Name (print):
ID number:
University of British Columbia
MIDTERM TEST: SCIE 001 MATHEMATICS
Date: February 14, 2012
Time: 8:30 a.m. to 9:20 a.m.
Number of pages: 7 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
Rules governing formal examinations:
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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:
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devices, other than those authorized by the examiners;
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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.
For examiners’ use only
Question
Mark
Maximum mark
1
9
2
7
3
6
4
5
5
7
Total
34
1. Evaluate the following integrals.
Z
1. (a) [3 marks]
x cos 5x dx
Z
1. (b) [3 marks]
0
Z
1. (c) [3 marks]
1
y2
y+1
dy
+ 2y − 5
sin3 ωt dt, where ω 6= 0 is a constant
2
2. [7 marks] Using the general slicing method, evaluate a definite integral to find the volume of a tetrahedron
(pyramid with four triangular faces), all of whose edges have a length L. Remember to give justification.
3
3. [6 marks] Prove that
100
X
k=1
1
ek/100
1
1
< 100 1 −
. (Hint: consider the function f (x) = x .)
e
e
4
4. [5 marks] For a mass m kg at a height h ≥ 0 m above the surface of the earth, the gravitational force,
in N, acting on the mass is given by F~ = −GM m(R + h)−2 r̂, where G is the gravitational constant in
N m2 /kg2 , M is the mass of the earth in kg, R is the radius of the earth in m, and r̂ is the unit vector
pointing away from the centre of the earth. Determine the work required to launch a rocket with a mass
of 500 kg to a height of 2500 km above the earth’s surface.
5
5. Let
f (x) =
sin
0
1
x
if x 6= 0
.
if x = 0
5. (a) [2 marks] Let δ be any number in the interval (0, 1]. Consider a partition P of the interval [0, 1]
whose first subinterval is [0, δ]. On that subinterval, what is the maximum value of δf (x)? What is
the minimum value of δf (x)?
5. (b) [5 marks] Prove that f is integrable on the interval [0, 1]. You may use, without proof, the fact that
if a function is continuous on [δ, b], then it is integrable on [δ, b].
6
This page may be used for rough work. It will not be marked.
7