1. You are sending a birthday present to your calculus instructor.
Fly-By-Night Delivery Service insists that any package they ship be such
that the sum of the length and the girth be at most 2m. What are the
dimensions of the largest box-shaped present you can send?
2. Find the maximum value of (x^2)(y^2)(z^2) subject to the constraint
that x^2 + y^2 + z^2 = a^2, where a^2 is a constant. Use this to show
that if u,v,w are positive numbers, then
(uvw)^(1/3) <= (u+v+w)/3 .
This is a special case of the arithmetic-geometric quantity, which says
that the geometric mean of a collection of numbers is less than or equal
to their arithmetic mean.
3. Stewart, Section 12.8, Exercises 21, 22.
4. Determine the dimensions of the largest cone that can be inscribed
in a sphere of radius a.
5. Draw the level curves and gradient vector field of the function
x^3 - x y^2. Describe the critical points. [Hint: Polar coordinates.]
6. [Optional brain teaser] Suppose that f is a function with continuous
second partial derivatives, and the level curves of f are as shown below.
Find the second partial derivatives of f at the point A. Explain your
answer.
[Picture of some level curves. The interesing feature is that
there is a 6-way crossing of level curves at the point A.]