
\input calc
\input epsf
\def\problemspace{0.75truein}

%% TITLE
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam II}\quad{\sl March 23, 1995}\hfill Section:\hbox to 2.5truein{\hrulefill}

\bigskip\bigskip
Record your answers to the multiple choice problems by placing an
$\times$ through one letter for each problem on this answer sheet.
There are 15 multiple choice questions worth 6 points each. You
start with 10 points.

\bigskip\bigskip

\def\answerspace{0.25in}


%1
%% PROBLEM
The force of gravity on a spacecraft located at $(x,y,z)$ is $F(x,y,z)
= 3125/(x^2+y^2+z^2)$. Suppose the spacecraft's position and velocity
at time $t=1$ are $\p = 9\j + 12\k$ and $\v = 10\i - 12\j + 90\k$,
respectively. Find ${dF\over dt}$ at time $t=1$.

%% CORRECT
{$-120$}
%% WRONG
{$0$}
%% WRONG
{$(0, -10/9, -40/27)$}
%% WRONG
{$(-0.001, -0.001, -0.008)$}
%% WRONG
{$-0.107$}

%2
%% PROBLEM
Let $f(x,y,z) = x^2 y - y^2z$. Compute the derivative of $f$
at the point $(1,2,0)$ in the direction $4\i+ 3\k$.
%% CORRECT
{$4/5$}
%% WRONG
{$4$}
%% WRONG
{$4/\sqrt{33}$}
%% WRONG
{$2$}
%% WRONG
{$4/\sqrt{14}$}

%3
%% PROBLEM
Let $f(x,y,z) = e^{-yz}\cos(xy)$. Compute $\grad(f)$ at the point $(\pi,1,0)$.
%% CORRECT 
{$\k$}
%% WRONG
{$-\k$}
%% WRONG
{$\0$}
%% WRONG
{$\j$}
%% WRONG
{$-\j$}


%4
%% PROBLEM
Determine the equation of the plane tangent to the ellipsoid
$$x^2 + 2y^2 + 3z^2 = 20$$ at the point $(3,2,1)$
%% CORRECT
{$3x + 4y + 3z = 20$}
%% WRONG
{$2x + 4y + 6z = 0$}
%% WRONG
{$2(x-3) + 4(y-2) + 6(z-1) = 0$}
%% WRONG
{$3x + 4y + 3z = 0$}
%% WRONG
{$2x(x-3) + 4y(y-2) + 6z(z-1) = 0$}

%5
%% PROBLEM
Find the critical points of the function $f(x,y)
= 2x^3y - 6 x y + 3y^2$.
%% CORRECT
{$(0, 0), (1,2/3), (-1,-2/3), (\sqrt{3}, 0), (-\sqrt{3}, 0)$}
%% WRONG
{$(0, 0), (1,2/3), (-1,2/3), (1,-2/3), (-1,-2/3)$}
%% WRONG
{$(1,2/3), (-1,2/3), (1,-2/3), (-1,-2/3)$}
%% WRONG
{$(0, 0), (\sqrt{3}, 0), (-\sqrt{3}, 0)$}
%% WRONG
{$(0, 0), (1,2/3), (-1,2/3), (1,-2/3), (-1,-2/3), (\sqrt{3}, 0), (-\sqrt{3}, 0)$}



%6
%% PROBLEM
Choose the statement below that applies to the function
$$f(x,y) = x^3 y - x y^2 + 2 x y$$
%% WRONG
{$f$ has a critical point at $(\sqrt{2},0)$}
%% WRONG
{$f$ has a local maximum at $(0,0)$}
%% WRONG
{$f$ has a local minimum at $(0,0)$}
%% CORRECT
{$f$ has a saddle point at $(0,2)$}
%% WRONG
{{\sl none of the above}}
%% STET

%7
%% PROBLEM
Find the maximum value of the function
$f(x,y) = (1-y^2)\log(1+x^2)$
on the rectangle $-1\le x \le 1$, $-1\le y \le 1$.
%% CORRECT
{$0.693$}
%% WRONG
{$0.285$}
%% WRONG
{$1.176$}
%% WRONG
{$0.104$}
%% WRONG
{$0.922$}

