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% *** COVER ***
\centerline{\huge \bf TEST I}
\vf \vf
Math 26L-04
February 11, 1998 \hfill
{\bf Name: } $\underbrace{\hspace{2.7in}}_{\mbox{\tiny by writing my
name i swear by the honor code}}$
\vf \vf \vf
{\bf Read all of the following information before starting the exam:}
\vs
\begin{itemize}
\item Show all work, clearly and in order, if you want to get full
credit. I reserve the right to take off points if I cannot see how you
arrived at your answer (even if your final answer is correct).
\item Justify your answers algebraically whenever possible to ensure
full credit. When you do use your calculator, sketch all relevant
graphs and explain all relevant mathematics.
\item Circle or otherwise indicate your final answers, when applicable.
Put a circle on the scrap page at the end of the exam for two points.
\item Please keep your written answers brief; be clear and to the point.
I will take points off for rambling and for incorrect or irrelevant
statements.
\item This test has 10 problems and is worth 100 points, plus some
extra credit at the end. It is your responsibility to make sure that you
have all of the pages!
\item Good luck!
\end{itemize}
\vf \vf \vf
\pg
% ***************************************************************************
% *** trig cheat sheet
\begin{center}
{\bf TRIGONOMETRY ``CHEAT SHEET''}
\vspace{3pc}
{\it You may not need all of the identities in this list, and you may
need identities not in this list.}
\vspace{.5pc}
{\it Please note that you can derive many other trigonometric
identities from the ones in this list.}
\vspace{6pc}
$\sin^2(x)+\cos^2(x)=1$
\vspace{4pc}
$\sin (u+v)=\sin u\cos v+\cos u\sin v$
\vspace{2pc}
$\cos (u+v)=\cos u\cos v-\sin u\sin v$
\vspace{2pc}
$\tan (u+v)=\displaystyle\frac{\tan u+\tan v}{1-\tan u\tan v}$
\vspace{4pc}
$$\sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1-\cos x}{2}}$$
\vspace{.5pc}
$$\cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1+\cos x}{2}}$$
\vspace{4pc}
$${\lim _{\theta\to 0} {\sin \theta \over \theta} = 1}$$
$${\lim _{\theta\to 0} {\cos \theta -1 \over \theta} = 0}$$
\end{center}
\newpage
% ***************************************************************************
% *** euler's method
\bp{10} % CSPK p.215 exercise 2; from lab 1/21
Suppose $\dd{y}{t} = \frac{2}{t}$ and $y(1)=0$. Use Euler's method to
approximate $y(2)$, using a time step of $\Delta t = \frac{1}{2}$.
Recall that Euler's formula is $y_k = y_{k-1} + y'_{k-1} \Delta t$.
Please show your work clearly and in order.
\vf
\ep
% *** checking solutions
\bp{10}
Is $y(x) = e^x \ln x$ a solution to the initial value problem
$\dd{y}{x} = \frac{e^x}{x} + y, \;\; y(1)=0$? Justify your answer.
\vf
\ep
\pg
% ***************************************************************************
% *** SAD data
\bp{12}
Consider the following table giving the length of day (\ie number of
hours of daylight) on various days of the year in Durham,
North Carolina:
%
\gotable{2}{1.2}{|crc|p{3.5in}}
\cline{1-3}
& & Hours of & \hspace{.5in}{\it Space for graphing and scratch work:} \\
Date & Day & Daylight \\
\cline{1-3}
01/18 & 18 & 10.0 \\
03/15 & 74 & 12.3 \\
05/26 & 146 & 15.3 \\
06/19 & 170 & 15.7 \\
07/29 & 210 & 14.8 \\
09/07 & 250 & 13.3 \\
11/26 & 330 & \phantom{1}9.9 \\
12/20 & 354 & \phantom{1}9.5 \\
\cline{1-3}
\stoptable
Find a function of the form $f(x) = A \sin[B(x-C)] + D$ that would be
a good model for the data. Justify how you obtained each of the constants
$A$, $B$, $C$, and $D$ in the spaces provided. {\it (Hint: you
do not need to use all of the data points! You may assume that the
maximum and minimum day lengths are amongst the data above.)}
\vs
$$f(x) = \underline{\hspace{2in}}$$
\vs
\npz
$A = \; \underline{\hspace{.5in}} \;$ because:
\vf
\npz
$B = \; \underline{\hspace{.5in}} \;$ because:
\vf
\npz
$C = \; \underline{\hspace{.5in}} \;$ because:
\vf
\npz
$D = \; \underline{\hspace{.5in}} \;$ because:
\vf
\ep
\pg
% ***************************************************************************
% *** prove deriv of inv trig fncn
\bp{10} % 4.6(C) p.221; from 2/8 reading and lecture
Prove that the derivative of $\arctan(x)$ is $\frac{1}{1+x^2}$. Be
sure to show all work clearly and in order.
