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Test 2
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<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, May 2019, Test 2</h3>
<p>You may use only your page of notes / calculator / writing instruments on this part of the 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>
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="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 continuous functions of two variables.  "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!"  
<ul>$$\frac{\del f}{\del x} = 3y^2+2x+\ln(x*y) + 1$$
$$\frac{\del f}{\del y} = 6xy+\frac xy$$
</ul>
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.

<li style="margin-top: 24ex">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="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">
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>
Determine whether the following derivatives are positive or negative or 0 at the point $C$.  <i>Explain</i> your answers.
<ul>$f_x$
<p>$f_y$
<p>$f_{xx}$
<p>$f_{yy}$
<p>$f_{xy}$
</ul>


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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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<p style="page-break-before: always; margin-top:0ex;padding-top:0;">&nbsp;</p>
<p style="page-break-before: always; margin-top:0ex;padding-top:0;">&nbsp;</p>
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<h3 style="page-break-before: always; margin-top:0ex;padding-top:0;">Name-<br>
Math 213, May 2019, Test 2</h3>
<p>You may use only your page of notes / calculator / writing instruments /
 <b><i>CoCalc</i> on one of the class computers</b> (or <i>Mathematica</i> on your own computer) on this part of the test.  Write out full responses to each question, but you may use software to do any calculations you wish.

<li>In the figure, the contour 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 $g(x,y)=-3$ (there may be more than one): Write out the 3 equations that such points
must satisfy.
<li>Solve the equations for $x$, $y$, and $\color{red}{\lambda}$. You may use software to solve these equations.  
Write down the coordinates of the points on the level curve which are solutions (to the nearest hundredth), and mark them on the diagram.  Which are maxima? Which are minima? Which are neither?
</ol>
<p><img src="0g/LM3.jpg">
</ol>
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