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\begin{document}
\noindent
Math 251 \hspace{0.65in}
Midterm 1\hspace{\fill}NAME:\underline{\hspace*{2.0in}}\\
F~20 Oct.\ 2017
\hspace{\fill}Student ID:\underline{\hspace*{2.0in}}\\
\underline{GENERAL INSTRUCTIONS}\\
\begin{enumerate}
\item
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
\item
The exam pages are {\textbf{two sided}}.
\item
Closed book, except for a $3 \times 5$ file card.
\item
The following are all prohibited:
Calculators (of any kind), cell phones, laptops, iPods,
% electronic dictionaries,
and any other electronic devices or communication devices.
All electronic or communication devices you have with you
must be turned completely off and put inside something
(pack, purse, etc.)\ and out of sight.
\item
The point values are as indicated in each problem; total 100 points.
\item
Write all answers on the test paper.
% Use the space below the extra credit problems
% Use the extra sheet at the back
% Use the back of the page with the extra credit problems
Use the back of the last page
for long answers or scratch work.
(If you do write an answer there,
indicate on the page containing the problem
where your answer is.)
\item
Show enough of your work that your method is obvious.
Be sure that every statement you write is correct.
Cross out any material you do not wish to have considered.
Correct answers with insufficient justification
or accompanied by additional incorrect statements
will not receive full credit.
Correct guesses to problems requiring significant work,
and correct answers obtained after a sequence of mostly incorrect steps,
will receive no credit.
\item
Be sure you say what you mean, and use correct notation.
Credit will be based on what you say, not what you mean.
\item
When exact values are specified, give answers such as $\frac{1}{7}$,
$\sqrt{2}$, $\ln (2)$, or $\frac{2\pi}{9}$.
Decimal approximations will not be accepted.
\item
Final answers must always be simplified unless otherwise specified.
\item
Grading complaints must be submitted in writing at the beginning of
the class period after the one in which the exam is returned
(usually by the Tuesday after the exam).
\item
Time: 50 minutes.
% (unless extended, with class agreement, by an early start).
\end{enumerate}
\vfill
\noindent
\hrulefill
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% \hspace{-1cm}
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\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\makebox[0.55in]{1} & \makebox[0.55in]{2} &
\makebox[0.55in]{3} & \makebox[0.55in]{4} & \makebox[0.55in]{5} &
\makebox[0.55in]{6} &
\makebox[0.55in]{7} &
\makebox[0.55in]{8} & \makebox[0.55in]{TOTAL} &
\makebox[0.55in]{EC} \\
\hline
1 & 20 & 18 & 9 & 27 & 8 & 8 & 9 & 100 & \\
\hline & & & & & & & & & \\
& & & & & & & & & \\
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}
\pagebreak
\noindent
1. (1 point) Are you awake?
\vspace{4ex}
\noindent
2. (a) (6 points)
State carefully the definition of the derivative of a function.
\vfill
(b) (14 points)
If $f (x) = {\displaystyle{\frac{1}{8 - x}}}$, compute the derivative
$f' (2)$ {\emph{directly from the definition}}.
(You can check your answer using a differentiation formula,
but no credit will be given for just using the formula.)
\vfill
\vfill
\vfill
\noindent
3. (9 points/part)
Differentiate the following functions.
(You need not compute the derivatives directly from the definition.)
\bigskip
(a) ${\displaystyle{ g (t)
= a e^t - \frac{7}{t^2} + \sqrt{t} + \pi^2 }}$.
($a$ is a constant.)
\vfill
(b)
$h (x) = \sin (6 x^2 - 11 x)$.
\vfill
\mbox{}
\pagebreak
\noindent
4. (9 points)
Find the equation of tangent line to the graph of $f (x) = x^2 - 2 x$
at $x = -3$.
You need not calculate the derivative directly from the definition.
