (1) = 0.
7
This page may be used for rough work. It will not be marked.
8