\documentclass[11pt]{amsart}
\parindent=0pt

% The next four lines override the settings in the amsart style file.
\voffset=-0.8in    % Moves text up
\hoffset=-1.2in    % Moves text left
\textwidth=6.8in   % Sets new page text width
\textheight=9.3in  % Sets new page text height

\usepackage{amssymb}
\usepackage{graphicx} % For including EPS files

\input{isomath}
\input{mathenv}

%% ================================================================
\begin{document}

\thispagestyle{empty} % Removes page numbering for this page only
\centerline{\textbf{\large Exam \#4 $\cdot$ Thurday, April 26, 2007}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 26 $\cdot$ Fall 2008  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 1.}
Let
\begin{eqnarray*}
	x(t) &=& 50 + \sin(2t)\cos(3t) \\
	y(t) &=& 20 + \sin^(4t).
\end{eqnarray*}

\textbf{Part (a).} Find the speed of the particle at $t=\pi/2$.

\textbf{Part (b).} Find an equation for the tangent line to this curve at
$t=\pi/2$.

%Parameteric equations:
%  * Write a parametric equation for a line between two specified points.
%  * Write a parametric equation for a line through a specified point,
%    in a specified direction.
%  * Given a curve specified by a parameteric equation and a point
%    on the curve, find an equation for the tangent line to the curve at
%    that point.
%  * Given a parameteric equation specifying the motion of a particle
%    along a curve, compute the speed of the particle at a given location or
%    time.
%  * Given a parameteric equation specifying the motion of a particle
%    along a curve, find the time(s) when the particle comes to a stop.
%  * Given parameteric equations specifying the motion of two particles,
%    find the time(s) when the particles collide.
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 2.}  The function $H(t)$ describes the growth rate in thousands
per month of flour beetles in a jar, where $t$ is measured in months since the
start of the year.

\textbf{Part (a).}  What are the units of $\int_{4}^{7} H(t) dt$?

\textbf{Part (b).}  Give a practical interpretation of $\int_{4}^{7} H(t) dt$.

%Numerical and graphical integration:
%  * Intuition for integrals:  Extend the familar concept of
%    $D=RT$ to handle non-constant $R$.  We sum up $RT$ products over small
%    $T$ intervals to obtain total distance.
%  * Approximate a definite integral using left-hand or
%    right-hand sums, with a specified rectangle width (e.g. $\Delta x =
%    0.25$) or specified number of rectangles (e.g. $n=10$).  Given input
%    units, give the units of the definite integral: units of $x$ times
%    units of $f(x)$.  Be able to do this when the integrand is specified by
%    an equation, a graph, or a data table.
%  * Remember that $\int_a^b f(x)dx$ is interpreted graphically as the
%    signed area between $f(x)$ and the $x$ axis, from $a$ to $b$.  Area
%    below the $x$ axis is taken to be negative.
%  * Describe the practical interpretation of a definite integral,
%    including units.  Example:  If $R(t)$ is rate of change of global
%    population in billions of people per year and $t$ is years,
%    then $\int_{2000}^{2004} R(t)dt$ is the population increase from 2000
%    to 2004 in billions of people.
%  * Remember that velocity is the first derivative of position:
%    $v = dx/dt$.
%  * Remember that acceleration is the first derivative of velocity:
%    $a = dv/dt$.
\vspace*{0.2in}

%% ----------------------------------------------------------------
%[Definite integral multiple ways]

%Fundamental theorem of calculus:
%  * To compute definite integrals:
%    $\int_a^b F'(x) dx = F(b) - F(a)$.
%    Point:  \emph{If} the integrand is the derivative of something (some
%    function $F$) \emph{then} we may replace all the rectangle sums etc.
%    with the very simple calculation $F(b)-F(a)$.
%  * For antiderivatives:
%    $F(b) = F(a) + \int_a^b F'(x) dx.$
%    Point:  Given a known value of the function at a point and
%    the known derivative of the function, you can find the value of
%    the function at subsequent points.
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 3.}  The air pressure within a chamber is given by
$$
	P(t) = 2.1 + 0.4 t^{0.5}
$$
where $P$ is in units called atmospheres and $t$ is measured in hours.
Find the average pressure from $t=2$ hours to $t=5$ hours.

%Average values:
%  * Definition: $\frac{\int_a^b f(x)dx}{b-a}$.
%  * Be able to compute an average value when the integrand is specified
%    by an equation, a graph, or a data table.
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 4.}  Consider the following integral:
$$
	\int_{0}^{2} \sqrt{3x^3+1} \,dx.
$$
I don't expect you to be able to compute it exactly.  Find a lower bound and
and an upper bound for the integral.  (Hint:  Graph the function.  Where are
its minimum and maximum values?)

