Name (print):
ID number:
Science One number:
University of British Columbia
APRIL EXAM: MATHEMATICS
Date: April 13, 2011
Time: 3:30 p.m. to 6:00 a.m.
Number of pages: 7 (including cover page)
Exam type: Closed book
Aids: No calculators or other electronic aids
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Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.
Question
Mark
Possible marks
1
8
2
10
3
9
4
8
5
17
6
11
Total
63
1. Recall that ex has the Maclaurin series expansion
ex = 1 + x +
x2
x3
+
+ ··· .
2!
3!
In this question, you will use this fact to prove Euler’s Identity.
(a) [2 marks] Write down the Maclaurin series for sin x and cos x. (You do not have to show your
work.)
(b) [4 marks] Recall that ex can be defined for complex values x by using its Maclaurin series expansion.
Prove that
eix = cos x + i sin x.
(c) [2 marks] Use your answer in part (b) to prove Euler’s Identity
eiπ + 1 = 0.
2
2. Let f be a function with Maclaurin series expansion
f (x) = a0 + a1 x + a2 x2 + a3 x3 + · · · .
Let Pn (x) denote the polynomial formed from the first n + 1 terms:
Pn (x) = a0 + a1 x + a2 x2 + · · · + an xn .
Taylor’s Theorem describes how closely P (n) approximates f (x); in particular, it states that
Z
1 x
(x − t)n f (n+1) (t) dt.
f (x) = Pn (x) +
n! 0
(f (n+1) (t) denotes the (n + 1)st derivative of f at t.) In this question, you will prove Taylor’s Theorem
by induction on n.
(a) [2 marks] In the case n = 0, Taylor’s Theorem is simply
Z x
f (x) = a0 +
f 0 (t) dt.
0
Explain why this is true. (Hint: use the Fundamental Theorem of Calculus.)
(b) [6 marks] Suppose Taylor’s Theorem is true in the case n = k; that is,
Z
1 x
f (x) = Pk (x) +
(x − t)k f (k+1) (t) dt.
k! 0
Show, by integrating by parts, that this implies the n = k + 1 case.
(c) [2 marks] Describe how induction works by explaining why your answers in part (a) and part (b)
imply that Taylor’s Theorem is true in the case n = 100.
3
3. Consider a game in which a bean-bag is thrown at a target consisting of a long thin trough extending
from x = −20 to x = 20. The game is set up so that the bean-bag always ends up in the trough and can
land at any point x along the trough. The score for any throw is S(x) = 100 − |x| points.
Fok has played the game numerous times and has determined that his own probability density for getting
the bean-bag to land at position x is p(x) = Ae−|x|/5 . The expected score is defined to be
Z
20
Sexp =
p(x)S(x)dx.
−20
(a) [2 marks] Given that the bean-bag always lands somewhere in the trough, what is the value of A?
(b) [2 marks] Set up an integral to calculate the probability that Fok will score higher than 95 points.
You do not need to solve the integral.
(c) [5 marks] Calculate Fok’s expected score in any one game.
4
4. [8 marks] Calculate the volume of air inside a bicycle tire where the radius of the of wheel is R and the
radius of the tire is r. (The picture on the right is the cross-section. The air inside the tire is shaded, and
the centre of the wheel is denoted by a black dot.)
5
5. Consider the system of differential equations
dx
= x − x2 − 2xy
dt
dy
= y − y 2 − 2xy.
dt
This is a standard model for two species competing for a common resource.
(a) [4 marks] Sketch the phase plane, including nullclines and direction vectors.
(b) [4 marks] List the coordinates of all the equilibrium points, indicating whether they are stable or
unstable.
(c) [2 marks] Sketch approximate trajectories for each of the following initial conditions:
1 1
(x(0), y(0)) =
,
3 10
3 1
(x(0), y(0)) =
,
4 4
(d) [2 marks] The following, altered system of differential equations models a situation where species
x is harvested:
dx
3
= x − x2 − 2xy − x
dt
4
dy
2
= y − y − 2xy.
dt
Sketch the phase plane, including nullclines and direction vectors.
(e) [2 marks] List the coordinates of all the equilibrium points (where x and y are nonnegative),
indicating whether they are stable or unstable.
(f) [2 marks] Sketch approximate trajectories for each of the following initial conditions:
1 1
(x(0), y(0)) =
,
2 4
1 1
(x(0), y(0)) =
,
16 16
(g) [1 mark] For the system modelled in part (d), describe the long-term behaviour of the species x
when both species x and y begin with a nonzero population. How does it compare to the long-term
behaviour of the species x in the absence of the species y?
6
6. Consider the system of differential equations
x0 (t) = y(t)
.
y 0 (t) = 2x(t) + y(t)
(a) [6 marks] Find the general solution.
(b) [3 marks] Suppose you are given the initial conditions x(0) = 3 and y(0) = −3. Find the particular
solution corresponding to these initial conditions.
(c) [2 marks] Sketch the graph of x(t) on x-versus-t axes.
7