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\centerline{\bf MAT 205 SPRING 2000}
\centerline{\bf FINAL EXAM}

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Name:
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SSN:
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\centerline{\bf THERE ARE TEN PROBLEMS.  SHOW YOUR WORK!}
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1.  Let $z=x\cos y$, $x=t^4$, $y=e^{2t}$.  Find $\frac{dz}{dt}$.
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2.  Find the equation of the plane containing the points $(0,2,0)$,
$(1,1,0)$, and $(-1,0,1)$.  
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3.  Find the derivative of the function $f(x,y)=e^{2x+y}$ in the direction
of the unit vector $\frac{\sqrt{3}}{2} {\bf i} +
\frac{-1}{2} {\bf j}$ at the point $(1,-1)$. 
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\noindent
4.  The velocity vector of a particle is given by ${\bf v(t)}=3t^2 {\bf i}
+
e^{2t} {\bf j} + t{\bf k}$.  Find ${\bf r(1)}$, the position vector of
the
particle when $t=1$,
given that ${\bf r(0)}={\bf i} + {\bf j}$.  
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\noindent
5.  Compute 
$$
\int \int_D x \cos x^2 dA
$$
where $D$ is bounded by $y=0$, $y= \pi$, $y=x^2$, and $x \geq 0$.  
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6.  Calculate
$$
\int \int_S {\bf F} \cdot {\bf dS}
$$ 
where ${\bf F}(x,y,z)=ye^{z^2} {\bf i} + y^2 {\bf j} + e^{xy} {\bf
k}$
and $S$ is the surface of the solid bounded by the cylinder $x^2+y^2=9$
and the planes $z=0$ and $z=3$  given the outward orientation.
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\noindent
7.  Calculate
$$
\int \int_S curl {\bf F} \cdot {\bf dS}
$$
where ${\bf F}(x,y,z)=yz {\bf i} + xz {\bf j} + x^2y^2 {\bf k}$ and $S$ is
the part of the paraboloid $z=x^2+y^2$, oriented upward, that lies inside the cylinder $x^2+y^2=4$.
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You will need to use the identity 
$\int \cos^2 t dt = \frac{1}{2} t + \frac{1}{4} \sin 2t + C$
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8.  Evaluate 
$$ 
\int_C xy \, dx + y^5 \, dy
$$ 
where $C$ is the triangle with vertices $(0,0)$, $(2,0)$, and $(2,1)$
positively oriented.
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\noindent
9.  Let ${\bf F} = (x+y^2){\bf i} + (2xy + y^2) {\bf j}$.\\
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1.  Is ${\bf F}$ conservative?
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2.  Find a function $f(x,y)$ such that $\nabla f = {\bf F}$.
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3.  Find $\int_C {\bf F} \cdot {\bf dr}$ where $C$ is any curve starting
at $(0,0)$ and ending at $(1,1)$.
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10.  Evaluate $$\int_C 3y \, ds$$ where C is the curve paramatrized by
$x=t$, $y=t^3$, $0 \leq t \leq 1$.


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