%Properties of definite and indefinite integrals:
%  * Constant multiples pull out.
%  * Integrals distribute through sums and differences of functions.
%  * $\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$.
%  * Area between $g(x)$ (above) and $f(x)$ (below) is
%    $\int_a^b (g(x)-f(x))dx$.
%  * Comparison: If $g(x) \ge f(x)$ for all $x$ between $a$ and $b$,
%    then $\int_a^b g(x)dx \ge \int_a^b f(x)dx$.
%  * Use this in particular for upper and lower bounds on functions:  $M \ge
%    f(x) \ge m$ for all $x$ in $[a,b]$ means $\int_a^b M dx \ge \int_a^b f(x)
%    dx \ge \int_a^b m dx$.  The first integral is just $M(b-a)$ and the last
%    integral is just $m(b-a)$.
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 5.}  Compute the following antiderivative:
$$
	\int \left( \frac{x^{2.1}}{3} - \frac{7}{x} + 0.2 e^x \right) \;dx.
$$

% one big messy integral?

%Antiderivatives obtained symbolically:
%  * Know antiderivatives for constant functions, polynomial functions,
%    $1/x$, exponential function, sine and cosine.
%  * Remember that you always get a family of functions:
%    e.g. $\int \cos x = \sin x + C$.
%  * Use antiderivatives (indefinite integrals) to compute definite
%    integrals.  Example:  find $\int_2^8 e^x dx$.  Since $\int e^x dx =
%    e^x+C$, we have $\int_2^8 e^x dx = e^8-e^2$.  (What happened to the
%    integration constant?  It canceled out: $(e^8+C)-(e^2+C) = e^8-e^2$.)
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 6.} Let $g'(x)$ be given by
[insert graph of $g'(x)$ here] and suppose $g(0) = 2$.

\textbf{Part (a).}  What are the $x$-coordinates of the critical points of $g(x)$?

\textbf{Part (b).}  What are the $x$-coordinates of the inflection points of $g(x)$?

\textbf{Part (c).}  Find the values of $g(x)$ at the critical and inflection points.

\textbf{Part (d).}  Using the space provided, sketch a graph of $g(x)$.

% Antiderivatives obtained graphically:
%  * Given a qualitative graph (i.e. no axis labels) of $f'(x)$ and a
%    specified value for $f(0)$, sketch $f(x)$.
%  * If axis labels are given in the graph of $f'(x)$, then be able to
%    specify coordinates of critical points and inflection points of $f(x)$.
%    This requires you to use the FTC from one critical/inflection
%    point to the next.  Then, determine concavity of $f(x)$ on various
%    intervals.
%  * Remember the following:
%    * If $f'(x) > 0$ on an interval then $f(x)$ is increasing on that
%      interval.
%    * If $f'(x) < 0$ on an interval then $f(x)$ is decreasing on that
%      interval.
%    * If $f'(x) = 0$ at a point then $f(x)$ has a critical point
%      at that point.
%    * If $f'(x)$ is increasing on an interval then $f(x)$ is concave up
%      on that interval.
%    * If $f'(x)$ is decreasing on an interval then $f(x)$ is concave
%      down on that interval.
%    * If $f'(x)$ has a maximum or minimum at a point then $f(x)$ has an
%      inflection point there.
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 7.}  You are given the following tabulated values
of $\omega(t)$:
$$
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline $t$         & 0.0  & 0.5  & 1.0  & 1.5  & 2.0  \\
\hline $\omega(t)$ & 0.01 & 0.41 & 0.35 &-0.07 &-0.36 \\
\hline
\end{tabular}
$$
Fill out the following table with approximated values for
$$
	\theta(t) = \int_0^t \omega(s)ds,
$$
given that $\theta(0) = 10.0$.
$$
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline $t$         & $\quad 0.0\quad$ & $\quad 0.5\quad$ & $\quad 1.0\quad$ & $\quad 1.5 \quad$ & $\quad 2.0\quad$ \\
\hline             &     &     &     &     &     \\
       $\theta(t)$ &     &     &     &     &     \\
                   &     &     &     &     &     \\
\hline
\end{tabular}
$$

%% ----------------------------------------------------------------
\textbf{Problem 8.}  The quantity $A$ varies with time as specified by
$$
	\frac{dA}{dt} = 7.3\cos(t)-0.04.
$$

\textbf{Part (a).}  Write down a general solution for $A$.

\textbf{Part (b).}  Given that $A(0) = 2.1$, write a specific solution for $A$.

%Initial-value problems:
%  * Given $f'(x)$ (which I might write in the form $dy/dx$) and
%    an initial value for $f(0)$ ($y(0)$, respectively) write down the
%    general solution for the differential equation.  Then apply the initial
%    condition to determine the value of the integration constant.
%    Specify the units of your solution.
\vspace*{0.2in}

%% ----------------------------------------------------------------
\textbf{Problem 9.}  Find the following derivative:
$$
	\frac{d}{dx} \int_1^{x^2} e^{x} dx.
$$

%% ================================================================
\end{document}

% <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
%
% * coverage of topics
%
% * % familiar/new:  not *identical* to HW but in the ballpark
%
% * questions might double up topics, but try not to test too many concepts
%   in a single problem
%
% * length!!
%
% * complexity of problems / multiple steps
%
% * rule of four:  many types of questions
%
% * different variable names:  not just x
%
% * 1-2 writing questions.
%
% <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
