\documentclass{article} \title{Exams from Calculus 223} \author{J. Robbin } \date{Fall 1994} \newcommand{\ds}{\displaystyle} \newcommand{\p}{\partial} \newcommand{\0}{\mathbf{0}} \newcommand{\bfa}{\mathbf{a}} \newcommand{\bfc}{\mathbf{c}} \newcommand{\bfh}{\mathbf{h}} \newcommand{\bfi}{\mathbf{i}} \newcommand{\bfj}{\mathbf{j}} \newcommand{\bfk}{\mathbf{k}} \newcommand{\bfr}{\mathbf{r}} \newcommand{\bfn}{\mathbf{n}} \newcommand{\bfv}{\mathbf{v}} \newcommand{\bfF}{\mathbf{F}} \newcommand{\bfN}{\mathbf{N}} \newcommand{\bfR}{\mathbf{R}} \newcommand{\bfT}{\mathbf{T}} \begin{document} \maketitle \begin{quote} \em Warning: Some of this material may not be included in Math 234 for 2001. \end{quote} \bigskip \centerline{\Large Calculus 223 First Exam \\ Thursday September 29, 1994} \bigskip \noindent{I} Find an equation for the tangent plane to the surface $$ x^2-y^2+z^2=4 $$ at the point $(2,-3,3)$. \bigskip \noindent{II} (a) Find the unique critical point of the function $$ f(x,y) = x^2+ 3xy + 2y^2 -8x-11y + 30. $$ \medskip \noindent(b) Is this critical point a minimum, maximum, or saddle? \medskip \noindent(c) Does the function $f(x,y)$ take negative values? (I.e. is there a point $(x,y)$ where $f(x,y)<0$?). {\em Justify your answer.} \bigskip \noindent{III} Find $\ds \left(\frac{\p w}{\p y}\right)_x$ at $(w,x,y,z)=(0,1,2,3)$ if $$ 4x+5y+6z=32+w \qquad \mbox{ and } \qquad 7x^2+8y^2+9z^2=w+120e^w. $$ \bigskip \noindent{IV} Find the polynomial of degree $2$ which best approximates $$ f(x,y)= \sin(xy) $$ near $(x,y)=(1,\pi)$. \bigskip \noindent{V} Find the absolute maximum and the absolute minimum of $$ f(x,y) = (x-1)(y-2) $$ in the closed triangle $0\le x$, $0\le y$, $x+y\le 7$ bounded by the $x$-axis, the $y$-axis, and the line $x+y=7$. \bigskip \noindent{VI} Find the point on the ellipse $2x^2+3y^2=11$ where the function $f(x,y)=8x-6y$ achieves its maximum. \bigskip \noindent{VI}\footnote{This problem was not included on the exam.} Let $T=f(x,y)$ be the temperature at the point $(x,y)$ on the circle $$ x= \cos\theta,\qquad y=\sin\theta, $$ and suppose that $$ \frac{\p T}{\p x} = 2x-y, \qquad \frac{\p T}{\p y} = 2y-x. $$ Find where the maximum temperature on the circle occurs. {\em Be sure not to specify where the maximum occurs and not the minimum. Be sure to justify your answer.} \bigskip \hrule \medskip \centerline{\Large Calculus 223 Second Exam \\ Tuesday November 1, 1994} \bigskip \noindent{I} Evaluate $ \ds \int_{x=0}^1 \int_{y=2x}^1 \int_{z=x^3+y}^{x^2+2y} y\, dz\, dy\, dx. $ \bigskip \noindent{II} The force at the point $(x,y)$ is $$ {\bf F}(x,y) = x^2y{\bf i} + 2xy^2{\bf j}. $$ Find the work $$ W = \int_C {\bf F}\cdot{\bf T}\, ds $$ done in moving a particle from $(0,0)$ to $(2,4)$ along the curve $y=x^2$. % % Make sure students have been assigned a problem like this one. % \bigskip \noindent{III} Evaluate the integral $\ds\int\!\int_R x\,dx\,dy$ where $R$ is the triangle with vertices $(1,2)$, $(3,3)$, $(4,5)$. \bigskip \noindent{IV} The transformation $(x,y)=T(u,v)$ is given by $$ x = 1+ 2v+u^2,\qquad y = \frac{u}{3}. $$ It transforms the triangle $$ G:\qquad 0\le u \le v \le 1 $$ in $(u,v)$ space to a region $R$ in $(x,y)$ space. The region $R$ is bounded by three curves. \medskip \noindent(a) Sketch the region $R$ on the axes provided. Indicate clearly which points of the boundary of $R$ correspond to the vertices $(0,0)$, $(0,1)$, $(1,1)$ of the triangle $G$. Find equations in $(x,y)$ for each of the boundary curves. (Indicate clearly which boundary curve of $R$ corresponds to which equation.) \medskip \setlength{\unitlength}{0.1in} \begin{picture}(50,20)(0,-5) \put(1,0){\vector(0,1){6}}\put(1,7){$v$} \put(0,1){\vector(1,0){6}}\put(7,1){$u$} \put(1,1){\line(1,1){4}} \put(1,5){\line(1,0){4}} \put(6,6){$G$} \put(20,0){\vector(1,0){28}} \put(50,0){$x$} \put(23,-5){\vector(0,1){18}} \put(23,14){$y$} \thicklines % \put(10,2.1){\line(1,0){5.8}} \put(10,2){\vector(1,0){6}} % \put(10,-1.9){\line(1,0){5.8}} \put(13,3){$T$} \end{picture} \medskip \noindent(b) Evaluate $\ds \int\!\int_R (x+y^2)\,dx\,dy$. \bigskip \noindent{V} Let $C$ be the cardioid with polar equation \message{Draw PICTURE!} $$ r = 1-\cos\theta, \qquad 0\le\theta\le 2\pi. $$ \medskip \noindent(a) Evaluate the integral $$ \oint_C x\,dy $$ using the definition of line integral. You may leave your answer in the form of a single definite integral. \bigskip \noindent(b) Evaluate the integral using Green's theorem. A numerical answer is required here. \bigskip \noindent{VI} The gravitational potential at the point $(0,0,z_0)$ due to a uniform mass distribution in the spherical shell $$ R:\qquad a\le \sqrt{x^2+y^2+z^2} \le b $$ is given by the triple integral $$ \int\!\int\!\int_R \frac{dx\,dy\,dz}{f(x,y,z)} $$ where $f(x,y,z)$ is the distance from the point $(x,y,z)$ to the point $(0,0,z_0)$. \medskip \noindent(a) Show that $f(x,y,z) = \sqrt{\rho^2+z_0^2-2\rho z_0\cos\phi}$. \medskip \noindent(b) Evaluate the potential as a function of $z_0$. Hint: $dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta$. {\em Leave your answer as a definite integral in $\rho$. You will be asked to evaluate this integral on the next page.} \medskip \noindent(c) Evaluate the integral in~(b) when $z_0>b$ \medskip \noindent(d) Evaluate the integral in~(b) when $0