**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.]

Thu Mar 12 22:15:17 EST 1998