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MAC 2313 Calculus 3 \hfill {\bf Test 2} \hfill 17 Oct 2002

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Show {\bf ALL} work for credit; be neat; and use only {\bf ONE}
side of each page of paper. Do {\bf NOT} write on this page.
Calculators can be used for graphing and calculating only. Give exact
answers when possible.

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1. Use the Chain Rule to find $\partial z/\partial u$ and
$\partial z/ \partial v$ when $z = \sin(x/y), x = \ln u$ and $ y = u^2-v^2$.

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2. Fixing Maple errors. Each of the following produced an error
or an empty graph, explain how to fix each.
\item {a} plot3d(exp\^{}x*sin(y),x=0..1,y=0..2*Pi);
\item {b} plot3d(x\^{}2-x*y+y\^{}2,x=0..1,y=1..1);
\item {c} f=sin(x)*y\^{}2+x\^{}2*sin(y);plot3d(f,x=-1..1,y=-1..1);
\item {d} plot3d(x y,x=-1..1,y=-1..1);
\item {e} plot3d(sin(x)*sin(y),x=0..pi,y=0..pi);

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3. For the function $f(x,y)=x^3+xy+y^2$
\item {a} Compute the quadratic Taylor polynomial for $f$ at the point $(-1,2)$. 
\item {b} Compute the equation of the normal line to $f$ at the point $(-1,2)$.

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4. The graph A is a plot of $\nabla f$, the gradient of $f$ and the graph B is
a contourplot of $g$. (Light regions have higher values than dark regions.]
Find the co-ordinates of all extrema of $f$ and $g$ and
{\bf LABEL} them as either local minimums,
local maximums or saddle points.
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5. Find the directional derivative of $f(x,y,z) = 3x^2y^2 + 2yz$ as you leave
the point $(1,-1,0)$ heading in the direction of the point $(0,1,1)$.

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6. The point $P$ is on the contour graph of
the function $f$ (below left) and the point $Q$ is on the surface of
the graph of the function $g$ (below right).
Let ${\bf u}$ be the unit vector ${\bf u} = (-{\bf i}-{\bf j})/\sqrt{2}$.

Find the sign (positive, negative or zero) of the partials of $f$:
$f_x(P), f_y(P), f_{xx}(P), f_{yy}(P), f_{xy}(P)$ and the partials of $g$:
$g_x(Q), g_y(Q), g_{xx}(Q)$
and the two directional derivatives $f_{\bf u}(P)$ and $g_{\bf u}(Q)$.


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\centerline{There is more test on the back.}
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\eject

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7. Check that the point $(-1, 1, 2)$ lies on the surface $\cos(x+y) = e^{xz+2}$
and find the equation of the tangent plane to this surface
at $(x,y,z)=(-1,1,2)$

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8. Sketch the region of integration, reverse the order of integration
and evaluate $$\int_0^1 \int_{e^x}^e {y \over \ln y} \ dy \ dx$$

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9. Find critical points of the function
$f(x,y)=(x+y)(x^2+y^2-2)$. Classify these local extrema by filling
out a table like the one below, with a separate line for each critical
point.
[Hint: Use your TI-89 to check that you got the correct collection of critical
points.]

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\vrule \hfill \quad $#$ \quad \hfill &
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\vrule \hfill \quad $#$ \quad \hfill &
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\vrule \hfill \quad # \quad \hfill \vrule \cr
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(x,y) & f_{xx} & f_{yy} & f_{xy} & big D & Classification \cr
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? & ? & ? & ? & ? & ? \cr
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}
}

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10. Use your TI-89 to plot the $z = 1$ contour of the function $z =
g(x,y) = x^2+xy+y^2$. On the same graph, plot some contour lines for
$f(x,y)=x+y$.
Use Lagrange Multipliers to find the maximum and minimum {\bf VALUES}
for $f(x,y)$ on the constraint $g(x,y)=1$.


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