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\centerline{\textbf{\huge Final Exam MAT 272 - Spring 2003 - Ruedemann}}

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\hspace*{3.0in} NAME \hrulefill

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{\hspace*{3.0in} I.D. \hrulefill}

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\noindent
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\textit{(Please show all work for credit.  Answers without work will not receive credit. \underline{Please 
         box your final answers.)} }
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\begin{enumerate}
 
\item
(25 points) Let $\vec{u} = 2 \,\vec{i} +3 \,\vec{j} +5 \,\vec{k}$ and let $\vec{v} = -1 \,\vec{i}  - 5 \,\vec{j} + 7 \,\vec{k}$.  
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(a) Find the angle $\theta$ in radians between  $\vec{u}$ and $\vec{v}$.

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(b) Find a vector orthogonal to both $\vec{u}$ and $\vec{v}$.

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(c) Find parametric equations for the line through $(1, -1, 0)$ parallel to $\vec{u}$.





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\hspace*{1.5in} Name:\hrulefill
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\item
(15 points)  Find an equation for the plane through the point $(1, 0, -1)$ and perpendicular to the line $x = 1+t, \; y = 13 + 7t, \; z = 8 - t$.
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\item
(10 points)  Let $\vec{v} = -1 \,\vec{i}  - 2 \,\vec{j} + 2 \,\vec{k}$.  Find a unit vector in the direction of $\vec{v}$.

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\hspace*{1.5in} Name:\hrulefill
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\item
(25 points)  Let $\vec{r}(t) =  2\sin (t) \,\vec{i} + 5t \,\vec{j} + 2\cos (t) \,\vec{k}$ represent a position function.
Find the following:

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(a) The velocity at time \textit{t} .

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(b) The unit tangent vector  $\vec{T}(t)$.

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(c) The length of the curve for the time interval $0 \leq t \leq 6$





 
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\hspace*{1.5in} Name:\hrulefill
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\item
(25 points)  Find all critical points of
$$f(x,y) = 3x^2y + y^3 -3x^2 - 3y^2 + 7  $$
an classify as local maximums, local minimums, or saddle points.
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\hspace*{1.5in} Name:\hrulefill
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\item
(25 points)  Find the volume of the solid beneath the surface $z = e^{-x^2-y^2} $ and
above the circle of radius 2, centered at the origin, in the \textit{xy}-plane.  

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\hspace*{1.5in} Name:\hrulefill
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\item
(15 points)  Find directional derivative of $ f(x,y)= x^2y^5$ at the point $(1, 2)$
in the direction of $\vec{v} = \vec{i} + 3\vec{j}$. 

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\item
(10 points)  Find the direction of greatest increase for the function $g(x,y) = x^2 + y^2\ln(x) $
at the point $(2,1)$.

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\hspace*{1.5in} Name:\hrulefill
\end{flushright} 

\item
(25 points)  Evaluate the line integral $$ \dfrac{1}{2}\oint_{C} \; -y \; dx \; + \; x \; dy$$ both
(a) directly and (b) using Green's Theorem.  C is the arc of the parabola from $(0,0)$ to
$(3,9)$ and the line segments from $(3,9)$ to $(0,9)$ and from $(0,9)$ to $(0,0)$ oriented
counterclockwise. 

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\hspace*{1.5in} Name:\hrulefill
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\item
(25 points) Calculate the flux of $ \vec{F}(x,y,z) = x \; \vec{i}+ y \; \vec{j} + z \; \vec{k}\;\;\;\;$ across the surface \textit{S}, where \textit{S} is the part of the plane $2x+y+4z=4$ in the first octant oriented upward. 



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