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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. ,  Makeup for Exam 1, Sunday, Oct. 25, 2020, 9:00-11:00am
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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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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).
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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.)  Find a parametric representation, in terms of the parameter, $t$,  of the line of intersection
of the planes $y=0$ and $z=0$. What is the usual name for that line?


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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) Use any method to find ${{\partial^2 h} \over {\partial r^2}}$ at $(q,r)=(1,1)$ where
$h(u,v)=u^5\,v^6$, $u=q^2\,r$, $v=q\,r^2$
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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 point  on the line $y\,=\,x+6$ (in the $xy$-plane) where the rate of change
of the function $f(x,y)=x^2+y^2$ in the direction $<1,1>$ is zero.
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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 4.} (10 points)
Find an equation of the tangent plane to the following surface at the given point
$$
z=x^2+y^3 \quad, \quad (1,2,9) \quad .
$$
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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) Compute $f_{xyy}(1,1)$ (aka $\frac{\partial^3 f}{\partial x \partial^2 y}(1,1)$,
if $f(x,y)=\ln(x^2 \,+ \, y)$.
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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) Use Linearization to apprximate
$\sqrt{(3.001)^2+(3.999)^2}$. Part of the challenge is to find the appropriate function $f(x,y)$ and
the appropriate `nice' point $(x_0,y_0)$.

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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) Decide whether the following limit exists. If it does find it.
If it does not {\bf Explain} why not?
$$
\lim_{(x,y,z)\rightarrow (1,1,1)} \,\, \frac{x+2y+3z-6}{3x+2y+z-6} \quad
$$
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{\bf Hint}: To prove that a limit {\bf exists} at a designated point you first try to plug it in. If there are
no issues you are done, and the value is the limit. Failing this, you try to use {\bf algebra} to simplify and then
plug it in, if possible. Failing this you try to prove that it does {\bf not} exist. One way of doing it
is to pick any two {\bf different} lines through the point in question and show that if you approach
the designated point via these lines you get {\bf different} limits (in the sense of single-variable calculus).

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{\bf 8.} (10 points)  Find the  minimum {\bf location(s)} (i.e the {\bf point(s)} where the function is the smallest)
of the function
$$
f(x,y,z) \, = \, x +2y-z
$$
in the {\bf closed pyramid} whose vertices are
$$
(1,1,0) \quad, \quad (-1,1,0) , \quad (1,-1,0) ,  \quad (-1,-1,0) ,  \quad (0,0,1)
$$

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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) A certain particle of mass $2$ kilograms has position function, expressed in
meters, where $t$ is time, expressed in seconts,
$$
{\bf r}(t) = \langle \, e^t, 2 \cos t, 2 \cos t \, \rangle \quad ,
$$
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(i) Find the {\bf magnitude} of the force acting on it at time $t=0$. {\bf Explain!}
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(ii) Find the {\bf unit direction} of that force.
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{\bf Reminder from physics}: $F=ma$, i.e. the force equals the mass  times the accelaration.



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The {\bf types} of the answers are (i) \qquad\qquad\qquad\qquad\qquad (ii)
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{\bf ans.}  (i) magnitude :    $\qquad\qquad\qquad\qquad$ Newtons \quad \quad ; \quad
(ii) unit direction:
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{\bf 10.} (10 points) A certain function $f(x,y,z)$ depends on three variables called $x$, $y$ and $z$. 
At a certain time
the rate of change of this function with respect to time happens to be $140$.
It is also known, that at that same time

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$\bullet$ The rate of change of $f(x,y,z)$ with respect to $x$ is $1$
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$\bullet$ The rate of change of $f(x,y,z)$ with respect to $y$ is $2$
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$\bullet$ The rate of change of $f(x,y,z)$ with respect to $z$ is $3$
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$\bullet$ The rate of change of  $y$ with respect to time is 
{\bf two times} The rate of change of  $x$ with respect to time.
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$\bullet$ The rate of change of  $z$ with respect to time is 
{\bf three times} The rate of change of  $x$ with respect to time.
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What is the rate of change of $x$ with respect to time at that time?


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