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\line{Math 174\hfill Exam II\hfill 3/5/99\hfill D.~Gottlieb}
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\leftline{Name\ {\hbox to 3truein{\hrulefill}}}
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\item{1.}a)\ Give an example of a set in the plane which is open

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AND
\itemitem{b)}Give an example of a set which is closed in the plane.

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\item{2.}Do one of the two questions.
\itemitem{a)}A function $f(x,y)$ is continuous at the point $(x_0,y_0)$ if what three conditions are satisfied?

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OR

\itemitem{b)}Consider the function $\ds\cases f(x,y)=\ds{x^4\over x^4+y^2}\\
f(0,0)=0.\endcases$ for $x$ and $y$ not both zero.
Is this function continuous at $(0,0)$?

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\item{3.}a)\ Suppose $f(x,y)$ and all partial derivatives are defined in an open neighborhood of a point $(a,b)$.
What does Euler's theorem say about $f_{xy}$?

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OR

\itemitem{b)}Let $f(x,y)=xy^3 e^x\cos y+y^4\ln (x)$.
What number does $f_{xx}(f_{xy}-f_{yx})$ equal when evaluated at $(\pi/2, 1)$?

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\item{4.}a)\ Suppose $f(x,y,z)$ is a differentiable function so that $f_x(a,b,c)=2,\ f_y(a,b,c)=-1$ and $f_z(a,b,c)=0$.
If the values of $b$ and $c$ are changed by $dy=1$ and $dz=.1$, what change $dx$ of $a$ will result in $df=0$?

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OR

\itemitem{b)}Let $f(x,y,z)=(e^{x+y})z+\sqrt{x^2+y^2} \ln z$.
Given $dy=.1,\ dz=.1$ and $df=0$ at the point $(3,2,1)$, what value must $dx$ have?

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\item{5.}Suppose $f(x,y,z)=x^2+y^2+z^2$ and $g(x,y,z)=x+y+z$.
Sketch the level surfaces $f(x,y,z)=f(1,-1,1)$ and $g(x,y,z)=g(1,-1,1)$.

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\item{6.}Find the gradient vectors of $f(x,y,z)=x^2+y^2+z^2$ and $g(x,y,z)=x+y+z$ at the point $(1,-1,1)$ and find a vector perpendicular to both of them.

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\item{7.}a)\ Which function $f(x,y,z)=x^2+y^2+z^2$, or $g(x,y,z)=x+y+z$ has the fastest rate of increase at $(1,-1,1)$.

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AND

\itemitem{b)}What is the tangent vector to the curve of intersection at $(1,-1,1)$?

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\item{8.}At what point is the maximum value of $g(x,y,z)=x+y+z$ subject to the constraint $f(x,y,z)=x^2+y^2+z^2=3$ taken on?

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\item{9.}Find the equation of the tangent plane to the sphere $x^2+y^2+z^2=\ds{1\over 3}$ at the point $\ds{\bigg({1\over 3}, {1\over 3}, {1\over 3}\bigg)}$.

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\item{10.}Use Taylor's formula for $f(x,y)=e^x\cos y$ to find a quadratic approximation near the origin.

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\bye