\vfill
\vfill
\vfill
\noindent
5. (9 points/part)
Find the exact values of the following limits (possibly including
$\infty$ or $- \infty$), or explain why they do not exist
or there is not enough information to evaluate them.
Give reasons in all cases.
\bigskip
(a)
${\displaystyle{ \lim_{x \to 1}
\frac{x - 1}{x^2 - x - 6} }}$.
\vfill
\vfill
\vfill
(b)
${\displaystyle{ \lim_{x \to 10}
\frac{x - 10}{3 \big(\sqrt{x} - \sqrt{10} \big)} }}$.
\vfill
\vfill
\vfill
(c)
${\displaystyle{ \lim_{x \to \infty}
\frac{x + 109}{7 x + 1} }}$.
(Be sure to show your work!)
\vfill
\vfill
\mbox{}
\pagebreak
\vspace*{-2cm}
\noindent
6.
For the function $y = k (x)$ graphed below, answer the following
questions:
\vspace{1ex}
\centerline{
\includegraphics[width=12cm]{WeirdGraph1}
}
\vspace{1ex}
% {\epsfxsize=12cm \epsffile{WeirdGraph1.eps} }
\bigskip
(a) (4 points.)
Find $\lim_{x \to - 5} k (x)$.
\vfill
(b) (4 points.)
Which of the following best describes $k' (4)$?
\hspace*{1em} (1) $k' (4)$ does not exist.
\hspace*{1em} (2) $k' (4)$ is close to $0$.
\hspace*{1em} (3) $k' (4)$ is positive and not close to $0$.
\hspace*{1em} (4) $k' (4)$ is negative and not close to $0$.
\vspace{2ex}
\noindent
7. (4 points/part)
A traffic reporter's helicopter is hovering over a freeway interchange.
Its height above the ground varies.
During the period from 8:00~am to 8:22~am,
its height $y (t)$ above the ground,
measured in meters,
at time~$t$, measured in minutes (min) after 8:00~am,
is given by $y (t) = t^3 - 5 t^2 + 110$.
\bigskip
(a)
Is the helicopter falling or rising $2$ minutes after 8:00~am?
How fast?
\vfill
\vfill
(b)
What is the average upwards velocity of the helicopter between
8:00~am and 8:02~am?
\vfill
\vfill
\noindent
8. (9 points)
If $x y = \cos (x + y) + \sin (6)$, find
${\displaystyle{ \frac{d y}{d x} }}$ by implicit differentiation.
(You must solve for ${\displaystyle{ \frac{d y}{d x} }}$.)
\vfill
\vfill
\mbox{}
\pagebreak
\noindent
Extra credit.
(Do not attempt these problems until you have done and
checked your answer to all the ordinary problems on this exam.
They will only be counted if you get a grade
of B or better on the main part of this exam.)
\bigskip
Do these problems below or on the the back of this page.
\bigskip
\noindent
EC1. (5 extra credit points)
Let $f (x) = \cos (3 x)$.
Find the $1033$th derivative $f^{(1033)} (x)$.
\bigskip
\noindent
EC2. (10 extra credit points)
We will see later this quarter that if $g$ is a differentiable function
on an open interval $(a, b)$,
and if $g' (x) = 0$ for all $x$ in $(a, b)$, then $g$ is constant.
By considering the function
$g (x) = {\displaystyle{\frac{f (x)}{e^x}}}$,
prove that if $f$ is a function on $(a, b)$ such that $f' (x) = f (x)$
for all $x$, then there is a constant $c$ such that $f (x) = c e^x$
for all $x$.
\bigskip
\noindent
EC3. (5 extra credit points/part)
For each of the following parts, find a function $f$
whose derivative is as given.
Check your function to be sure its derivative really
is what you think it is.
(Caution: These are tricky.)
\bigskip
(a)
$f' (x) = x e^{-x^2}$.
\bigskip
(b)
$f' (x) = x \sin (x)$.
\end{document}