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\def\Tilde{\char126\relax}
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{\bf NAME: (print!)}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
\quad\quad
{\bf RUID: (print!)}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\
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{\bf SSC:} (circle) None / I / II / I and II
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MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. ,   Final Exam , Tue., Dec. 15, 2020
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Email the completed test, renamed as {\tt  finalFirstLast.pdf } to DrZcalc3@gmail.com no later than
3:30pm, (or, in case of conflict, three and half hours after the start).

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WRITE YOUR FINAL ANSWERS BELOW
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Sign the following declaration:
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I  \quad \quad \quad \quad \quad \quad \quad \quad Hereby declare that all the work was done by myself. 
I was allowed to use Maple (unless  specifically told not to), calculators, the book, and all the material in the
web-page of this class but {\bf not} other resources on the internet.
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I only spent (at most) $3$ hours on  doing the exam. The last 30 minutes were spent in checking
and double-checking the answers.
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I also understand that I may be subject to a random short chat to verify that I actually did it all by myself.
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Signed:

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{\bf Possibly useful facts from school Geometry} (that you are welcome to use) : (i) The area of a circle radius $r$ is $\pi r^2$. 
(ii) The circumference of a circle radius $r$ is $2\pi r$ (iii) The parametric equation of an ellipse with
axes $a$ $b$ and parallel to the $x$ and $y$ axes respectively is $x=a \cos \theta$, $y=b\cos \theta$, $0<\theta<2\pi$.
(iv) The area of an ellipse with axes $a$ and $b$ is $\pi a b$ (v) The volume and surface area of a sphere radius $R$ are
$\frac{4}{3} \pi R^3$ and $4\pi R^2$ respectively (vi) The volume of a cone is the area of the base times the height over $3$.
(vii) The volume of a pyramid is the area of the base times the height over $3$. (viii) The area of a triangle is
base times height over $2$.


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{\bf Formula that you may (or may not) need}
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If the surface $S$ is given in {\bf explicit} notation $z=g(x,y)$, above the region of the $xy$-plane , $D$, then
$$
\int\int_S {\bf F} \cdot d {\bf S}=
$$
$$
\int\int_D \left (
-P {{\partial g} \over {\partial x}}
-Q {{\partial g} \over {\partial y}}
+R \right ) \, dA \quad .
$$


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{\bf 1}. (12 pts.)   {\bf Without using Maple (or any software)} Compute the {\bf vector-field line integral}
$$
\int_C \, (cos\, ( e^{\sin x})  +5y) \, dx \, + \, ( sin\, ( e^{\cos y} ) \,+ \, 11 x\,)\, dy \,  \quad,
$$
over the path consisiting of the  five line segments (in that order)
$$
(1,0) \rightarrow (-1,0) \rightarrow (-1,1) \rightarrow (0,2) \rightarrow (1,1) \rightarrow  (1,0) \quad .
$$

Explain!

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{\bf ans.} 
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{\bf 2.} (12 points) Change the order of integration
$$
\int_{\frac{1}{4}}^{1} \int_{0}^{\sqrt{x}} \, f(x,y)\, dy \, dx \quad \quad .
$$
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{\bf ans.} 
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{\bf 3.} (12 points) Find the equation of the tangent plane at the point $( \frac{\pi}{6}, \frac{\pi}{6}, \frac{\pi}{6} )$ to the surface
given implicitly by
$$
2 \cos (x+y) + 4 \cos (x+z)+ 8 \cos (y+z) \, = \, 7 \quad \quad .
$$
Express you answer in {\bf explicit} form, i.e in the format $z=ax+by+c$.

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{\bf ans.} $z=$ 
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{\bf 4.} (16 points) Let ${\bf a},{\bf b},{\bf c}$ be three vectors such that

$$
{\bf a} \times {\bf b}\,= {\bf i} \,+ \,{\bf j} - {\bf k} \quad , \quad
{\bf b} \times {\bf c} \,= {\bf i} \,- \,{\bf j} + {\bf k} \quad , \quad
{\bf a} \times {\bf c} \,= 2{\bf i} \,+ \,{\bf j} + 2{\bf k} \quad .
$$
What is
$$
({\bf a}+{\bf b}+{\bf c}) \times (2{\bf a}-{\bf b}+3 {\bf c})   \quad \quad ?
$$
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{\bf ans.} 
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{\bf 5.} (12 points) Find the three angles of the triangle $ABC$ where
$$
A=(0,0,0) \quad, \quad
B=(1,0,1) \quad, \quad
C=(1,1,0) \quad \quad .
$$
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{\bf ans.}  The angle at $A$ is: \qquad \qquad radians  \quad;\quad
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The angle at $B$ is: \qquad \qquad radians  \quad;\quad
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The angle at $C$ is: \qquad \qquad radians  \quad;\quad
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{\bf 6.} (12 points)  Find the directional derivative of
$$
f(x,y,z)=x^3+y^3+z^3+xyz \quad ,
$$
at the point $(1,1,1)$ in a direction pointing to the point $(-1,-1,-1)$ \quad .
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{\bf ans.} 
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{\bf 7.} (12 points)  Using the Chain Rule (no credit for other methods), find 
$$
\frac{\partial g}{\partial u}
$$ 
at $(u,v)=(0,1)$, where 
$$
g(x,y)=3x^2-3y^2 \quad ,
$$ 
and
$$
x=e^u \cos v \quad, \quad y=e^u \sin v \quad .
$$

