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

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MATH 251 (4-6,11 ), Dr. Z. ,  Exam 1, Thurs., Oct. 8, 2009, SEC 117, 12:00-1:20pm
[Blue Version]
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{\bf FRAME YOUR FINAL ANSWER(S) TO EACH PROBLEM}

Do not write below this line
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1. \qquad  (out of 10)    
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2.  \qquad  (out of 10)    
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3.   \qquad (out of 10)    
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4.   \qquad (out of 10)    
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5.   \qquad (out of 10)    
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6.    \qquad (out of 10)    
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7.    \qquad (out of 10)    
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8.    \qquad (out of 10)    
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9.    \qquad (out of 10)    
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10.    \qquad (out of 10)    
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{\bf Types:} Number, Function of $variable$(s), 2D vector of numbers, 3D vector of numbers,
2D vector of functions, 3D vector of functions, equation of a plane, parametric equation of a line,
equation of a line, equation of a surface, equation of a line, DNE (does not exist).

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{\bf 1.} (10 points) Use the chain rule to find $f_u$ and $f_v$ if
$$
f(x,y)=x^3+y^3 \quad , \quad x=e^{u+v} \quad , \quad y=2u+3v \quad .
$$
Express your answer in terms of $u$ and $v$.

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The {\bf types} of the answer are:
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{\bf 2.} (10 points) Find an equation of the tangent plane to the surface
$$
xz+2x^2y+y^2z^3=11 \quad
$$
at the point $(2,1,1)$.

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The {\bf type} of the answer is:


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{\bf 3.} (10 points)
Find the maximal rate of change of $f(x,y,z)=x^2y^3z^4$ at $(1,1,1)$,
and the {\bf unit} direction where it occurs.
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The {\bf types} of the answers are:


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{\bf 4.} (10 points) Compute $f_{xx}$, $f_{xy}$, and $f_{yy}$ if
$$
f(x,y)=e^{x^2+y^2} \quad .
$$
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The {\bf types} of the answers are:

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{\bf 5.} (10 points) Use implicit differentiation to find ${{\partial z} \over {\partial x}}$ and
${{\partial z} \over {\partial y}}$ if
$$
z^3+x^3+y^3+3xyz= 6 \quad .
$$

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The {\bf types} of the answer are:

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{\bf 6.} (10 points) Find a parametric equation of the line of intersection of the planes
$x+2y+3z=6$ and $3x+2y+z=6$.
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The {\bf type} of the answer is:


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{\bf 7.} (10 points) A certain particle has law of motion
$$
{\bf r}(t) = \langle \, \sin t \, , \, \cos 2t \, , \, e^t \, \rangle \quad ,
$$
Find its velocity, acceletartin, and speed at $t=\pi/6$.
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The {\bf types} of the answers are:

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{\bf 8.} (10 points) Write a definite integral that describes the length of the curve
$$
{\bf r}(t)= \langle \, \sin 5t \, , \, e^{2t} \, ,  \, e^{3t} \, \rangle \quad , \quad 0 \leq t \leq \pi \quad .
$$
{\bf Do not try to evaluate the integral!}
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The {\bf type} of the answer is:

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{\bf 9.} (10 points, altogether) Do the following limits exist? If they do, find them. Explain!
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The {\bf types} of the answers are: \qquad\qquad\qquad                         and    \qquad\qquad\qquad .
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{\bf a.} (5 points)
If
$$
\lim_{(x,y,z) \rightarrow (1,2,3)} f(x,y,z)=1 \quad, \quad
\lim_{(x,y,z) \rightarrow (1,2,3)} g(x,y,z)=2 \quad
$$
compute
$$
\lim_{(x,y,z) \rightarrow (1,2,3)} (f(x,y,z)+g(x,y,z))^3 e^{g(x,y,z)} \quad
$$

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{\bf b.} (5 points)
$$
\lim_{(x,y) \rightarrow (0,0)} { {x^4+y^4}  \over {2x^4+y^4}} \quad .
$$

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{\bf 10}. (10 pts.)  A force with magnitude $500N$ is moving a body of mass $5kg$ in the direction
$\langle 3,4,0 \rangle$. If at $t=0$ the body is at location $(1,2,3)$ and it is moving with
velocity $\langle 2,1,4 \rangle$, 

(i)find its position vector ${\bf r}(t)$ at time $t$ ;

(ii) find its speed at time $t$.
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The {\bf types} of the answers are:

\end
