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.
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:
For each value of , how do the images of C compare ? What appears to be hapening as decreases?
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.