** Problem 1**

Let Using Maple, for
and ,
plot the gradients of **f** for
b=1, 2, and 4. Print these out. On each of these printouts, draw in by hand
(roughly) several level curves.

** Problem 2**

We can consider the derivative Df of a function f at a point as
an approximation of f
by an affine linear function. (The word affine here refers to the fact that
these approximations have constant terms in them.) Here we explore this
linear approximation
aspect of the derivative.

Consider the function

The corresponding first order approximation to f at the point is given by

where is the derivative of **f** at the point

Let C be a square of side centered at the point , i.e.

For taking on each ofthe values and , do the following:

- Plot C and the image of C under the function f.
- Plot C and the image of C under the first order approximation S at .
- Combine these two pictures and estimate roughly the ratio of
the area of to the area of .

Do you have a guess at how this ratio might depend on ?

The Maple procedure * transform_rect* available in the Math 222
folders will be useful here. The output of * transform_rect* is
a Maple graphics structure, so you can assign the result to a variable
and then display several of these simultaneously.