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\centerline{\textbf{Exam \#4 study guide $\cdot$
Math 124 $\cdot$ Calculus I $\cdot$ Section 8 $\cdot$ Spring 2007}}

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Disclaimers about the study guide:

\begin{itemize}

\item Exam 3 covers sections 4.8 through 6.4.  While all \emph{topics} on the
exam will be taken from this study guide, the specific \emph{questions} on the
exam will not be identical to the ones you see here.

\item In addition to consulting this guide, please review all homework problems
for sections 4.8 through 6.4.  In particular, look at unassigned problems
nearby.  For example, if I assigned \#14, see if you can do \#13 and \#15.

\item For reference, you can: (*) use the back of the book; (*) use the student
solution manual; (*) make use of the tutor center in Math East 145; (*) ask
questions in class; (*) talk to me after class, or during office hours.

\end{itemize}

John Kerl \\
\texttt{kerl.john.r@gmail.com} \\
2007-04-19

\begin{center}* * *\end{center}

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\textbf{Differentiation}:
	\begin{itemize}
	\item You still need to know all your differentiation rules, i.e.
		derivatives of elementary functions, product and quotient rules, chain
		rule, etc.  See the exam 2 and exam 3 study guides for more
		information.
	\end{itemize}

\textbf{Parameteric equations}:
	\begin{itemize}
	\item Write a parametric equation for a line between two specified points.
	\item Write a parametric equation for a line through a specified point,
		in a specified direction.
	\item 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.
	\item 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.
	\item Given a parameteric equation specifying the motion of a particle
		along a curve, find the time(s) when the particle comes to a stop.
		(Hint:  solve $dx/dt = 0$ and $dy/dt = 0$ as simultaneous equations,
		i.e. find $t$ such that $dx/dt=0$ and $dy/dt=0$.)
	\item Given parameteric equations specifying the motion of two particles,
		find the time(s) when the particles collide.
	\end{itemize}

\textbf{Numerical and graphical integration}:
	\begin{itemize}
	\item 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.
	\item The integral $\int_a^b f(x)dx$ 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.
	\item 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.
	\item 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.
	\item Remember that velocity is the first derivative of position:
		$v = dx/dt$.
	\item Remember that acceleration is the first derivative of velocity:
		$a = dv/dt$.
	\end{itemize}

\textbf{Fundamental theorem of calculus}:
	\begin{itemize}
	\item 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 relatively simple calculation $F(b)-F(a)$.
	\item 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.
	\end{itemize}

\textbf{Average values}:
	\begin{itemize}
	\item Definition: $\frac{\int_a^b f(x)dx}{b-a}$.
	\item Be able to compute an average value when the integrand is specified
		by an equation, a graph, or a data table.
	\end{itemize}

\textbf{Properties of definite and indefinite integrals}:
	\begin{itemize}
	\item Constant multiples pull out.
	\item Integrals distribute through sums and differences of functions.
	\item $\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$.
	\item Area between $g(x)$ (above) and $f(x)$ (below) is
		$\int_a^b (g(x)-f(x))dx$.
	\item Comparison: If $g(x) \ge f(x)$ for all $x$ between $a$ and $b$,
		then $\int_a^b g(x)dx \ge \int_a^b f(x)dx$.
	\item 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
		$m(b-a)$.
	\end{itemize}

\textbf{Antiderivatives obtained symbolically}:
	\begin{itemize}
	\item Know antiderivatives for constant functions, polynomial functions,
		$1/x$, exponential function, sine and cosine.
	\item Remember that you always get a family of functions:
		e.g. $\int \cos x = \sin x + C$.
	\item 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: $(e^8+C)-(e^2+C) = e^8-e^2$.)
	\end{itemize}

