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

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{\bf MATH 251 (1-6,10-11), Dr. Z. ,  Second Practice 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} \qquad \qquad (out of 100)
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{\bf 1}. (10 pts.)  
Show that the line integral
$$
F(x,y,z)=(e^x +yz) \, {\bf i} + (e^y +xz)\, {\bf j} + (e^z+xy)\, {\bf k} 
$$ 
is conservative. If it is conservative, find a function $f$ such that
${\bf F}=\nabla f$.

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{\bf Ans. }: 
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{\bf 2.} (10 points) 
Evaluate the line integral
$$
\int_C {\bf F} \cdot d{\bf r} \quad,
$$
where $C$ is given by the vector function ${\bf r}(t)$.
$$
{\bf F} (x,y,z)= \sin(x+2y+3z)\, {\bf i} +2 \sin(x+2y+3z)\, {\bf j} + 3 \sin(x+2y+3z) \, {\bf k} \quad,
$$
$$
{\bf r} (t)= 3t \, {\bf i} + t \, {\bf j} +  2t \, {\bf k} \quad,
\quad 0 \leq t \leq \pi/26 \quad .
$$
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{\bf Ans.}:
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{\bf 3.} (10 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!

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{\bf Ans.}:
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{\bf 4.} (10 points)
Evaluate the iterated integral
$$
\int_{0}^{2} \int_{2x}^{3x} \int_{0}^{x+y} z^2 \,dz \, dy \, dx \quad.
$$

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{\bf Ans.}:
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{\bf 5.} (10 points) 
Use polar coordinates to compute the double
integral
$$
\int \, \int_D (x+2y) \, dA \quad ,
$$
where 
$$
D= \{ (x,y) \, \vert \, x^2+y^2 \leq 1 \, , \,
x \geq 0 \, , \, y \leq 0 \, \} \quad .
$$
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{\bf Ans.}:
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{\bf 6.} (10 points) 
Evaluate  the iterated integral
by converting to polar coordinates.
$$
\int_{-3}^{0} \int_{-\sqrt{9-x^2}}^{0} 3(x^2+y^2)^5 \,dy \,dx
$$

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{\bf Ans.}:
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{\bf 7.} (10 points)  
Find the volume of the solid bounded by the cylinder $y=x^2$
and the planes $z=0$ and $y+z=1$. Simplify as much as you can.

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{\bf Ans.}:
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{\bf 8.} (10 points) 
Calculate the double integral
$$
\int\int_{R} {{(3x^2+2)y^2} \over {x^3+2x+1}} \, dA \quad, \quad
$$
$$
R=\{(x,y)\, \vert \, 
0 \leq x \leq 1 \, , \, -2 \leq y \leq 0 \, \} \quad.
$$

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{\bf Ans.}:
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{\bf 9.} (10 points)  
Change the order of integration in
$$
\int_{e}^{e^3} \int_{0}^{\ln x} f(x,y) \, dy \, dx \quad .
$$

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{\bf Ans.}:
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{\bf 10.} (10 points) 
Find the local maximum and minimum {\bf values},
the local maximum and minimum {\bf points} (i.e. locations),
and saddle point(s) of the function $f(x,y)=x^4+y^4-xy-2$.

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{\bf Local maximum value(s):}\quad         \qquad {\bf at}
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{\bf Local minimum value(s):} \quad   \qquad {\bf at}
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{\bf saddle point(s):} 
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