Download University of British Columbia Examination: Mathematics 121, April 2005 and more Exams Mathematics in PDF only on Docsity!
THE UNIVERSITY OF BRITISH COLUMBIA
Sessional Examinations.April 2005
MATHEMATICS 121
Closed book examination Time:2 1/2 hours Calculators are not allowed in this examination
I-[21] SHORT ANSWERS QUESTIONS Each question is worth 3 marks, but not all questions are of equal difficlty. Full marks will be given for correct answers, but at most one mark will be given for in- correct answers. Simplify your answers as much as possible.
a) Evaluate
R
(x^2 + e^2 x)dx.
b) Find the average value of sin x on rthe interval [0, π].
c) Find limn→∞
Pn j=
1 n sin
(^2) (j/n).
d) Find the general solution y = y(x) of the differential equation y^00 − 2 y^0 + y = 0.
e) Find the general solution y = y(x) of the differential equation y^00 − 2 y^0 + y = x.
f ) Evaluate
R ∞
0 (1 +^ x)
− (^3) dx.
g) A continuous random variable X is exponentially distributed with a mean of 4. Find the probability that X ≥ 8.
In Questions II-IX, justify your answers and show all your work. Unless otherwise indicated, simplification of answers is not required.
II-15] Let R be the finite region bounded above by the curve y = 4 − x^2 and below by y = 2 − x.
a)-[4] Carefully sketch R and find its area explicitly. 1
2
b)-[3] Express the volume of the solid obtained by rotating R about the x-axis as a definite integral. You do not need to evaluate this integral.
c)-[4] Express the volume of the solid obtained by rotating R about the vertical line x = 2 as a definite integral. You do not need to evaluate this integral
d)-[4] Express the length of the upper curve that bounds R as a definite integral, and using an appropriate substitution express your answer as an integral involving trigonometric functions. You do not need to evaluate this trigonometric integral.
III-[16] Evaluate the following integrals.
a)-[5]
R (^4) x+ x(x+1)^2 dx.
b)-[4]
R
(x + 1) ln xdx.
c)-[7]
R (^) dx (5− 4 x−x^2 )^3 /^2.
IV-[8] Find f (x) such that
f (x) = 1 +
Z (^) x
0
tf (t) 1 + t + t^2
dt.
V-[8] A child initially at O walks along the edge of a pier, towing a sailboat by a string of length L. The pier is taken to be the y axis. (a) If the boat starts at Q and the string always remains straight, show that the equation of the curved path y = f (x) followed by the boat must satisfy the differential equation
dy dt
L^2 − x^2 x
(b) Find y = f (x).
VI-[8] The vertical face of a dam across a river has the shape of a parabola 36m across the top and 9m deep at the center. What is the
