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{\bf NAME: (print!)}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
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{\bf Section:}  \_\_\_\_
\qquad
 {\bf E-Mail address:}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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MATH 251 (1-3), Dr. Z. ,  FINAL, 8:00-11:00am , Friday, Dec. 15, 2006
[Blue Version]
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{\bf FRAME YOUR FINAL ANSWER(S) TO EACH PROBLEM}

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1. \qquad  (out of 13)    
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16.    \qquad (out of 12)    
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{\bf 1}. (13 pts.)  Find the curvature of
the curve 
$$
{\bf r}(t)=\langle t,t^2, {{2} \over {3}} t^3 \rangle
$$
at the point $(1,1, {{2} \over {3}})$.
\vfill\eject
\vfill\eject
{\bf 2.} (13 points) By using Stokes's Theorem, or otherwise,
evaluate $\int_{C} {\bf F} \cdot d {\bf r}$ where
$$
F(x,y,z)=yz^2 \, {\bf i}+xz^2 \, {\bf j}+ 2xyz \, {\bf k} \quad ,
$$ 
and $C$ is the curve of intersection of the plane $x+y+z=1$ and
the cylinder $x^2+y^2=9$ oriented counterclockwise as viewed
from above. Be sure to explain everything.
\vfill\eject
{\bf 3.} (13 points) Find ${{\partial z} \over {\partial x}}$
and ${{\partial z} \over {\partial y}}$ if 
$$
x^3+y^3+z^3=5xyz+1 \quad .
$$

\vfill\eject
{\bf 4.} (13 points) Find an equation for the tangent plane to the
parametric surface: 
$$
x=u^2 \quad , \quad y=u+v \quad , \quad z=v^2 \quad ,
$$ 
at the point  $(1,2,1)$. Simplify as much as you can!
\vfill\eject
{\bf 5.} (13 points) 
Change the order of integration in
$$
\int_1^2 \int_0^{\ln x} f(x,y) \, dy \, dx \quad .
$$
\vfill\eject
{\bf 6.} (13 points) Let 
$$
{\bf F}(x,y,z)= 
$$
$$
\langle \cos(\sqrt{1+x}+x^3) \, , \, 
\tan((1+\cos(\sqrt{1+x}+x^3))^7) \, , \,
\tan^{-1} ((e^{x^2} +\cos(\sqrt{1+x}+x^3))^7 \rangle \quad ,
$$
and let $\langle \, P \, , \, Q \, , \, R \, \rangle = curl \,{\bf F}$.
Compute
$$
{{\partial P} \over {\partial x}}+
{{\partial Q} \over {\partial y}}+
{{\partial R} \over {\partial z}} \quad .
$$
Be sure  to explain everything.
\vfill\eject
{\bf 7.} (13 points) Let $C$ be the line segment from
$(0,1)$ to $(3,5)$, find $\int_C 2xy \, ds$.
\vfill\eject
{\bf 8.} (13 points) Evaluate
$$
\int_C (5y-\sin(e^x)) \, dx +(10x-e^{\cos^2 y}) \, dy \quad,
$$
where $C$ is the closed curve
consisting of the boundary of the rectangle 
$$
\{ \, (x,y) \, \vert \, 0 \leq x \leq 4 \, , \, 0 \leq y \leq 3 \, \} \quad .
$$

\vfill\eject
{\bf 9.} (12 points) Find the Jacobian of the transformation
$$
x=u+v+w \quad , \quad y=u^2+v^2+w^2 \quad , \quad z=u^3+v^3+w^3 \quad .
$$ 
Simplify as much as you can!
\vfill\eject
{\bf 10.} (12 points)
Find the volume
of the solid bounded by the 
cylinder $y=x^2$ and planes $z=0$ and $y+z=1$.
Simplify as much as you can!
\vfill\eject
{\bf 11.} ( 12 points) Use Lagrange multipliers 
(no credit for other methods!) to find
the largest value that
$x+3y+5z$ can be,  given that $x^2+y^2+z^2=35$.
\vfill\eject
{\bf 12.} (12 points) Find an equation of the tangent plane to the
surface $z=e^{2x-3y}$ at the point $(3,2,1)$.
Simplify as much as you can!
\vfill\eject
{\bf 13.} (12 points) Find the local maximum and minimum points,
the local maximum and minimum values,
and saddle point(s) of the function $f(x,y)=4x^2+y^2+2x^2y-1$.
\vfill\eject
{\bf 14.} (12 points) Find the velocity and position vectors of a 
particle whose acceleration is ${\bf a}(t)= {\bf i}+{\bf j}$, 
and at $t=0$ the velocity is ${\bf i}-{\bf j}$ and
the position is $\bf k$.
\vfill\eject
{\bf 15.} (12 points) Find an equation for the plane
through the point $(1,0,2)$ that contains the line
$$
{\bf r}(t)= \langle 1,1,1 \rangle + t\langle 1,-1,0 \rangle \quad .
$$
Simplify as much as you can!
\vfill\eject
{\bf 16.} (12 points) Compute the limit
$$
\lim_{(x,y,z) \rightarrow (1,1,1)} \,\, e^{-xy} \sin(\pi z/2) \quad ,
$$
or prove that it does not exists.

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