PHZ 3113 Fall 2011 – Homework 3
Due at the start of class on Wednesday, September 14. Half credit will be available
for homework submitted after the deadline but no later than the start of class on Friday,
September 16.
Answer all questions. Please write neatly and include your name on the front page of your
answers. You must also clearly identify all your collaborators on this assignment. To gain
maximum credit you should explain your reasoning and show all working.
1. A square pyramid
with height h and base sides of length b has volume b2 h/3 and surface
p
2
area b + 2b h2 + b2 /4. Use the method of Lagrange multipliers to find the maximum
volume for a given surface area A.
2. Find the shortest distance from the origin to any point at the intersection of the planes
x + y − 3z = 2 and x + 2y + z = −1.
3. Transform the partial differential equation
5
∂2f
∂2f
∂2f
−
9
=0
−
2
∂x2
∂x∂y
∂y 2
using u = 2x + y, v = x − 5y, and hence find the general form of f (x, y).
4. Find the derivative with respect to x of
Z
1
1/x
ext
dt.
t
5. Integrate x2 + y over the interior of a circle of radius 2 centered on x = 0, y = 0.
6. Evaluate
Z
1
Z
0
2−x
0
(x + y)1/2 (x − 3y) dy dx
by transforming to the variables u = x + y, v = x − 3y.
7. Evaluate
Z
0
∞
Z
0
∞
x2 + 2y 2
e−xy/3 dy dx
1 + (x2 − 2y 2 )2
by transforming to the variables u = x2 − 2y 2 , v = 13 xy.