Name (print):
ID number:
University of British Columbia
DECEMBER EXAM: SCIE 001 MATHEMATICS
Date: December 12, 2011
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
Question
Mark
Maximum mark
1
9
2
7
3
8
4
6
5
16
6
7
7
8
8
5
9
8
10
6
Total
80
Rules governing formal examinations:
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.
Unless stated otherwise, δ − ε proofs are not necessary. You do not have to simplify your
answers; in particular, mumerical answers can be left unsimplified in a “calculator-ready”
form. You should show your calculations and your solutions should be clear. Good luck!
This page may be used for rough work. It will not be marked.
2
sin (2x − 6)
, or explain why the limit does not exist.
x→3
x2 − 9
1. (a) [3 marks] Evaluate lim
1. (b) [3 marks] Find the derivative of g(x) = ln
x+3
.
x+4
1. (c) [3 marks] Find the slope of the curve x4 − x2 y + y 4 = 1 at the point (1, 1).
3
2. (a) [3 marks] Describe, in precise δ − ε terms, what it means for lim f (x) = 0.
x→0
2. (b) [4 marks] Let
f (x) =
1
0
1
1
1
1
, 100
, 1000
, 10000
,...
if x = 10
otherwise
Prove, using your δ − ε definition from part (a), that lim f (x) 6= 0.
x→0
4
3. Let f (x) =
x2 sin
0
1
x
if x 6= 0
.
if x = 0
3. (a) [4 marks] Prove that f is differentiable at 0.
3. (b) [4 marks] Prove that f 0 is not continuous at 0.
5
4. [6 marks] Let f (x) = ln(x). Let the nth derivative of f (x) be denoted f (n) (x). Prove, using mathematical
induction, that
f (n) (x) = (−1)n−1 (n − 1)! x−n for n = 1, 2, 3, . . . .
(Remember that 0! = 1.)
6
5. Let f (x) =
x2 + x
x2 − x
5. (a) [2 marks] What is the domain of f ?
5. (b) [2 marks] Find the x- and y-intercepts of f .
5. (c) [3 marks] Determine if f has any vertical or horizontal asymptotes.
5. (d) [3 marks] Identify where f is increasing and where it is decreasing
7
5. (e) [3 marks] Identify where f is concave up and where it is concave down.
5. (f) [3 marks] Sketch the graph of the function, including all of the characteristics identified in parts (a)
through (e).
8
6. (a) [3 marks] Find the Taylor polynomial of degree 2 for ln(x) about x = 1.
6. (b) [4 marks] Let P (x) be the Taylor polynomial from part (a), and let
R(x) = ln(x) − P (x).
Prove that R(1.1) ≤
1
.
3000
9
7. [8 marks] Determine the radius and height of a cylindrical can with a volume of 500 cm3 that minimizes
the surface area. The can has a circular top, a circular bottom, and the sides are constructed from a
rectangle.
10
8. [5 marks] As a woman walks away from a street lamp, her shadow lengthens. Prove that it does so at a
rate which depends on her speed but not on her distance from the lamppost.
11
9. A mass of m = 2 kg is attached to a spring that obeys Hooke’s Law with a spring constant k = 8 N·m−1
and moves without friction along a straight line. Let x(t) denote its displacement (in m) at time t (in s),
where x = 5 is the equilibrium position (the position where no force from the spring acts on the mass; if
the mass was at rest at x = 5 then it would remain there forever).
10. (a) [2 marks] Write down the second-order differential equation satisfied by x(t), obtained by applying
Newton’s Second Law of Motion to the mass-and-spring system.
10. (b) [2 marks] Let u(t) = x(t) − 5 so that u = 0 corresponds to x = 5. Write down the second-order
differential equation satisfied by u(t).
10. (c) [4 marks] If x(0) = 6 m and x0 (0) = 2 m·s−1 , determine the maximum and minimum values attained
by x(t), for all 0 ≤ t < ∞.
12
10. [6 marks] Show that there is a number c in the open interval
13
π π
4, 3
√
2−1
√ .
such that tan c = √
3− 2