
\input calc
\input epsf


%% TITLE
{\bf Math 225: Calculus III}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam II}\quad{\sl November 1, 1990}\hfill Score:\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 16 multiple
choice questions worth 6 points each. You start with 6 free points.

\bigskip\bigskip



%1
%% PROBLEM
Find the limit $\lim_{(x,y)\to(1,1)}{x^2-y^2\over x-y}$.
%% CORRECT
{$2$}
%% WRONG
{$0$}
%% WRONG
{$1$}
%% WRONG
{$-1$}
%% WRONG
{{\sl does not exist}}

%2
%% PROBLEM
Let $f(x,y) = y^x$. Compute $f_{xy}(2,e)$.
%% CORRECT
{$3e$}
%% WRONG
{$1$}
%% WRONG
{$e$}
%% WRONG
{$2e$}
%% WRONG
{$0$}

%3
%% PROBLEM
A rectangular sheet of metal is expanding. When the width is 2~inches and the length is 3~inches, the width is increasing at the rate of 0.25~in/hr and the length at 0.5~in/hr. At what rate is the area of the rectangle increasing?
%% CORRECT 
{$1.75 {\rm\ in^2/hr}$}
%% WRONG
{$0.75 {\rm\ in^2/hr}$}
%% WRONG
{$3 {\rm\ in^2/hr}$}
%% WRONG
{$4.5 {\rm\ in^2/hr}$}
%% WRONG
{$6 {\rm\ in^2/hr}$}


%4
%% PROBLEM
Suppose that $z$ is a function of $u$ and $v$, and that $u = \cos(xy)$ and $v = \sin(x/y)$. If $z_u(-1,0) = 3$ and $z_v(-1,0) = 1$, find $\d{z}/dy$ when $x = \pi$ and $y = 1$.
%% CORRECT
{$\pi$}
%% WRONG
{$1$}
%% WRONG
{$0$}
%% WRONG
{$-\pi$}
%% WRONG
{$3\pi$}

%5
%% PROBLEM
Which of the following represents the graph of the function $f(x,y) = 12y - y^3 - 3x^2$?
%% CORRECT
{\epsfxsize=1.5truein\epsfbox{exam2.1.eps}}
%% WRONG
{\epsfxsize=1.5truein\epsfbox{exam2.3.eps}}
%% WRONG
{\epsfxsize=1.5truein\epsfbox{exam2.2.eps}}
%% WRONG
{\epsfxsize=1.5truein\epsfbox{exam2.4.eps}}
%% WRONG
{\epsfxsize=1.5truein\epsfbox{exam2.5.eps}}




%6
%% PROBLEM
Suppose $w(u,v)$ is a function of $u$ and $v$ and that
$\d{w}/du = 4u^3v^2 - 2uv^4$, 
$\d{w}/dv = 2u^4v - 4u^2v^3$.
The equation $w(u,v) \equiv 0$ defines $u$ as a function of $v$. Find ${du\over dv}$.
%% CORRECT
{${u(2v^2 - u^2)\over v(2u^2 - v^2)}$}
%% WRONG
{$0$}
%% WRONG
{${2v(v^2 - u^2)\over u(u^4 - 6v^2)}$}
%% WRONG
{${v(v^2 - 6 u^2)\over 4u(u^2 - v^2)}$}
%% WRONG
{$-{2u^3v^2 - 2uv^3\over 2u^4v - 4u^2v^3}$}

%7
%% PROBLEM
Compute the derivative of $f(x,y) = \ln(x^2y - y^2 - 2)$ in the direction $3\i + 4\j$ at the point
$(2,1)$.
%% CORRECT
{$4$}
%% WRONG
{$12\i + 8\j$}
%% WRONG
{$4\i + 2\j$}
%% WRONG
{$20$}
%% WRONG
{{\sl does not exist}}

%8
%% PROBLEM
Find the direction in which the function
$f(x,y) = x^3 - y^2 x - 2 x y^2$
increases most rapidly at the point $(1,-1)$.
%% CORRECT
{$\j$}
%% WRONG
{$\i$}
%% WRONG
{$3x^2\i - 2xy \j$}
%% WRONG
{$(x^2 - y)\i - x\j$}
%% WRONG
{$2\i -\j$}

%9
%% PROBLEM
Determine the equation of the tangent plane to the paraboloid
$z = 9 - 4x^2 - y^2$
at the point $(1,1,4)$.
%% CORRECT
{$8x + 2y + z = 14$}
%% WRONG
{$x + y + 4z = 18$}
%% WRONG
{$4x + 2y + z = 10$}
%% WRONG
{$x + 8y + 4z = 25$}
%% WRONG
{$x + 4y + z = 9$}


