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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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MATH 251 (1-3,10 ), Dr. Z. ,  Final Exam , Fri., Dec. 18, 2009, SEC 111, 8:00-11:00am
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{\bf WRITE YOUR FINAL ANSWER TO EACH PROBLEM IN THE INDICATED PLACE (right under the question)}

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tot. \qquad \qquad   (out of 200)
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{\bf 1.} (10 points) Compute the line-integral
$$
\int_C 7y \, dx + 3x \, dy \quad,
$$
where $C$ is the closed path that consists  from the line segment from $(0,0)$ to $(1,2)$
followed by the line segment from $(1,2)$  to $(2,0)$ followed by the line segment from $(2,0)$ back to $(0,0)$.
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{\bf Ans.:} 
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{\bf 2.} (10 points) Find an equation of the tangent plane to the surface
$$
\Phi(u,v)= ( u^2+v^2, 2uv, u^3) \quad ,
$$
at the point $(5,4,1)$.
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{\bf Ans.:} 
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{\bf 3.} (10 points)
Find the  absolute maximum value and the absolute minimum value
of the function $f(x,y)=x^2+y^2$ in the region
$$
\{ (x,y) \, \vert \, -1 \leq x \leq 1,  -1 \leq y \leq 1 \} .
$$
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{\bf Absolute minimum value:} 
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{\bf Absolute maximum value:} 
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{\bf 4.} (10 points) Compute $f_{xy}$ if
$$
f(x,y)=\sin(x^2+xy+y^2) \quad .
$$
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{\bf Ans.:} 
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{\bf 5.} (10 points) Find ${{\partial z} \over {\partial x}}$
at the point $(1,1,1)$ if $(x,y,z)$ are related by:
$$
x^3y^3z^3+3xyz=4
$$
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{\bf Ans.:} 
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{\bf 6.} (10 points) Find an equation for the plane that contains
both the  line
$$
x=1+t \, , \, y=2+t \, , \, z= 3 +t \quad (- \infty < t < \infty )
$$
and the line
$$
x=-t \, , \, y=1+t \, , \,  z= 2 +t \quad (- \infty < t < \infty )
$$

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{\bf Ans.:} 
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{\bf 7.} (10 points) A certain particle has law of motion
$$
{\bf r}(t) = \langle \, 2\sin t \, , \, 2\cos 2t \, , \, 2\sin 3t \, \rangle \quad ,
$$
Find its acceleration at $t=\pi/6$.
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{\bf Ans.:} 
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{\bf 8.} (10 points) Compute the (scalar-function) line-integral
$$
\int_C y \, ds
$$
where the curve $C$ is given by the parametric equation:
$$
{\bf r}(t)= \langle \, 1+ \sin 5t \, , \, 1+ \cos 5t  \, ,  \, \sqrt{75} t \, \rangle \quad , \quad 0 \leq t \leq 1 \quad .
$$
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{\bf Ans.:} 
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{\bf 9.} (10 points) 
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If
$$
\lim_{(x,y,z) \rightarrow (1,2,3)} f(x,y,z)=1 \quad, \quad
\lim_{(x,y,z) \rightarrow (1,2,3)} g(x,y,z)=\pi/6 \quad
$$
compute
$$
\lim_{(x,y,z) \rightarrow (1,2,3)} (f(x,y,z)+ \cos (2g(x,y,z)))^2 \sin(g(x,y,z)) \quad
$$
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{\bf Ans.:} 
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{\bf 10.} Compute
$$
\int\int_S  {\bf F} \cdot d{\bf S} \quad,
$$
where
$$
{\bf F}= \langle x^2+\sin(y+z) \, , \, y^2+xz^3 \, , \, z^2+e^{xy}  \, \rangle
$$
and where $S$ is the boundary (consisting of all six faces) of the cube
$$
\{(x,y,z) \, \vert \, 0 \leq x,y,z \leq 1 \}
$$
with the normal pointing {\bf outward}.
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{\bf Ans.:} 
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{\bf 11.} By finding a function $f$ such that ${\bf F}=\g f$, evaluate
$\int_C {\bf F} \cdot d {\bf r}$ along the given curve $C$.
$$
{\bf F}(x,y,z)= 2e^{2x+3y+4z} \, {\bf i}+ \, 3e^{2x+3y+4z}{\bf j} +
4e^{2x+3y+4z}\, {\bf k} \quad,
$$
$$
C: x=t^2 \quad , \quad y=t^4 \quad , \quad z=-t^9 \quad, 
\quad 0 \leq t \leq 1 \quad .
$$
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{\bf Ans:}
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{\bf 12.} 
Evaluate the line integral
$$
\int_{C} 5y\,dx+ 5x \, dy + 6z \,dz \quad ,
$$
where $C: x=t^2 \, , \,  y=t \, , \, z=t^2 \, , \, 0 \leq t \leq 1$.
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{\bf Ans.:} 
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{\bf 13.} 
Evaluate
$$
\int\int\int_{E}  {{1} \over {x^2+y^2+z^2}} \, dV \quad ,
$$
where $E$ is the portion of the ball $x^2+y^2+z^2 \leq 4$
that lies in the octant 
$$
\{(x,y,z) \vert x \geq 0 \, , \, y \geq 0 \, , \, z \geq 0 \} \quad .
$$
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{\bf Ans.:} 
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{\bf 14.} 
Evaluate the triple integral
$$
\int\int\int_{E} \,\, \, 10yz\, dV \quad ,
$$
where
$$
E=\{ (x,y,z) \, \vert \, 0 \leq z \leq 1 \, , \, 0 \leq y \leq z
\, ,  \, y+z \leq x \leq y+2z \, \} \quad .
$$
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{\bf Ans.:} 
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{\bf 15.} Find the Jacobian of the transformation from $(u,v,w)$-space to
$(x,y,z)$-space.
$$
x=2u+3v+w^2 \quad, \quad y=-u+2v \quad, \quad z=v+3w  ,
$$
at the point $(u,v,w)=(1,1,0)$.
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{\bf Ans.:} 
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{\bf 16.} Evaluate the integral
$$
\int\int_{D} \,\, \frac{8e^8}{\pi}e^{-2x^2-2y^2} \, dA \quad,
$$
where $D$ is the region 
$$
\{ (x,y)  \, \vert \, x^2+y^2 \leq 4, x \geq 0 , y \leq 0 \}
$$
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{\bf Ans.:} 
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{\bf 17.}.
Calculate the iterated integral
$$
\int_{0}^{1} \int_{0}^{1} \int_0^{1} (2x+2y+2z)\, dx \, dy \, dz
\quad .
$$
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{\bf Ans.:} 
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{\bf 18.} Find the maximum
value of the function
$$
f(x,y)=3x+4y \quad ,
$$ 
subject to the
constraint $x^2+y^2=25$.

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{\bf Ans.}:
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{\bf 19.} Find the local maximum and minimum {\bf values}
and saddle point(s) of the function $f(x,y)=x^3+y^3-3xy$

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