%8
%% PROBLEM
Find the extreme values of the function $f(x,y) = 1 + x^2 y$
on the unit circle $x^2 + y^2 = 1$.
%% CORRECT
{1.385 and 0.615}
%% WRONG
{1.570 and 0.301}
%% WRONG
{1.125 and 0.978}
%% WRONG
{1.404 and 0.839}
%% WRONG
{1.829 and 0.532}


%9
%% PROBLEM
Let $R$ be the region between the $x$-axis and $y = x$ for $0 \le x \le 1$.
Compute $\iint_R 6 y e^{x^3} \, dA$.
%% CORRECT
{1.718}
%% WRONG
{1.415}
%% WRONG
{2.843}
%% WRONG
{0.286}
%% WRONG
{1.031}


%10
%% PROBLEM
Reverse the order of integration of the integral
$$\int_1^5\int_{\sqrt{y-1}}^2 f(x,y)\,dx\,dy$$
%% CORRECT
{$\int_0^2\int_1^{1+x^2} f(x,y)\,dy\,dx$}
%% WRONG
{$\int_{\sqrt{y-1}}^2\int_1^5 f(x,y)\,dy\,dx$}
%% WRONG
{$\int_0^1\int_2^{1+x^2} f(x,y)\,dy\,dx$}
%% WRONG
{$\int^{\sqrt{y-1}}_0\int_1^2 f(x,y)\,dy\,dx$}
%% WRONG
{$\int_1^2\int_{1+x^2}^5 f(x,y)\,dy\,dx$}

%11
%% PROBLEM
Find the area of the region inside the cardiod $r = 1 + \sin(\theta)$ in
the first quadrant.
$$\epsfysize=1.75truein\epsfbox{exam2.1.eps}$$

%% CORRECT
{$1 + {3\pi\over 8}$}
%% WRONG
{$1 + {\pi\over 2}$}
%% WRONG
{$1 + {\pi\over 4}$}
%% WRONG
{$1 + {3\pi\over 4}$}
%% WRONG
{$1 + {\pi\over 8}$}


%12
%% PROBLEM
Compute the volume of the solid region under the graph of $f(x,y) = 4 - x^2 - y^2$
over the triangular region defined by $x + y \le 1$ in the first quadrant.
$$\epsfysize=2.5truein\epsfbox{exam2.2.eps}$$
%% CORRECT
$11/6$
%% WRONG
$4/3$
%% WRONG
$17/12$
%% WRONG
$5/2$
%% WRONG
$9/4$

%13
%% PROBLEM
Find the average value of the function $f(x,y) = y\sin(x y)$
over the region $0 \le x \le \sqrt{\pi}$, $0 \le y \le \sqrt{\pi}$.
%% CORRECT
{${1\over\sqrt{\pi}}$}
%% WRONG
{${1\over\pi\sqrt{\pi}}$}
%% WRONG
{$0$}
%% WRONG
{$\sqrt{\pi}$}
%% WRONG
{$\pi\sqrt{\pi}$}


%14
%% PROBLEM
Suppose $z$ is a function of $u$ and $v$ and that $u = x^2 - y^2$ and
$v = \log(x - y)$. If $z_u(3,0) = -3$ and $z_v(3,0) = 5$, compute 
$\d{z}/dx$ when $x = 2$ and $y = 1$.
%% CORRECT
{$-7$}
%% WRONG
{$28$}
%% WRONG
{$-1$}
%% WRONG
{$-23$}
%% WRONG
{$8$}

%15
%% PROBLEM
Choose the function below that has the following graph.
$$\epsfysize=3truein\epsfbox{exam2.3.eps}$$
%% CORRECT
{$f(x,y) = y^2 + x^2y - y$}
%% WRONG
{$f(x,y) = y^2 + x^2$}
%% WRONG
{$f(x,y) = x - y^2 - x^2$}
%% WRONG
{$f(x,y) = y^2 + 2xy - x$}
%% WRONG
{$f(x,y) = y^2 - x^2y$}




%% END
\bye
