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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., Fall 2009, First Practice Test for Exam 2}
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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) Is the following 
$$
{\bf F} \, = \, \langle \, 2\cos(2x+3y+5z) + y^3 +1 \, ,
\, 3\cos(2x+3y+5z)+ 3xy^2+1 \, , \, 5 \cos(2x+3y+5z) \, \rangle
$$
a conservative vector field? 
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(ii) If it is, find a function $f(x,y,z)$ such that ${\bf F}= \g f$.
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(iii) Compute $3f(1,1,-1)-3f(0,0,0)-2$

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{\bf Ans. to (iii)}: 
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{\bf 2.} (10 points) 
Show that the line integral
$$
\int_C 2x \sin y \, dx \, + \, (x^2 \cos y -3 y^2) \, dy \quad ,
$$
is independent of the path $C$, and evaluate it if
$C$ is {\it any} path from $(1,0)$ to $(0,2)$.



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{\bf Ans.}:
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{\bf 3.} (10 points)
Evaluate
$$
\int\int\int_{E} 10(x^2+y^2)^4 \, dV \quad ,
$$
where $E$ is the solid that lies within the cylinder
$x^2+y^2=4$, below the plane $z=0$, and above the cone
$z^2=x^2+y^2$.

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{\bf Ans.}:
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{\bf 4.} (10 points)

Use the transformation 
$$
x=2u+v \quad , \quad y=u+2v \quad ,
$$
to evaluate the integral
$$
\int \, \int_R \, (2x-y) \, dA
$$
where $R$ is the triangular region with vertices $(0,0)$,
$(2,1)$, and $(1,2)$.

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{\bf Ans.}:
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{\bf 5.} (10 points) 
Use the given transformation
to evaluate the integral
$$
\int\int_{R} 4 \sqrt{2x+y} \, dA \quad ,
$$
where $R$ is the triangular region with vertices
$(0,0)$,$(4,-6)$, $(6,-10)$; 
$x=3u-v$, $y=-5u+2v$.

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{\bf Ans.}:
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{\bf 6.} (10 points) 
Evaluate the iterated integral by converting it to polar coordinates
$$
\int_0^3 \, \int_0^{\sqrt{9-x^2}} e^{x^2+y^2} \, dy \, dx 
\quad .
$$

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{\bf Ans.}:
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{\bf 7.} (10 points)  Let $A$ be the {\bf number}
$$
A= \int_{0}^{9} \int_{y/3}^{3}  e^{x^2} \, dx \, dy \quad .
$$
What is $2A+3$ ?
(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) 
Find the volume of the solid bounded by the cylinder $y=x^4$
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 9.} (10 points)  Use Lagrange multipliers (no credit for other methods) to find the maximum
value of the function
$f(x,y)=x^2y^3$  subject to the constraint $x+y=10$.

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{\bf Ans.}:
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{\bf 10.} (10 points) 
Use Largange multipliers (no  credit for other methods) to
find the largest value that $x+3y+5z$ can be, given that
$x^2+y^2+z^2=35$

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{\bf Ans.}:
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