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Test 2
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<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, May 2018, Test 2</h3>
<p>You may use only your page of notes / calculator / writing instruments on this part of the test.
<ol>




<li> A bacterium only has local knowledge about its surroundings.
It senses the nutrient concentration on different sides of its (microscopic) body, and swims in the direction of most food.  

It 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>
<li>Consider the function
$$f(x,y)=\frac{x+y}{|x|+|y|}$$
(a) Evaluate the following:
<ol type="i">
<li>$f(1,1)$
<li>$f(1,-1)$
<li>$f(-1,1)$
<li>$f(-1,-1)$
</ol>
(b) Does this function have a limit at (0,0)?
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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 style="page-break-before: always">
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 centimeters.
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 on the plate were
 measured and recorded. A portion of those measurements is shown in the table below.<br>
<img src="0g/T2.temptable2.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>Using these partial derivative estimates, write down the linear approximation of $T(x,y)$ for points near $(6,4)$.
<li>
 Using your linear approximation, calculate the value of $T(7,5)$.
</ol>
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<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 $k=f(x,y)$ on the each of the (not necessarily equally spaced) level
curves shown? Label each level curve with the value of $k$ <i>on</i> that level curve.
<li>
Find a formula for the level curve that passes through the point $(0,1)$.

<li>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>
On the graph, 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>
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> 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 integral 
$\int_0^2 \int_{y=\frac12 x^2}^{x} \left(  x+3y^2 \right)\,dy\,dx$.
<ol type="a">
<li>
Sketch the region of integration, labeling all important points.

<li> 
Evaluate the integral by hand, showing all steps in the computation, including
antiderivatives.

<li>
 Write the integral that reverses the order of integration. (You <b>do not need to evaluate it</b>.)
</ol>

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<h3 style="page-break-before: always; margin-top:0ex;padding-top:0;">Name-<br>
Math 213, May 2018, Test 2</h3>
<p>You may use only your page of notes / calculator / writing instruments /
 <b><i>Mathematica</i> on one of the class computers</b> on this part of the test.

<li>In the figure, the countour plot of $f(x,y)=x^2+xy+y^2$ and the level curve $g(x,y)=-3$ are shown, where $g(x,y)=y-x^2$.
<ol type="a"><li>Find a global maximum or a global minimum of $f$.  Prove that this is the only critical point of the function, and decide whether it is a maximum or a minimum.
<li>Calculate the gradients of $f(x,y)$ and $g(x,y)$
<li>Using the method of Lagrange multipliers, look for maxima or minima along the level curve $g(x,y)=-3$: Write out the 3 equations that such points
must satisfy.
<li>Use <i>Mathematica</i> to solve these equations.  (Actually, use <tt>NSolve</tt> rather than <tt>Solve</tt>.)
<li>Write down the coordinates of the points on the level curve which are solutions, and mark them on the diagram.  Which are maxima? Which are minima?
</ol>
<p><img src="0g/LM3.jpg">
</ol>
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