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

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{\bf MATH 251 (1-3,10 ), Dr. Z. ,  Exam 2, Thurs., Nov. 19, 2009, 
10:20-11:40am, SEC 118}
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{\bf No Calculators!}
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Do not write below this line (office use only)
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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.)  
(i) Prove that
$$
{\bf F} \, = \, \langle \, 2e^{2x+3y+5z} + y^2 +1 \, ,
\, 3e^{2x+3y+5z}+ 2xy+1 \, , \, 5 e^{2x+3y+5z} \, \rangle
$$
is a conservative vector field. 
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(ii) Find a function $f(x,y,z)$ such that ${\bf F}= \g f$.
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(iii) Compute $f(1,1,1)-f(0,0,0)-e^{10}+1$

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{\bf Ans. to (iii)}: 
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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 ${\bf r}(t)$.
$$
{\bf F} (x,y,z)= yz {\bf i} + xz {\bf j} + xy {\bf k} \quad,
$$
$$
{\bf r} (t)= t^3 {\bf i} + t^2 {\bf j} +  {\bf k} \quad,
\quad 0 \leq t \leq 1 \quad .
$$
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{\bf Ans.}:
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{\bf 3.} (10 points)
Evaluate
$$
\int\int\int_{E} {{9} \over {\pi}} (x^2+y^2)^3 \, dV \quad ,
$$
where $E$ is the solid that lies within the cylinder
$x^2+y^2=1$, above the plane $z=0$, and below the cone
$z^2=4x^2+4y^2$.

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{\bf Ans.}:
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{\bf 4.} (10 points)
Evaluate the iterated integral
$$
\int_{0}^{1} \int_{x}^{2x} \int_{0}^{x+y} {{24 z} \over {19}} \,dz \, dy \, dx \quad.
$$

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{\bf Ans.}:
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{\bf 5.} (10 points) 
Use the given transformation
to evaluate the integral
$$
\int\int_{R} 20 (2x+y)^2 \, dA \quad ,
$$
where $R$ is the triangular region with vertices
$(0,0)$,$(2,-3)$, $(3,-5)$; 
$x=3u-v$, $y=-5u+2v$.

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{\bf Ans.}:
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{\bf 6.} (10 points) 
Evaluate  the iterated integral
by converting to polar coordinates.
$$
\int_{-2}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} {{27} \over {64 \pi}} (x^2+y^2)^2 \,dy \,dx
$$

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{\bf Ans.}:
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{\bf 7.} (10 points)  Let $A$ be the {\bf number}
$$
A= \int_{0}^{4} \int_{y/2}^{2}  e^{x^2} \, dx \, dy \quad .
$$
What is $1+e^4-A$ ?
(Hint: Not even Dr. Z. can do $\int e^{x^2} \, dx$, so you must be clever,
and first change the order of integration.)
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{\bf Ans.}:
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{\bf 8.} (10 points) 
Calculate the double integral
$$
\int\int_{R} {{27x^2y^2} \over {(\ln 2) (x^3+1)}} \, dA \quad, \quad
$$
$$
R=\{(x,y)\, \vert \, 
0 \leq x \leq 1 \, , \, -1 \leq y \leq 1 \, \} \quad.
$$

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{\bf Ans.}:
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{\bf 9.} (10 points)  Use Lagrange multipliers to find the maximum
value of the function
$f(x,y)=x^2y-27$  subject to the constraint $x+y=6$.

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{\bf Ans.}:
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{\bf 10.} (10 points) 
Find the local maximum and minimum {\bf values},
and saddle point(s) of the function $f(x,y)=x^4+y^4-4xy+5$.

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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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