
%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\Tilde{\char126\relax}
\def\g{\bigtriangledown}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
%end macros
 
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{\bf NAME:} 
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{\bf Section:} 
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{\bf MATH 251 (1-6,10-11), Dr. Z. ,  Third Practice Test for Exam \#2 }

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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 total}:
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{\bf 1.} (10 points)
By finding a function $f$ such that ${\bf F}=\g f$, evaluate
$\int_C {\bf F} \cdot d {\bf r}$ along the given curve $C$.
$$
{\bf F}(x,y,z)= xe^{x^2+y^2+z^2} \, {\bf i}+ ye^{x^2+y^2+z^2} \, {\bf j} +
ze^{x^2+y^2+z^2} \, {\bf k} \quad,
$$
$$
C: x=2t \quad , \quad y=t^2 \quad , \quad z=t^5 \quad, 
\quad 0 \leq t \leq 1 \quad .
$$

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{\bf Ans:}
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{\bf 2.} (10 points)
(a) Find a function $f$ such
that ${\bf F}=\g f$ and (b) use part (a) to evaluate
$\int_C {\bf F} \cdot d {\bf r}$ along the given curve $C$.
$$
{\bf F}(x,y,z)= (2xz+y^2) \, {\bf i}+ 2xy \, {\bf j} +
(x^2+3z^2) \, {\bf k} \quad,
$$
$$
C: x=t^2 \quad , \quad y=t+1 \quad , \quad z=2t-1 \quad, 
\quad 0 \leq t \leq 1 \quad .
$$


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{\bf Ans.:} 
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{\bf 3.} (10 points) 
Evaluate
$$
\int\int\int_{E}  \, (x^2+y^2+z^2)^{9} \, dV \quad ,
$$
where $E$ is bounded by the $xy$-plane and the hemispheres
$z=-\sqrt{1-x^2-y^2}$ and $z=-\sqrt{9-x^2-y^2}$.
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{\bf Ans.:} 
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{\bf 4.} (10 points) 
Plot the vector field ${\bf F}= \langle \, -x, -y \rangle $.

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{\bf Ans.:} 
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{\bf 5.} (10 points)
Find the Jacobian of the transformation from $(u,v,w)$-space to
$(x,y,z)$-space.
$$
x=u^3+v^2+w \quad, \quad y=u^2+v \quad, \quad z=u+v+w  ,
$$
at the point $(u,v,w)=(1,2,3)$.
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{\bf Ans.:} 
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{\bf 6.} (10 points)
Evaluate  the iterated integral
by converting to polar coordinates.
$$
\int_{0}^{3} \int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}} x^2y \,dx \,dy
$$


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{\bf Ans.:} 
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{\bf 7.} (10 points)
Calculate the iterated integral
$$
\int_{0}^{1} \int_{0}^{y} (x^2y+xy^2)\, dx \, dy
\quad .
$$
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{\bf Ans.:} 
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{\bf 8.}  (10 points)
Evaluate the triple integral
$$
\int\int\int_{E} x^2 y^2 \, dV \quad,
$$
where $E$ is bounded by $z=1-y^4$
and the planes $z=0$, $x=1$ and $x=-1$.

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{\bf Ans.:} 
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{\bf 9.} (10 points) Find the local maximum and minimum {\bf values}
and saddle point(s) of the function $f(x,y)=(9+xy)(x+y)$

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{\bf local maximum value(s):} 
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{\bf local minimum value(s):} 
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{\bf saddle point(s):} 
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{\bf 10.} (10 points)
Evaluate the iterated integral
$$
\int_{0}^{1} \int_{x}^{2x} \int_{0}^{y} 2xyz \,dz \, dy \, dx \quad.
$$


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{\bf Ans.:} 
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