Math 53 - Hutchings
GSI: Nicolas Bray
Final, Fall 2003, Solutions
1. The chain rule tell us that
d
dt f (r(t)|t=0
=
f (r(0) r (0) = 2e2 , e2 3, 4 = 10e2 .
2. (a) A useful fact about spheres is that if the point x, y, z is on a sphere S then
the vector x,
Math 53 - Hutchings
GSI: Nicolas Bray
Final, Spring 2007, Solutions
1. Letting g (x, y, z ) = x2 + y 2 + z 2 , we want to nd solutions to g = 1 and
f = g which
works out to the four equations:
2(x 2) = 2x (1 )x = 2 x =
2
1
2
1
1
2(z 1) = 2z (1 )z = 1 z =
Math 53 Midterm 2
Monday, Nov 21, 2011 1:10 - 2:00
Directions: Do all the work on these pages; use reverse side if needed. Answers without
accompanying reasoning may only receive partial credit. No books, notes, calculators, or
electronic devices. Please
Perfect Substitutes
1. this means that it is an all or nothing problem so demand is either 0 if none is
being consumed or m/p if all is being consumed
2. This also means that you dont have to compare utilities b/c marginal utility
doesn have a variable in
Max and Quasi Linear Concave
1. these are all or nothing problems so demand will be 0 for nothing and m/p
for other
2. check if mu of both x and y are integers, if so you can take shortcut and solve
as a perfect substitutes problem
3. If not then you will