Vector calculus: test 2

Grading

These are exercises for Vector calculus: course.

1. Find the point on the surface $z=x^{2}-y^{2}$ nearest to the origin. Explain.

2. Classify the extrema of the following function.

3. Suppose $f,g:\mathbf{R}\rightarrow \mathbf{R}$ are continiuous functions. Let the map $H:\mathbf{R}\rightarrow \mathbf{R}^{2}$ be given by $% H(x)=(f(x),g(x)).$ Prove that $H$ is continuous.

4. Find the volume of the region bounded by the surface $z=1-x^{2},$ the $xy$-plane and the planes $y=0$ and $y=1.$