Review Problems for Test II - Part I

The University of Akron
Department of Mathematics
Calculus III
Spring 2026, Golovaty

REVIEW PROBLEMS FOR TEST II - Part II

Find $D_{\bf u}f$ at

.

$f(x,y)={(1+xy)}^{3/2};\ P(3\,,1);\ {\bf u}=\frac{1}{\sqrt{2}}{\bf i}+\frac{1}{\sqrt{2}}{\bf k}$ .
$f(x,y,z)=\ln{\left(x^2+2y^2+3z^2\right)};\ P(-1\,,2\,,4);\ {\bf u}=\left<-\frac{3}{13}\,,-\frac{4}{13}\,,-\frac{12}{13}\right>$ .

Find the directional derivative of $f(x\,,y)=\frac{x}{x+y}$ at $P(1\,,0)$ in the direction of $Q(-1\,,1)$ .

Find a unit vector in the direction in which $f(x\,,y\,,z)=\frac{x+z}{z-y}$ decreases most rapidly at $P(5\,,7\,,6)$ and find the rate of change of

at

in this direction.

Let $r(x,y)=\sqrt{x^2+y^2}$ .

Show that $\nabla r=\frac{{\bf r}}{r}\,,$ where ${\bf r}=<x\,,y>$ .
Verify that ${\bf r}_1(t)=<x(t)\,,y(t)>=<\cos{t}\,,\sin{t}>$ is a level set for corresponding to .
Show that the gradient of at ${\bf r}_1(t)$ is orthogonal to the level curve ${\bf r}_1(t)$ for any .

Find all relative extrema and saddle points, if any.

.
$g(x,z)=z\sin{x}$ .

Find the absolute extrema of the function $f(x\,,y)=x^2+2y^2-x$ on the triangular region with vertices (0,0), (2,0), and (2,2).

Find the dimensions of the rectangular box of maximum volume that can be inscribed in a sphere of radius

.

Dmitry Golovaty