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{\it Math 32A - Midterm 1}

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1. (10 points) Suppose that a swan (not a duck!!) is swimming in a
circle described by the path $$c(t) = (cos t, sin t) ,  \, \, \, \,
t \in \mathbb{R}^+$$

and that the water temperature is given by $$T(x,y) = x^2e^y \, \,
\, \, \, \, x,y \in \mathbb{R}$$


(a) Find $DT(x,y)$ and explain what this represents.

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(b)  Find  $D(T\circ c)(t)$ and explain what this represents.

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2.  (10 points)  Determine whether the following statements are true
or false. If a statement is true, indicate how you could show it. If
a statement is false,  provide a counterexample.

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(a)  If $f(x,y,z)=x^4 + xy + z^3$, then $\nabla$$f$(1,0,1) is
perpendicular to the surface $f$=2 at the point (1,0,1).

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 This statement is (circle one): \,\,\,\,\,\,\,\,\,\,true
\,\,\,\,\,\,\,\,\,false

Explanation or counterexample:

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(b)  If $f(x,y)$=ln$y$, then $\nabla$$f(x,y)$=$\frac{1}{y}$

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This statement is (circle one): \,\,\,\,\,\,\,\,\,\,true
\,\,\,\,\,\,\,\,\,false

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Explanation or counterexample:

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3. (10 points) Find the absolute maxima and minima of the function
$$f(x,y)=5x^2-2y^2+2$$

on the disk $x^2+y^2\leq1$

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4. (10 points) Pictured are a contour map of $f$ and a (dashed)
curve with equation $g(x,y)=8$. Estimate the maximum and minimum
values of $f$ subject to the constraint that $g(x,y)=8$. Explain
your choices.





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