Math 296 problems 9

Igor Kriz

Regular problems:

1.

Consider the function given by . Suppose is an inverse function to f, which is defined and has total differential on an open set . Find the total differential of g at the point .

2.

Find all the partial derivatives of the function up to order 3.

3.

Find the tangent plane to the surface in given in parametric form by

at the point s=2, t=3.

4.

Let and be two functions given by:

Compute Df, Dg, , .

Challenge problems:

5.

If f(0,0)=0 and

prove that the partial derivatives of f exist at every point of , although f is not continuous at (0,0).

6.

Suppose where U is an open set in . Prove that if all partial derivatives exist and are bounded in U (i.e. all their values are in an interval (-M,M) for some constant M;SPMgt;0), then f is continuous. [Hint: We proved in 295 that if is a one-variable function in an interval satisfying everywhere, then |f(x)-f(y)|;SPMlt;M|x-y|. Use this fact.]

Igor Kriz
Thu Mar 12 22:15:17 EST 1998