\textbf{Antiderivatives obtained graphically}:
	\begin{itemize}
	\item Given a qualitative graph (i.e. no axis labels) of $f'(x)$ and a
		specified value for $f(0)$, sketch $f(x)$.
	\item 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.
	\item Remember the following:
		\begin{itemize}
		\item If $f'(x) > 0$ on an interval then $f(x)$ is increasing on that
			interval.
		\item If $f'(x) < 0$ on an interval then $f(x)$ is decreasing on that
			interval.
		\item If $f'(x) = 0$ at a point then $f(x)$ has a critical point
			at that point.
		\item If $f'(x)$ is increasing on an interval then $f(x)$ is concave up
			on that interval.
		\item If $f'(x)$ is decreasing on an interval then $f(x)$ is concave
			down on that interval.
		\item If $f'(x)$ has a maximum or minimum at a point then $f(x)$ has an
			inflection point there.
		\end{itemize}
	\end{itemize}

\textbf{Antiderivatives obtained numerically}:
	\begin{itemize}
	\item Given a data table for $f'(x)$ and a specified value for $f(0)$,
		fill in a data table for values of $f(x)$.  (Here there isn't enough
		information for you to determine concavity of $f(x)$.)  Here too you
		need to use the FTC.
	\end{itemize}

\textbf{Initial-value problems}:
	\begin{itemize}
	\item 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.
	\end{itemize}

\textbf{Second fundamental theorem of calculus}:
	\begin{itemize}
	\item Statement:  If $f(x)$ is continuous, then
		$$
			\frac{d}{dx} \int_0^x f(t)dt = f(x).
		$$
	\item Point:  this means that $F(x)=\int_0^x f(t)dt$ is an antiderivative
		of $f(x)$.
	\item Given a graph of $f(x)$, sketch $F(x) = \int_0^x f(t)dt$.
	\item Given a data table for $f(x)$, fill in a data table for values of
		$F(x) = \int_0^x f(t)dt$.
		Example:
		$$
			\begin{tabular}{|c||c|c|c|c|c|}
			\hline $x$    & 0   & 2   & 4   & 6   & 8   \\
			\hline $f(x)$ & 1.1 & 1.3 & 1.4 & 1.2 & 0.9 \\
			\hline
			\end{tabular}
		$$
		First, $F(0) = 0$ since $\int_0^0 f(t)dt = 0$.  Then,
		\begin{eqnarray*}
			F(2) &=& F(0) + \int_0^2 f(t)dt                          \approx 0 + 2\cdot 1.1 = 2.2 \\
			F(4) &=& F(0) + \int_0^4 f(t)dt = F(2) + \int_2^4 f(t)dt \approx 2.2 + 2\cdot 1.3 = 4.8 \\
			F(6) &=& F(0) + \int_0^6 f(t)dt = F(4) + \int_4^6 f(t)dt \approx 4.8 + 2\cdot 1.4 = 7.6 \\
			F(8) &=& F(0) + \int_0^8 f(t)dt = F(6) + \int_6^8 f(t)dt \approx 7.6 + 2\cdot 1.2 = 10.0. \\
		\end{eqnarray*}
	\item Chain-rule problems involving the SFTC:
	We write
	$$
		F(x) = \int_a^{x} f(t) dt,
	$$
	and the SFTC says that
	$$
		\frac{d}{dx} F(x) = \frac{d}{dx} \int_a^{x} f(t) dt = f(x).
	$$
	i.e.  $F'(x) = f(x)$.
	So, it is legitimate to write
	$$
		F(u(x)) = \int_a^{u(x)} f(t) dt \\
	$$
	and ask about
	$$
		\frac{d}{dx} F(u(x)).
	$$
	Using the chain rule, this is
	\begin{eqnarray*}
		\frac{d}{dx} F(u(x)) &=& F(u(x)) u'(x) \\
		&=& f(u(x)) u'(x).
	\end{eqnarray*}
	Example:
	$$
		\frac{d}{dx} \int_2^{x^3} \sin(t^2) \, dt
		= \sin(x^6) \cdot 3x^2.
	$$
	If both endpoints are allowed to move, we have
	\begin{align*}
		\frac{d}{dx} \int_{\ell(x)}^{u(x)} f(t)\,dt
		&= f(u(x)) u'(x) - f(\ell(x)) \ell'(x).
	\end{align*}
	Example:
	$$
		\frac{d}{dx} \int_{x^2}^{x^3} \sin(t^2) \, dt
		= \sin(x^6) \cdot 3x^2 - \sin(x^4) \cdot 2x.
	$$
	\end{itemize}

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