%10
%% PROBLEM
Compute the differential of the function
$f(x,y,z) = x \cos({y\over z})$.
%% CORRECT
{$\cos({y\over z})\,dx-{x\over z}\sin({y\over z})\,dy+{xy\over z^2}\sin({y\over z})\,dz$}
%% WRONG
{$\cos({y\over z})+{x\over z}\sin({y\over z})+{xy\over z^2}\sin({y\over z})$}
%% WRONG
{$\cos({y\over z})\,dx-\sin({y\over z})\,dy-\sin({y\over z})\,dz$}
%% WRONG
{$\sin({y\over z})-{x\over z}\cos({y\over z})+{xy\over z^2}\sin({y\over z})$}
%% WRONG
{$\sin({y\over z})\,dx-{y\over z}\cos({x\over z})\,dy$}

%11
%% PROBLEM
Determine the critical points of the function
$f(x,y) = 3x^2 - 3xy^2 + 2y^3$.
%% CORRECT
{$(0,0)$, $(2,2)$}
%% WRONG
{$(0,0)$, $(2,0)$, $(2,2)$}
%% WRONG
{$(0,0)$, $(2,0)$}
%% WRONG
{$(0,0)$, $(2,2)$, $(-2,2)$}
%% WRONG
{$(0,0)$, $(2,0)$, $(2,2)$, $(-2,2)$}


%12
%% PROBLEM
The function
$f(x,y) = 2x^3 - 6x + 2y^3 - 3xy^2$
has a critical point at $(\sqrt{2},\sqrt{2})$. Use the Second Partials Test to determine which of the following is true at this point:
%% CORRECT
{{\sl $f$ has a relative minimum}}
%% WRONG
{{\sl $f$ has a relative maximum}}
%% WRONG
{{\sl $f$ has a saddle point}}
%% WRONG
{{\sl test is inconclusive}}
%% WRONG
{{\sl none of the above}}

%13
%% PROBLEM
Find the maximum of the function
$f(x,y) = y^3 + 3x^2 y$
in the region
$x^2 + y^2 \le 4$.
%% CORRECT
{$8\sqrt{2}$}
%% WRONG
{$8$}
%% WRONG
{$0$}
%% WRONG
{$16$}
%% WRONG
{$16\sqrt{2}$}


%14
%% PROBLEM
Determine which of the following sets of equations must be solved to find the extreme values of the function
$f(x,y) = x^3 y + y^2 x + xy^4$
subject to the constraint
$x^4 + 4xy + y^4 = 4$.
%% CORRECT
{\vtop{
\hbox{$3x^2y + y^2 + y^4 = \lambda(4x^3 + 4y)$}
\hbox{$x^3 + 2xy + 4xy^3 = \lambda(4x + 4y^3)$}
\hbox{$x^4 + 4xy + y^4 = 4$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$3x^2y + y^2 + y^4 = \lambda(4x^3 + 4y^3)$}
\hbox{$x^3 + 2xy + 4xy^3 = \lambda(4x^3 + 4y^3)$}
\hbox{$4x^3 + 4y + 4y^3 = 0$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$3x^2y + y^2 + y^4 = \lambda(4x^3 + 4y + 4y^3)$}
\hbox{$3x^2 + 2y + 4y^3 = \lambda(4x + 4y^3)$}
\hbox{$x^4 + 4xy + y^4 = 4$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$3x^2y + y^2 + y^4 = \lambda(4x^3 + 4y^3)$}
\hbox{$x^3 + 2xy + 4xy^3 = \lambda(4x^3 + 4y^3)$}
\hbox{\strut}
}}
%% WRONG
{\vtop{
\hbox{$3x^2y + y^2 + y^4 = \lambda(4x^3 + y)$}
\hbox{$x^3 + 2xy + 4xy^3 = \lambda(x + 4y^3)$}
\hbox{\strut}
}}

%15
%% PROBLEM
Let $R$ be the triangle in the plane with vertices at the points $(0,0)$,
$(0,1)$, and $(1,1)$. Compute the double integral
$\iint_R x^2 y \,dA$.
%% CORRECT
{$1\over 10$}
%% WRONG
{$1\over 15$}
%% WRONG
{$1\over 2$}
%% WRONG
{$1\over 3$}
%% WRONG
{$1\over 5$}


%16
%% PROBLEM
Reverse the order of integration of the iterated integral
$\int_0^1\int_{\sqrt{y}}^1f(x,y)\dxdy$.
%% CORRECT
{$\int_0^1\int_0^{x^2}f(x,y)\dydx$}
%% WRONG
{$\int_1^{\sqrt{x}}\int_0^1f(x,y)\dydx$}
%% WRONG
{$\int_0^1\int_{\sqrt{x}}^1f(x,y)\dydx$}
%% WRONG
{$\int_{\sqrt{y}}^1\int_0^1f(x,y)\dydx$}
%% WRONG
{$\int_0^1\int_{x^2}^1f(x,y)\dydx$}



%% END
\bye