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{\bf ans.}  
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{\bf 8.} (12 points)  Without using Maple (or any other software), compute the vector-field surface integral $\int_S {\bf F}.d{\bf S}$ 
if
$$
{\bf F}= \langle \, 3x +cos(y^3+yz) \, , \, -2y+ e^{x+z^2} \, , \, 5z+\sin(xy^3+e^x) \, \rangle \quad \quad ,
$$
and $S$ is the closed surface   in 3D space bounding  the region
$$
\{(x,y,z): x^2+y^2+z^2<4 \quad and \quad  x>0 \quad and \quad  y<0 \quad and \quad z>0 \} \quad .
$$

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{\bf ans.}
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{\bf 9.} (12 points) Compute the vector-field surface integral $\int \int_S {\bf F}.d{\bf S}$ 
if 
$$
{\bf F} \, = \, \langle \, 3z \, , \, 2x \, , \, y+z \, \rangle \quad, 
$$
and $S$ is the oriented surface
$$
z=2x+3y \quad, \quad 0<x<1, \quad 0<y<1  \quad ,
$$
with {\bf upward pointing} normal.


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{\bf ans.}  
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{\bf 10.} (12 points)  Without using Maple or software,
find the critical point(s) of 
$$
f(x,y)=4x-y^2-\ln(2x+y) \quad,
$$ 
and decide for each whether it
is a local maximum, local minimum, or saddle point. Explain.

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{\bf ans.}  
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{\bf 11.} (12 points)  Without using Maple or software, using a {\bf Linearization} around the point $(1,1,2)$, approximate $f(1.001,0.999,2.001)$ if
$$
f(x,y,z)\,=\, \sqrt{2x^2+3y^2+z^2} \quad .
$$
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{\bf ans.}  
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{\bf 12.} (12 points) Without using Maple (or any other software) and by using polar coordinates (no credit for doing it
directly) find
$$
\int_0^{\frac{\sqrt{2}}{2}} \, \int_0^x x \, dy \, dx \, + \,
\int_{\frac{\sqrt{2}}{2}}^1 \, \int_0^{\sqrt{1-x^2}} x \, dy \, dx \, \quad .
$$
Explain!
 
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{\bf ans.}  
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{\bf 13.} (12 points)  Convert the triple iterated integral
$$
\int_{0}^2 \int_0^{\sqrt{4-z^2}} \int_{-\sqrt{4-z^2-y^2}}^{0} \,\,\, x^2\,y\,z \,dx\,dy\,dz 
$$
to {\bf spherical coordinates}. {\bf Do not evaluate.}

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{\bf ans.}  
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{\bf 14.} (12 points)  Find the curvature of the curve
$$
{\bf r}(t)= \langle 5, 3 \sin t, 3 \cos t \rangle 
$$
at the point where $t=\frac{\pi}{3}$.
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{\bf ans.}  
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{\bf 15.} (12 points)   Set-up an iterated double integral, in type I format, but do not compute, for the surface area of the
surface given parameterically by
$$
{\bf r}(u,v) \,= \, \langle u^2 \, ,\,  uv \, , \, v^2 \rangle \quad, \quad  0<u<v<1 \quad .
$$
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{\bf ans.}  
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{\bf 16.} (12 points)  Let 
$$
f(x,y,z)\,=\, xy^2z^3  \quad ,
$$
and let 
$$
g(x,y,z)\,=\, x+y^2+z^3 \quad.
$$

compute   the dot-product
$$
grad(f)\,.\,grad(g)    \quad \quad \quad .
$$ 

at the point $(1,1,1)$.

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{\bf ans.}  
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{\bf 17.} (8 points) Decide whether the following limit exists. If it does ,find it. If it does not,  explain why it does
not exist.
$$
\lim_{(x,y,z,w) \rightarrow (0,0,0,0) } \, \frac{(x+y)^2-(z+w)^2}{x+y-z-w} \quad .
$$  
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{\bf ans.}  
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\end

