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\line{Math 174\hfill Exam II\hfill 3/31/2000\hfill D.~Gottlieb}
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\item{(5)\quad 1.}Suppose that $f(x,y)$ is a real valued function defined everywhere on the plane.
Suppose its gradient $\overset\longrightarrow\to{\nabla f}$ is not defined at $(0,0)$.

\itemitem{a)}Is it possible to determine whether or not $f$ is continuous at $(0,0)$ given the above information?
(if yes, is it continuous at $(0,0)$ or not)

\itemitem{b)}Is it possible to determine whether or not $f(x,y)$ is differentiable at $(0,0)$?
(if yes, is it differentiable at $(0,0)$)

\item{}HINT:\ If you don't know the answers for a) or b), write down the definitions of continuity for a) or differentiability for b) for some partial credit.

\vfill
\item{(20)\quad 2.}Let $f(x,y,z)=xyz$ and $P$ be the point $(2,-3,4)$.
\itemitem{a)}Calculate the gradient vector field $\overset\longrightarrow\to{\nabla f}$.
\itemitem{b)}Find the gradient vector of $f$ at the point $P$ $(2,-3,4)$.
\itemitem{c)}Find the directional derivative $(D_{\vec u} f)_P$ at the point $P$ $(2,-3,4)$ in the direction $\vec u=\ds{1\over \sqrt{2}}\ \vec i-\ds{1\over \sqrt{2}}\ \vec k$.
\itemitem{d)}In which direction $\vec v$ does the directional derivative
$(D_{\vec v} f)_p$ have its minimum value at the point $P$.

\itemitem{e)}Give the equation of the level surface of $f(x,y,z)$ through the point $P$.

\itemitem{f)}Find the normal vector to the level surface at $P$.

\itemitem{g)}Find the tangent plane to the level surface at $P$.

\itemitem{h)}Give a critical point of $f(x,y,z)$ and state whether or not it gives a maximum or minimum value of $f(x,y,z)$.

\vfill
\item{(10)\quad 3.}Let $w=f(x,y,z)$ be a differentiable function and $x,y,z$ be differentiable functions of $t$.

\itemitem{a)}Express $\ds{dw\over dt}$ by the chain rule.

\itemitem{b)}Express $\ds{dw\over dt}$ as the dot product of a gradient vector field and a velocity vector field of a path.

\itemitem{c)}What is the rate of change with respect to $t$ of $f(x,y,z)=x+y^2+z^3$ along the path $t\vec i+t\vec j+2\vec k$ at the point $(0,0,2)$?

\vfill\eject
\item{(10)\quad 4.}Consider the curve defined by the intersection of the surfaces $x^2+y^2+z^2=2$ and $x^2+2y^2=1$.
Find a tangent vector $\vec T$ for this curve at the point $(1,0,1)$.

\vfill
\item{(5)\quad 5.a)}Write down the second order (quadratic) Taylor's formula for the function $f(x,y)$ about $(0,0)$.

\itemitem{b)}Suppose $f(0,0)=0,\ \overset\rightarrow\to{\nabla f}|_{(0,0)}=\vec 0$ and the second order partial derivatives are all equal to 3 at $(0,0)$.
Write down the second order Taylor's formula in this case.

\vfill
\item{(50)\quad 6.}Find the volume under the paraboloid $z=x^2+y^2$ and above the triangle in the $xy$ plane enclosed by the lines $y=x$ and $x+y=z$ and the $y$--axis.

\itemitem{a)}Sketch the region of integration.

\itemitem{b)}Set up the integral.

\itemitem{c)}Evaluate the integral.

\vfill\eject
\bye
