

%begin macros
%\magnification=\magstephalf
\magnification=\magstep1
\def\g{\bigtriangledown}
\def\L{{\cal L}}
%\baselineskip=14pt
%\parskip=10pt
\def\Tilde{\char126\relax}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\R{{\cal R}}
\parindent=0pt
\overfullrule=0in
\def\frac#1#2{{#1 \over #2}}
%end macros

\medskip
{\bf NAME: (print!)}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\bigskip

{\bf Section:}  \_\_\_\_
\qquad
 {\bf E-Mail address:}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\bigskip
MATH 251 (4-6,11 ), Dr. Z. ,  Final Exam , Mon., Dec. 21, 2009, SEC 111, 
12:00-3:00pm
\bigskip

{\bf WRITE YOUR FINAL ANSWER TO EACH PROBLEM IN THE INDICATED PLACE (right under the question)}

Do not write below this line
\medskip
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\bigskip
1. \qquad  (out of 10)    
\medskip
2.  \qquad  (out of 10)    
\medskip
3.   \qquad (out of 10)    
\medskip
4.   \qquad (out of 10)    
\medskip
5.   \qquad (out of 10)    
\medskip
6.    \qquad (out of 10)    
\medskip
7.    \qquad (out of 10)    
\medskip
8.    \qquad (out of 10)    
\medskip
9.    \qquad (out of 10)    
\medskip
10.    \qquad (out of 10)    
\medskip
11.    \qquad (out of 10)    
\medskip
12.    \qquad (out of 10)    
\medskip
13.    \qquad (out of 10)    
\medskip
14.    \qquad (out of 10)    
\medskip
15.    \qquad (out of 10)    
\medskip
16.    \qquad (out of 10)    
\medskip
17.    \qquad (out of 10)    
\medskip
18.    \qquad (out of 10)    
\medskip
19.    \qquad (out of 10)    
\medskip
20.    \qquad (out of 10)    
\medskip
\_\_\_\_\_\_\_
\smallskip
tot. \qquad \qquad   (out of 200)
\vfill\eject
This page left blank
\vfill\eject
{\bf 1}. (10 pts.)  Find the curvature of
the curve 
$$
{\bf r}(t)=\langle e^t, \sin t, \cos t \rangle
$$
at the point $(1,0,1)$.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 2.} Sketch the region of integration
and change the order of integration.
$$
\int_{0}^{1} \int_{0}^{x} F(x,y) \, dy \, dx \, + 
\,   \int_{1}^{2} \int_{0}^{1} F(x,y) \, dy \, dx
\,+ \, \int_{2}^{3} \int_{0}^{3-x} F(x,y) \, dy \, dx
$$
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 3.} (10 points)
Find the absolute maximum value of the function $f(x,y)=x^2+y^2-2x-2y+2$ in the triangular region
$$
\{ (x,y) \, \vert \, x \geq 0 \, , \, y \geq 0 \, , \, x+y \leq 4 \, \} \quad .
$$
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule



\vfill\eject
{\bf 4.} (10 points)
Find an equation of the tangent plane to the surface
$$
z^5+x^4-y^3=1 \quad ,
$$
at the point $(1,1,1)$
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 5.} (10 points) Compute $f_{xy}(1,1)$ if $f(x,y)=e^{x^2+xy+y^2y}$.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 6.} (10 points) Use the linearization of 
$f(x,y)=\sqrt{x^2+y^2+1}$ to approximate
$f(2.01,1.98)$.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 7.} (10 points) Does the following limit exist? 
If it does, find it. If it does not, explain why it
does not exist.
$$
\lim_{(x,y,z) \rightarrow (0,0,0)} { {x^3+y^3+z^3}  \over {2x^3+3y^3+4z^3}}
$$
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule

\vfill\eject
{\bf 8.} (10 points) Find the length of the curve
$$
{\bf r}(t)= \langle \, 4t \, , \, 3 \sin t \, ,  \, 3 \cos t \, \rangle \quad 0 \leq t \leq \pi \quad .
$$

\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule



\vfill\eject
{\bf 9.} (10 points) A certain particle has acceleration
$$
{\bf a}(t) = \langle \, 16e^{4t} \, , \, 9\sin(3t) \, , \, 6t\, \rangle \quad ,
$$
and at $t=0$ its velocity is $\langle 4,-3,0 \rangle$ and its position vector is $\langle 1,0,0 \rangle$, find
its  position vector at any time $t$.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule

\bigskip


\vfill\eject
{\bf 10.} (10 points) 
Let $\Phi(u,v)=(3u+v,u-2v)$. Use the Jacobian to determine the
area of $\Phi(\R)$ for $\R=[2,5] \times [1,7]$.