\vf \vf \vf
\ep
% *** derivs of trig and inv trig fncns
\bp{8}
Differentiate the following functions with respect to $x$.
\vs
\npz %4.5(C) #27, from 2/1 hw
$h(x) = x \cos x + \tan x$
\vf
\npz %4.6(C) #18, from 2/8 hw
$g(x) = e^{\arctan(3x^2)}$
\vf
\ep
\pg
% ***************************************************************************
% *** computing trig fncns exactly
\bp{12}
In the problems below, do all computations by hand; do not use your calculator.
Justify your answers by showing all work clearly and in order. Include
pictures of relevant angles, triangles, and the unit circle where appropriate.
\vs
\npz % CSPK trig review wksht; from hw
Find the exact value of $\cos(\frac{5\pi}{6})$.
\vf
\npz % 6(P) #8b; from 1/25 hw
If $-\frac{\pi}{2} < t < 0$ and $\sin(t) = -\frac{2}{3}$, find $\tan(t)$.
\vf
\npz % 1.9(C) #11; from 2/8 hw
Given that $\sin(\frac{\pi}{12}) = 0.259$ and
$\cos(\frac{\pi}{5}) = 0.809$, compute $\sin(\frac{11\pi}{12})$.
\vf
\ep
\pg
% ***************************************************************************
% *** newt's law of cooling IVP
\bp{10} % CSPK p.106, example 2, from 1/22 reading
Newton formulated the principle that the rate of change of the
temperature of an object is proportional to the difference between
the object's temperature and the temperature of the surroundings.
Suppose that a cup of coffee, which has temperature $110^{\circ}$~F, is
placed in a room which is at temperature $72^{\circ}$~F. Suppose that
we measure time in minutes after the placement of the coffee in the
room and that the constant of proportionality is .7.
\vs
\np{0}
Construct an initial value problem whose solution is the temperature
$T(t)$ of the coffee at time $t$.
\vf
\np{0}
Solve this IVP (show your work clearly please), and use the solution to find the
temperature of the coffee after 2 minutes.
\vf \vf \vf
\ep
\pg
% ***************************************************************************
% *** trig word problem
\bp{8} % 4.5(C) #37; from 2/1 hw
A lighthouse is 2 km from the long, straight coastline shown in the
figure below. Find the rate of change of the distance of the spot of
light from the point $O$ with respect to the angle $\theta$.
\vs
% MUST HAVE PICTURE! right triangle with height 2 km, shoreline along
% base, beam of light on hypontenuse, spot of light at intersection of
% these. theta is angle at top from point ``O'' to spot of light
\begin{figure}[h]
\epsfig{file=lighthouse.eps,height=2in,width=3.1in}
\end{figure}
\vf
\ep
% *** true or false about trig
\bp{8} % CSPK p.124 #13abcdef; from 5/2 hw (except for last two)
Label each of the items below as true (write TRUE) or false (write FALSE).
\vs \vs
\npz
\blank $\sin({\pi\over 2}-x)=\cos x$
\vs \vs
\npz
\blank $\sin x=\sqrt{1-\cos^2 x}$
\vs \vs
\npz
\blank $\sec x$ is undefined at $x={\pi\over 2}$
\vs \vs
\npz
\blank $\tan(-x)=\tan x$
\vs \vs
\npz
\blank $\displaystyle\cot x={{\cos x}\over{\sin x}}$
\vs \vs
\npz
\blank $3+\cos(2x)$ is an even function
\vs \vs
\npz
\blank The range of $\cos^{-1}(x)$ is $[-\frac{\pi}{2},\frac{\pi}{2}]$.
\vs \vs
\npz
\blank Elvis discovered the number ``$e$''.
\ep
\pg
% ***************************************************************************
% *** IVPS
\bp{12}
Solve the following initial value problems.
\vs
\npz % CSPK p.100 #2c from 1/15 hw
$y'(x) = e^{-x} - 7x, \quad y(0)=3$
\vf
\npz
$\dd{r}{s} = 2s, \quad r=4 \mbox{ when } s=1$
\vf
\npz % CSPK p.100 # 4b from 1/15 hw
$\dd{r}{s} = 2r, \quad r=4 \mbox{ when } s=1$
\vf
\npz
$\dd{y}{t} = \cos(3t) + 1, \quad y(0)=1$
\vf
\ep
\pg
% ***************************************************************************
% *** SURVEY
{\large \bf Survey Question (2 Extra Credit Points):}
\vs \vs
What did you think of this test? Were you prepared for the kinds of questions
that were on this test?
\vf
\pg
% ***************************************************************************
% *** SCRAP PAGE
{\large \bf Scrap Page}
(please do not remove this page from the test packet)
\showpoints
\end{document}