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\def\L{{\cal L}}
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\def\Tilde{\char126\relax}
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\def\P{{\cal P}}
\def\Q{{\cal Q}}
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{\bf NAME: (print!)}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
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{\bf Section:}  \_\_\_\_
\qquad
 {\bf E-Mail address:}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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MATH 251 (4,6,7 ), Dr. Z. ,  Exam 1, Thurs., Oct. 16, 2017, SEC 118
[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 pts.)  Find the curvature of
the curve 
$$
{\bf r}(t)=\langle e^{2t}, \cos 3t, 2\,\sin 4t \rangle
$$
at the point $(1,1,0)$.
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The {\bf type} of the answer is:


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{\bf 2.} (10 points) Find ${{\partial h} \over {\partial r}}$ at $(q,r)=(2,1)$ where
$h(u,v)=ue^{v^2}$, $u=q^3+q$, $v=q^2r^3$
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The {\bf type} of the answer is:
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{\bf 3.} (10 points)
Find the directional derivative of $f(x,y,z)=x^3y^4z^5$ at $P=(1,-1,1)$ in the direction pointing to the
point $Q=(1,2,2)$.
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The {\bf type} of the answer is:


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{\bf 4.} (10 points)
Find an equation of the tangent plane to the following surface at the given point
$$
xy+2yz+3xz \,= \, 6 \quad, (1,1,1) \quad .
$$
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The {\bf type} of the answer is:


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{\bf 5.} (10 points) Compute $f_{xy}(1,1)$ if $f(x,y)=x^5 \ln (x+y)$.
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The {\bf type} of the answer is:

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{\bf 6.} (10 points) Use the linearization of $f(x,y,z)=\sqrt{2x \, + \, 3y \, + \, 4xz}$ to approximate
$f(1.01,0.99,1.02)$.
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The {\bf type} of the answer is:


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{\bf 7.} (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.} (3 points)
$$
\lim_{(x,y,z) \rightarrow (1,1,1)} { {\ln (x^2+y^2+z^2)}  \over {x+y+z}}
$$

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{\bf b.} (3 points)
$$
\lim_{(x,y,z) \rightarrow (0,0,0)} { {x+y+z}  \over {x^3+y^3+z^3}} \quad .
$$

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{\bf c.} (4 points)
$$
\lim_{(x,y,z) \rightarrow (0,0,0)} \sin(\frac{1}{x+y+z}) (x+y+z)  \quad .
$$

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{\bf 8.} (10 points)  Find the  local maximum and minimum point(s), and saddle points (if they exist) of the functions
$$
f(x,y) \, = \, 6x^2 \, - \, 2x^3 \, + \, 3y^2 \, + \, 6xy \quad .
$$

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The {\bf type}(s) of the answer are:


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{\bf 9.} (10 points) A certain particle has acceleration
$$
{\bf a}(t) = \langle \, e^t, \cos t, \sin 2t \, \rangle \quad ,
$$
and at $t=0$ its velocity is $\langle 1,1,1 \rangle$ and its position vector is $\langle 0,1,2 \rangle$, find
its velocity and position vector at time $t=2$.
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The {\bf type}(s) of the answer are:

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{\bf 10.} (10 points) Find an equation to the plane that passes through the points $(6,0,0)$, $(0,4,0)$, $(0,0,3)$.
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The {\bf type} of the answer is:








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