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

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MATH 251 (22,23,24 ) [Fall 2020], Dr. Z. ,   Exam 2, Monday, Nov. 23, 2020, 8:40-10:40am
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Email the completed test, renamed as {\tt  mt2FirstLast.pdf } to DrZcalc3@gmail.com no later than
10:40am, (or, in case of conflict, two hours after the start).

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WRITE YOUR FINAL ANSWERS BELOW
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1. \qquad  
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2.  \qquad 
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3.   \qquad 
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4.   \qquad 
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5.   \qquad 
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6.    \qquad 
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7.    \qquad  (a) \qquad \qquad  \qquad (b) \qquad \qquad  \qquad (c) \qquad \qquad  \qquad (d) \qquad \qquad  \qquad 
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8.    \qquad 
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9.    \qquad 
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10.    \qquad
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{\bf Types:} Number, Function of $variable$(s), 2D vector of numbers, 3D vector of numbers,
2D vector of functions, 3D vector of functions, equation of a plane, parametric equation of a line,
equation of a line, equation of a surface, equation of a line, DNE (does not exist),
parametric equation of surface, double integral of an abstract function.
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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, 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) 90 minutes 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 1}. (10 pts.)   Compute the line integral
$$
\int_C \, yz \, dx \, + \, (xz+z) \, dy \, + \,(xy+y+1) \, dz \quad,
$$
over the path
$$
{\bf r}(t) = \langle e^{t^3},  t^2 e^{t^4}, t e^{t^7} \rangle \quad, \quad 0 \leq t \leq 1 \quad .
$$
Explain!


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The {\bf type} of the answers is:
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{\bf ans.} 
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{\bf 2.} (10 points) By changing the order of integration, if necessary, evaluate the double-integral
$$
\int_{0}^{5} \int_{(y/5)^{1/3}}^{1} \sin \, x^4 \, dx \, dy
$$
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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 3.} (10 points) Find the equation of the tangent plane at the point $(1,1,1)$ to the surface
given parametrically by
$$
x(u,v)= u^3 v \quad , \quad  y(u,x)=uv \quad \, , \quad z(u,v)=uv^3  \quad , \quad
-\infty < u < \infty \quad , \quad
-\infty < v < \infty \quad .
$$
Express you answer in {\bf explicit} form, i.e in the format $z=ax+by+c$.
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The {\bf type} of the answer is:
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{\bf ans.} $z=$ 
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{\bf 4.} (10 points) Let $f(x,y,z)=e^{\cos x^2 \, + \, \sin xyz + \cos xz}$, and let

$$
{\bf F}= \langle \frac{\partial f}{\partial x} , \frac{\partial f}{\partial y} , \frac{\partial f}{\partial z} \rangle
$$

Let $C$ be the  curve
$$
r(t)= \langle \cos t , t, \sin t \rangle \quad, \quad 0 \leq t \leq 2\pi \quad .
$$
Find the value of the line-integral
$$
\int_C {\bf F} . d{\bf r} \quad .
$$
Explain! Just giving the answer will give you no credit.

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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 5.} (10 points) Evaluate the triple integral
$$
\int_R \, (x^2+y^2+z^2)^3 \, dx \, dy \, dz \quad,
$$

where $R$ is the region in 3D space given by

$$
\{ (x,y,z) \, | \, x^2+y^2+z^2 \leq 1 \quad, \quad x,y,z \geq 0 \} \quad .
$$


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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 6.} (10 points)  Evaluate the double integral
$$
\int_{-3}^{0} \int_0^{\sqrt{9-x^2}} (x^2+y^2)^2 \, dy \, dx
$$

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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 7.} (10 points altogether) Decide whether the following limits exist. If it does find them.
If it does not {\bf Explain} why not?
$$
(a) \,\, (2 \,\, points) \, \lim_{(x,y)\rightarrow (\pi/2, \pi/2)} \,\, \frac{\cos x + \sin x}{x+y} \quad, \quad
(b)\,\, (2 \,\, points) \, \lim_{(x,y)\rightarrow (0,0)} \,\, \frac{x^2-y^2}{x-y} \quad, \quad
$$
$$
(c) \,\, (2 \,\, points) \, \lim_{(x,y)\rightarrow (0,0)} \,\, \frac{x-y}{x^2-y^2} \quad, \quad
(d)\,\, (4 \,\, points) \,\lim_{(x,y)\rightarrow (1,1)} \,\, \frac{x+y-2}{2x+y-3} \quad, \quad
$$
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{\bf ans.}  (a) \qquad \qquad (b) \qquad \qquad (c) \qquad \qquad (d) \qquad \qquad
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{\bf 8.} (10 points)  Compute the line integral $\int_C f \, ds$ where
$$
f(x,y,z)=xyz \quad
$$
and $C$ is the line segment starting at  $(0,0,0)$ and ending at $(1,2,-3)$

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The {\bf type} of the answer(s) is:
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{\bf ans.}
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{\bf 9.} (10 points) Compute the vector-field surface integral $\int \int_S {\bf F}.d{\bf S}$ 
if ${\bf F}$ is 
$$
{\bf F} \, = \, \langle z,z,x \rangle \quad, 
$$
and $S$ is the oriented surface
$$
z=9-x^2-y^2 \quad, x \geq 0, y \geq 0 , z \geq 0
$$
with {\bf downward pointing} normal.



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The {\bf type} of the answer is
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{\bf ans.}  
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{\bf 10.} (10 points)  Find the {\bf point } on the plane $x+2y+3z=18$ where the function $f(x,y,z)=xyz$ is 
{\bf as large as possible}.



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The {\bf type} of the answer is:
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{\bf ans.}  
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