UNIVERSITY OF DUBLIN
2112
TRINITY COLLEGE
Faculty of Science
school of mathematics
SF Engineers
SF MSISS
Foundation Scholarship
Hilary Term 2000
Course 2E1/2E2
Thursday, March 16
Exam Hall
Professor T.T. West
09.30 — 12.30
Dr. P. Tran-Ngoc-Bich
Answer SECTIONS A and B in separate answer books.
Engineers must answer FIVE questions from SECTION A and THREE questions from
SECTION B, including questions B11 and B13.
MSISS students must answer EIGHT questions from SECTION A
SECTION A
1. By using a Lagrange multiplier or otherwise find the dimensions of a rectangular
box, open at the top, having a volume of 32f t3 and requiring the least amount of
material for its construction.
2. Find the normal vector and the equation of the tangent plane to the surface
x2 + 4y 2 + z 2 = 18.
at the point (1, 2, −1).
3. Let f be a differentiable function of one variable, let w = f (r) where r2 = x2 +y 2 +z 2
Show that
2 2 2 2
∂w
∂w
dw
∂w
+
+
=
∂x
∂y
∂z
dr
4. Two points are given in polar co-ordinates in R2 by (r1 , θ1 ) and (r2 , θ2 ). Prove that
the square of the distance between them is
r12 + r22 − 2r1 r2 cos(θ1 − θ2 )
2
2112
5. Write down a formula for the length element of a curve given in polar co-ordinates
(r, θ)
Find the length of the curve given by
r = 1 + cos θ from θ = 0 to θ = π.
6. A tetrahedron of constant density in the positive octant is bounded by the coordinate planes and the plane x + y + z = 1.
Find its Mass and Centre of Gravity.
7. Sketch the volume of integration of the integral
Z
3
−3
Z
√
+ 9−x2
−
√
9−x2
9−x2 −y 2
Z
x2 dzdydx.
0
By changing to cylindrical polar co-ordinates evaluate the integral.
8. If u and v are any two vectors in an inner product space, prove that
(a)
| hu, vi | ≤ || u || . || v ||,
(b)
|| u + v ||2 + || u − v ||2 = 2 || u ||2 +2 || v ||2 ,
(c)
hu, vi =
1
1
|| u + v ||2 − || u − v ||2 .
4
4
9. C[−1, 1] denotes the vector space of continuous real valued functions defined on the
interval [−1, 1] with
Z
1
hf, gi =
f (x)g(x)dx.
−1
Prove that hf, gi is an inner product for C[−1, 1] and derive an orthonormal set
{f0 , f1 , f2 , f3 , } of functions via the Gram-Schmidt process from the set of functions
{1, x, x2 , x3 }.
SECTION B
10. Let f (t) be a continuous function. We suppose that f (t) is periodic of period 1.
Show that the Laplace transform of f (t) is given by the formula
Z 1
1
L(f (t)) =
e−st f (t)dt
1 − e−s 0
3
2112
11. Laguerre’s differential equation of order 3 is
ty 00 + (1 − t)y 0 + 3y = 0
Using the formula L(tf (t)) = −F 0 (s) if F (s) = L(f (t)), show that Y (s) = L(y(t))
satisfies the differential equation
s(1 − s)Y 0 + (4 − s)Y = 0
The solution of this equation is:
Y (s) =
(s − 1)3
s4
Find Y (t) by taking the inverse Laplace transform of Y (s).
12. We consider the system of differential equations
~x0 (t) = A~x(t) + f~(t)
where
A=
−3 −4
1
1
,
~x(t) =
x1 (t)
x2 (t)
,
(1)
f~(t) =
f1 (t)
f2 (t)
.
Keeping arbitrary the input f~(t), we intend to solve the eq. (1) for the initial
conditions:
1
~x(0) =
1
~
Find the equation satisfied by the Laplace transform X(s)
of ~x(t). Deduce by inverse
Laplace transform that
~x(t) = L−1 ((sI − A)−1 )~x(0) + L−1 ((sI − A)−1 ) ∗ f~(t)
where ∗ stands for the convolution operation. Compute L−1 ((sI − A)−1 ). Write
down the solution ~x(t). (Do not compute the integral in the convolution).
13. We consider the differential equation yy 00 + 1 = 0, for the initial conditions y(0) = 1
and y 0 (0) = 1. We assume that this equation admits a power series solution.
Compute y 00 (0). By differentiating the differential equation, compute y (3) (0), y (4) (0)
and y (5) (0). Write up to the term in x5 the power series solution of yy 00 + 1 = 0.
c UNIVERSITY OF DUBLIN 2000