\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule

\vfill\eject

{\bf 11.} (10 pts.)  
Compute $curl \,\, {\bf F}$ if
$$
{\bf F} \, = \, \langle \, 2e^{2x+3y+5z} + y^2 +2y+1 \, ,
\, 3e^{2x+3y+5z}+ 2xy+3x+1 \, , \, 5 e^{2x+3y+5z} \, \rangle
$$
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.:}: 
\bigskip
\bigskip
\hrule

\vfill\eject
{\bf 12.} (10 points) 
Evaluate the line integral
$$
\int_C {\bf F} \cdot d{\bf r} \quad,
$$
where $C$ is given by the vector function ${\bf r}(t)$.
$$
{\bf F} (x,y,z)= y {\bf i} + z {\bf j} + y {\bf k} \quad,
$$
$$
{\bf r} (t)= t^2 {\bf i} + t {\bf j} +  2t {\bf k} \quad,
\quad 0 \leq t \leq 1 \quad .
$$
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.}:
\bigskip
\bigskip
\hrule

\vfill\eject
{\bf 13.} (10 points)
Evaluate
$$
\int\int\int_{E} \frac{1}{\sqrt{x^2+y^2}} \, dV \quad ,
$$
where $E$ is the solid that lies within the cylinder
$x^2+y^2=1$, above the plane $z=0$, and below the cone
$z^2=4x^2+4y^2$.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.}:
\bigskip
\bigskip
\hrule



\vfill\eject
{\bf 14.} (10 points)
Evaluate the iterated integral
$$
\int_{0}^{2} \int_{x}^{3x} \int_{0}^{x+y} x \,dz \, dy \, dx \quad.
$$
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.}:
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 15.} (10 points) 
Use the given transformation
to evaluate the integral
$$
\int\int_{R} (2x+y)^2 \, dA \quad ,
$$
where $R$ is the triangular region with vertices
$(0,0)$,$(2,-3)$, $(3,-5)$; 
$x=3u-v$, $y=-5u+2v$.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.}:
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 16.} (10 points) 
Evaluate  the iterated integral
by converting to polar coordinates.
$$
\int_{0}^{1} \int_{-\sqrt{1-x^2}}^{0} 10(x^2+y^2)^4 \,dy \,dx
$$

\bigskip
\bigskip
\hrule
\bigskip
{\bf Ans.}:
\bigskip
\hrule



\vfill\eject
{\bf 17.} (10 points)  Compute the (vector-field) line-integral
$$
\int_C \, {\bf F} \, \cdot d {\bf r} \quad ,
$$
where $C$ is an ellipse with semi-major-axis $4$ and semi-minor-axis $3$, travelled {\bf clockwise},
and the vector-field ${\bf F}$ is
$$
{\bf F}= \langle \, 2y \, , \, 3x \, \rangle \quad .
$$

\bigskip
(Reminder: the area of an ellipse with semi-major axis $a$ and semi-minor-axis $b$ equals $\pi ab$).
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.}:
\bigskip
\bigskip
\hrule
\vfill\eject
{\bf 18.} (10 points) 
Compute the vector-field surface integral:
$$
\int\int_S  {\bf F} \cdot d{\bf S} \quad,
$$
where
$$
{\bf F}= \langle 5x \, , \, 2y \, , \, 3z  \, \rangle
$$
and where $S$ is the bounardy of a pyramid whose base is a $2 \times 2$ square
and whose height is $3$, with the normal pointing {\bf outward}.
\bigskip
(Reminder: the volume of a pyramid is the area of the base times the height divided by $3$)
\bigskip
\bigskip
\hrule
\bigskip
{\bf Ans.}:
\bigskip
\hrule

\vfill\eject
{\bf 19.} (10 points)  Find the maximum
value of the function
$f(x,y)=x^2y^4$  subject to the constraint $x^2+2y^2=6$.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
{\bf Ans.}:
\bigskip
\bigskip
\hrule


\vfill\eject
{\bf 20.} (10 points) 
Find the local maximum and minimum {\bf values},
and saddle point(s) of the function $f(x,y)=e^x-xe^y$.
\bigskip
\bigskip
\hrule
\bigskip
{\bf Local maximum value(s):} 
\bigskip
{\bf Local minimum value(s)} 
\bigskip
{\bf saddle point(s)}: 
\bigskip
\hrule

\end





