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\noindent{\bf Department of Mathematics} \hfill 21-259 Calculus in 3-D

\noindent {\bf CARNEGIE MELLON UNIVERSITY}\hfill W. Hrusa Spring 2003\ 

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\begin{center}
{\bf Review Problems for Test 2}

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\begin{enumerate} 
\item  Consider the function $f$ defined by $f\left( x,y,z\right) =xy+y^2z.$

\begin{enumerate}
\item[a)]  Find $\stackrel{\rightharpoonup }{\nabla }f.$

\item[b)]  Find the directional derivative of $f$ at the point $\left( 2,3,-1\right) $ in the direction of the vector $\stackrel{\rightharpoonup }{v}\,=\left\langle 2,4,-3\right\rangle .$
\end{enumerate}

\item  Suppose that $f$ is a smooth function with the following property: At the point $\left( 0,0,0\right) ,$ $f$ decreases most rapidly in the direction of $\left\langle 6,-1,2\right\rangle $ and the rate of change of $f $ in this direction is $-5.$ Find $\stackrel{\rightharpoonup }{\nabla }f\left( 0,0,0\right) .$

\item  Find the tangent plane and normal line to the surface $z=x^2+4y^2$ at the point $P_0=\left( -1,2,17\right) .$

\item  Let $\stackrel{\rightharpoonup }{u}\,={\displaystyle \frac{1}{\sqrt{2}}}\left\langle 1,1\right\rangle $ and $\stackrel{\rightharpoonup }{v}\,={\displaystyle \frac{ 1}{\sqrt{2}}}\left\langle 1,-1\right\rangle .$

Assume that $f_x$ and $f_y$ are continuous. If $D_{\stackrel{\rightharpoonup }{u}}f\left( 2,1\right) =2\sqrt{2}$ and $D_{\stackrel{\rightharpoonup }{v}}f\left( 2,1\right) =-8\sqrt{2},$ find $\stackrel{\rightharpoonup }{\nabla }f\left( 2,1\right) .$

\item  Find all points on the surface $x^2+4y^2-z^2=4$ at which the tangent plane is parallel to the plane $x+2y+z=6.$

\item  Suppose that $w=x^3y+y^2-4z^3,\;x=rs,\;y=\left( r^2-s^2\right) ^3,$ and $z=2r+s.$ Use the chain rule to compute ${\displaystyle \frac{\partial w}{\partial s}}.$  (Express your answer in terms of $r$ and $s$ alone. Algebraic simplication is not necessary.)

\item  Suppose that $w=f\left({\displaystyle  \frac{ y}{y-x}}\right) $ where $f$ is continuously differentiable. Show that
\[
x{\displaystyle  \frac{\partial w}{\partial x}}+y{\displaystyle \frac{\partial w}{\partial y}}=0,\;x\not =y. 
\]

\item  Assume that $u$ and $v$ have continuous second order partial derivatives and that 
\[
{\displaystyle  \frac{\partial u}{\partial x}}={\displaystyle \frac{\partial v}{\partial y}},\;{\displaystyle \frac{\partial u}{\partial y}}={\displaystyle \frac{-\partial v}{\partial x}}. 
\]
Show that 
\[
{\displaystyle \frac{\partial ^2u}{\partial x^2}}+{\displaystyle \frac{\partial ^2u}{\partial y^2}}=0\ {\rm and}\ {\displaystyle \frac{\partial ^2v}{\partial x^2}}+ {\displaystyle \frac{\partial ^2v}{\partial y^2}}=0. 
\]

\item  Assume that $w=f\left( u\right) ,$ where $u=x^2+y^2$ and $f$ is a smooth function. Find an expression for ${\displaystyle \frac{\partial ^2w}{\partial x^2}}+ {\displaystyle \frac{\partial ^2w}{\partial y^2}}$ in terms of $u,\;{\displaystyle \frac{dw}{du}},$ and ${\displaystyle \frac{d^2w}{du^2}}.$

\item  Find all local maxima, local minima, and saddle points for the
function $f$ defined by

\begin{enumerate}
\item[a)]  $f\left( x,y\right) =x^2+xy+y^2+x-y+2,$

\item[b)]  $f\left( x,y\right) =3x^2-3xy^2+y^3+3y^2.$
\end{enumerate}

\item  Let $R$ be region described by 
\[
0\leq y\leq \sqrt{16-x^2},\;0\leq x\leq 4 
\]
Find the absolute maximum and the absolute minimum values of 
\[
f\left( x,y\right) =x^2+y^2-4x-2y+1 
\]
on $R.$
\item Use Lagrange multipliers to find all possible maxima and minima of $f$ subject to the given constraint.

\begin{enumerate}
\item[a)] $f(x,y,z) = \frac{2}{3}\, x^3 + y - 3z$; \newline
$x^2+y^2+z^2 = 10$

\item[b)] $f(x,y) = xy;\ 9x^2+y^2=4$
\end{enumerate}
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\item  Write an equivalent double integral with the order of integration
reversed.

\begin{enumerate}
\item[a)]  ${\displaystyle \int_0^1} {\displaystyle \int_y^{\sqrt{y}}}f\left( x,y\right) dxdy$

\item[b)]  ${\displaystyle \int_0^2}{\displaystyle \int_1^{e^{2x}}}f\left( x,y\right) dydx$

\item[c)]  ${\displaystyle \int_0^2}{\displaystyle \int_{2x}^4}f\left( x,y\right) dydx$
\end{enumerate}

\item  Evaluate each of the following.

\begin{enumerate}
\item[a)]  ${\displaystyle \int_0^1}{\displaystyle \int_0^{x^3}}4x^2y\;dydx$

\item[b)]  ${\displaystyle \int_0^1}{\displaystyle \int_x^13x\sqrt{x^2+y^2}}\,dydx$
\end{enumerate}
\end{enumerate}

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