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Test 2
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<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, Fall 2016, Test 2</h3>
<ol>




<li> A bacterium only has local knowledge about its surroundings.  It
will always swim in the direction of the steepest nutrition gradient
at the point where it finds itself, and then re-adjust its
direction as it moves.
Below is a contour map indicating the nutrient concentration in
a petri dish.  Sketch the path that a bacterium starting at point $P$
would follow.

<br><img src="0g/T2.nutrition.jpg" style="width: 400px"><br>


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<li style="margin-top: 2ex;">Meh!
Draw a contour map for the function $f(x,y)=x+2y-1$, with at least 4 labeled contours.
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<li>
A metal plate is situated in the $xy$ plane and occupies the rectangle
$0 \leq x \leq 10,\ 0\leq y \leq 8$
where $x$ and $y$ are measured in meters.
The temperature at the point $(x,y)$ in the plate is
$T(x,y)$, where $T$ is measured in degrees Celsius.
Temperatures at equally spaced points were
 measured and recorded. A portion of those measurements is shown in the table below.<br>
<img src="0g/T2.temptable.jpg" style="width: 400px;"><br>
<ol type="a">
<li>Estimate the values of the partial derivatives $T_x(6,4)$ and $T_y(6,4)$.
Give <i>units</i> with your
answers.

<li style="margin-top: 20ex">
 Estimate the value of $T(7,5)$.  Preferably, use the values you calculated above in a) to do this. Show your work.

</ol>
<li>*OR*
Suppose that $f(x, y) = e^{x − y}$ and $f (\ln 2, \ln 2) = 1$. Use the technique of linear approximation (not just your calculator!) to
estimate $f (\ln 2 + 0.1, \ln 2 + 0.04)$.

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<li style="page-break-before:always">
Several level curves for the function $f(x,y)=e^x +y$ are shown below.
<img src="0g/T2.contours.jpg" class="rightalign"></br>
<ol type="a">
<li>What is the value of $f(x,y)$ on the each of the (not necessarily equally spaced) level
curves shown? Label each level curve with the value of $f(x,y)$ <i>on</i> that level curve.
<li>
Find a formula for the level curve that passes through the point $(0,1)$.

<li style="margin-top:10ex;">At the point $(0,1)$ draw and label the vector
$\myv \grad f(0,1)$. Be accurate as to the direction and
length of this vector.


<li style="margin-top:10ex;">
On the graph above draw and label a vector that starts at $(0,1)$
and points in a direction
in which $f(x,y)$ remains constant.
What are the components of the vector you found?

<li style="margin-top:10ex;">
Find the directional derivative of $f(x,y)$ at the point (0,1)
in the direction of $\myv v=3\hat i +4\hat j$.

<li style="margin-top:10ex;"> What is the maximum possible value of the
directional derivative at $(0,1)$?

</ol>

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<li style="page-break-before:always">
Consider the surface $f(x,y)=4+x^3+y^3-3xy$. A contour plot for this function is shown.
<ol type="a">
<li><img src="0g/T2.funkylandscape.jpg" class="rightalign">Find $f_x$ and $f_y$.
<li style="margin-top:24ex;"> 
Find the critical points for this function.

<li style="margin-top:20ex;"> 
Determine whether each critical point is a local and/or global maximum; local and/or global minimum, or saddle point. (No preferable method, but justify your answer).

<li style="margin-top:18ex;"> 
Label some point on the graph where $f_{xy}<0$ if possible.<br>
Label some point on the graph where $f_{xy}>0$ if possible.<br>

</ol>

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<li style="page-break-before:always">
Consider the integral 
$\int_0^2 \int_{y=x^2}^{2x} \left(  x^2+4y \right)\,dy\,dx$.
<ol type="a">
<li>
Sketch the region of integration, labeling all important points.

<li style="margin-top:30ex;"> 
Evaluate the integral by hand, showing all steps in the computation, including
antiderivatives.

<li style="margin-top:22ex;"> 
 Write the integral that reverses the order of integration. (You <b>do not need to evaluate it</b>.)
</ol>

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<li style="page-break-before:always">
Find the extreme values of $f(x,y)=9-x^2-y^2$ subject to the constraint
$x+y=3$ by
performing the following steps.
<ol type="a">
<li>
Give the Lagrange equations that need to be satisfied.

<li style="margin-top:22ex;">
 Solve the Lagrange equations. (There should be a unique critical point, P.)

<li style="margin-top:22ex;">
Find the value of $f(x,y)$ at the point $P$ you found in part b.)

<li style="margin-top:16ex;">
Determine if $P$ corresponds to a minimum or maximum value of $f$.<br>
<p style="margin-top:6ex;"><img src="0g/T2.Lagrangehint.jpg" style="width:200px" class="rightalign"><i>Hint</i>: Do the values of $f(x,y)$ increase or decrease
as $(x,y)$ moves away from P along the line $x+y=3$? The figure shows some level curves of $f(x,y)$, together with the graph of the line $x+y=3$ representing the constraint.
</ol>


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