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Test 2
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<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, May 2022, Test 2</h3>
<p>You may use your page of notes / calculator / writing instruments on this test.




<ol>



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<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>

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<li>
(21 pts) 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.
<div class="result justme">
<p>$T_x\approx \frac{90-64}{8-4}=$<span class="answer">6.5 C/cm</span><br>
$T_y\approx \frac{72-84}{6-2}=$<span class="answer">-3.0  C/cm</span>
</div>
<li>Using these partial derivative estimates, write down the linear approximatiof $T(x,y)$ for points near $(6,4)$.
<div class="result justme">
$$\begineq T(x,y)&=T(6,4)+T_x(x-6)+T_y(y-4)\\
&=76 +6.5(x-6)-3(y-4)\\
\endeq$$


<li>
 Using your linear approximation, estimate the value of $T(5,3)$.
<div class="result justme">
$T(5,3)=76 +6.5(5-6)-3(3-4)=76-6.5+3=$<span class="answer">72.5 C</span>
</div>
</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="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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<!--
<b>Ernie's Emporium of Empiricism</b> is having a 50% off sale on lightly used functions of two variables from the first quadrant.  "Hey kid", Ernie tells you, "I got a great function for ya.  Impress your friends". You ask what it is, but he tells you that you have to pay first. "But get a load of these first derivatives!"
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(9) You are told that a certain function, $f(x,y)$ has these partial derivatives:
<ul>$$\frac{\del f}{\del x} = 3y^2+2x+\ln(x*y) + 37$$
$$\frac{\del f}{\del y} = 6xy+\frac xy+\cos(y^2)$$
</ul>
Carry out Clairaut's test to decide if $f(x,y)$ is a continuous function or not.  Show the results of your test, and state your conclusion about whether it is continuous or not.
<!--
It suddenly occurs to you that, even without knowing what $f(x,y)$ is, there's a test you can carry out on the first derivatives to see if $f(x,y)$ is really a continuous function, as Ernie claims, or not. Carry out the test and state whether it's safe to buy or not.
-->

<div class="result justme">
For a continuous function, Clairaut says $f_{xy}=f_{yx}$.
Calculating:
$$f_{xy}=\frac{\del}{\del y}f_x = \frac{\del}{\del y}3y^2+2x+\ln(x*y) + 37=6y+0+\frac{1}{xy}x +0.$$
$$f_{yx}=\frac{\del}{\del x}f_y = \frac{\del}{\del x}6xy+\frac xy+\cos(y^2)
=6y+\frac{1}{y}+0.$$
They are equal, so <span class="answer">$f(x,y)$  is a continuous function</span>. (We know this, even without knowing what $f(x,y)$ is!)

</div>
<li style="margin-top: 42ex">(12) Consider the function
$$f(x,y)=\frac{x+y}{|x|+|y|}$$
(a) Evaluate the function at the  following locations:
<ol type="i">
<li>$f(0.1,0.1)$<span class="answer justme">=1</span>
<li>$f(0.1,-0.1)$<span class="answer justme">=0</span>
<li>$f(-0.1,0.1)$<span class="answer justme">=0</span>
<li>$f(-0.1,-0.1)$<span class="answer justme">=1</span>
</ol>
(b) Either find $\lim_{(x,y)\to (0,0)}f(x,y)$ or show that it doesn't exist.
<p class="result justme">Replacing each 0.1 above with 0.0[increasing number of zeroes]1 above changes none of the answers, but all of these calculations are getting ever closer to (0,0). For each approach  their limits are identical to the numbers above, but the limits are not unique, so <b class="answer">the limit does not exist</b></p>



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<li style="page-break-before:always">(24)
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  each of the level
curves shown? Label each level curve with the value of $k$ <i>on</i> that level curve.
<li>
Find a formula, $y=f(x)$, for the level curve shown 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 $\myv u=3\hat i +4\hat j$.


<li> What is the maximum possible value of any
directional derivative  at $(0,1)$?

</ol>

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<li style="page-break-before:always">(15)
The following is a contour graph of a function $z=f(x,y)$.  Values $z=c$ are indicated on the plot.<br>
<img src="0g/CPslopes.jpg" style="width: 500px"><br>
Estimate from the contour plot the values of these partial derivatives at point $C$.
<ul>$f_x=?$
<p style="margin-top: 10ex">$f_y=?$
</ul>
<p style="margin-top: 10ex">Determine whether the following derivatives are positive or negative or 0 at the point $C$.  <i>Explain</i> your answers.
<ul>$f_{xx}$
<p>$f_{yy}$
<p>$f_{xy}$
</ul>


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<li style="page-break-before:always">(12)
<ol type="a">
<li>Find $A$ and $B$ so that $f(x,y)=x^2+Ax+y^2+B$ has a critical point at (7,0) with $z$-coordinate 5.
<li>Is the critical point a maximum, a minimum, or neither?
</ol>
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$\int_0^2 \int_{y=\frac12 x^2}^{x} \left(  x+3y^2 \right)\,dy\,dx$.
<ol type="a">
<li>
Sketch the region of integration in the $xy$ plane, labeling all important points, and functions you graph. 

<li>Evaluate the integral by hand, showing all steps in the computation, including
antiderivatives.  Your answer should be a single fraction or decimal number (to the nearest thousand'th).

<li>
 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; margin-top:0ex;padding-top:0;">
(20) Evaluate the integral below by hand, showing all steps in the computation, including
antiderivatives.  Your answer should be a single fraction or decimal number (to the nearest thousand'th).
$$\int_0^2\int_{y=-1}^{1} (3x+y^2)\,dy\,dx$$




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<li style="page-break-before: always; margin-top:0ex;padding-top:0;">
(17)
In the figure, the contour plot of $f(x,y)=x^2+xy+y^2$ is shown.  (Dark blue are regions where $f$ is low and in orange-yellow-colored regions, $f$ is high.)
Also, a path (in red) given by $g(x,y)=y-x^2=-3$ is drawn in.

<ol type="a"><li>Find the coordinates of all the critical points of the function $f(x,y)$.  For each one, state whether it is a maximum, a minimum, or neither.  If a maximum or minimum, state whether it is a global or local one.  Mark the position of any that would appear within the bounds of the diagram.

<li>The red line is a path through the "landscape" of the surface, $f$, going up and down, according to the values of $f$ as indicated by the contour plot. Estimate the position(s) of any critical points $f(x,y)$ along the red path. Mark and label each one on the path on the diagram.  Determine whether each is local maxima, local minima, or neither and write down your determination for your labelled points(s).
</ol>
<img src="0g/LM3.jpg">


</ol>
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