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\centerline{\textbf{Exam \#2 study guide $\cdot$
Math 124 $\cdot$ Calculus I $\cdot$ Section 26 $\cdot$ Fall 2008}}

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Disclaimers about the study guide:

\begin{itemize}

\item Exam 2 covers sections 2.2 through 3.5.  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 2.2 through 3.5.  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} \\
2008-10-06

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

Topics:

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\begin{itemize}
\item Differentiability at a point $x=c$:  As you take secant lines through
	that point $c$ and neighboring points getting close to $c$, from either
	side, do their slopes approach some value?  If so, the function is
	differentiable there.

	Reasons for non-differentiability at a point, given a graph:
	\begin{itemize}
	\item The original function isn't defined at that point.
	\item Discontinuity at that point.
	\item A corner at that point.
	\item A vertical tangent line at that point.
	\end{itemize}

	Standard example:  $f(x)=|x|$ at $x=0$.  Secant lines through $0$ and any
	point to the right have slope $+1$; secant lines through $0$ and any point
	to the left have slope $-1$.  So, the secant lines aren't coming to any
	consensus as to what the slope of the derivative ought to be.
	(Graphically, $|x|$ has a corner at $x=0$.)

	Another example:  $f(x)=\sin(x)/x$.  This looks nice and level at $x=0$,
	so you'd think $f'(0)=0$, but this function isn't defined at 0 so we don't
	allow ourselves to define its derivative there either.

\item Numerically approximate a difference quotient:
	$$
	\frac{f(x+h)-f(x)}{h}
	$$
	for small $h$.

	Example: $f(x) = x^2$ at $x=3$.  (For this function, we know $f'(x)=2x$ so
	$f'(3)=6$.  But you can do this for all sorts of functions,
	even if you don't know a formula for the derivative.)

	$h=0.1$: $ [f(x+h)-f(x)]/h = (3.1^2 - 3^2)/0.1 = (9.61-9)/0.1 = 6.1$.

	$h=0.01$: $ [f(x+h)-f(x)]/h = (3.01^2 - 3^2)/0.01 = (9.0601-9)/0.01 = 6.01$.

	$h=0.001$: $ [f(x+h)-f(x)]/h = (3.001^2 - 3^2)/0.001 = (9.006001-9)/0.001 =
	6.001$.

\item Interpretations of the derivative:
	\begin{itemize}
	\item $f'>0$ when $f$ is increasing
	\item $f'<0$ when $f$ is decreasing
	\item $f'=0$ when $f$ is constant
	\item $f''>0$ when $f$ is concave up
	\item $f''<0$ when $f$ is concave down
	\item $f''=0$ when $f$ is linear
	\item Units of $f$, $f'$, and $f''$:
		If $x$ is in feet and $f(x)$ is in seconds, then $f'(x)$ is in feet per
		second and $f''(x)$ is in feet per second per second (which we also
		call feet per second squared).
	\item Verbal interpretation of $f$, $f'$, and $f''$.
	\end{itemize}

\item Sum rule: $(f+g)'=f'+g'$.
\item Difference rule: $(f-g)'=f'-g'$.
\item Power rule: $d/dx(x^r)=rx^{r-1}$.
\item Exponential rule: $d/dx(a^x)=\ln(a)a^x$.
\item Product rule: $(fg)'=f'g + fg'$.
\item Quotient rule: $(f/g)'=(f'g - fg')/g^2$.
\item Chain rule:
	\begin{itemize}
	\item $f(g(x))' = f'(g(x))g'(x)$.
	\item $\textrm{outer}'(\textrm{inner}) \textrm{inner}'$.
	\item $(d(\textrm{outer})/d\Box) \cdot (d\Box/dx)$.
	\end{itemize}
\item Trig rules:
	\begin{eqnarray*}
		\frac{d}{dx}(\sin(x)) &=& \cos(x) \\
		\frac{d}{dx}(\cos(x)) &=& -\sin(x) \\
		\frac{d}{dx}(\tan(x)) &=& \frac{1}{\cos^2(x)} \\
%		\frac{d}{dx}(\sin^{-1}(x)) &=& \frac{1}{\sqrt{1-x^2}} \\
%		\frac{d}{dx}(\tan^{-1}(x)) &=& \frac{1}{(1+x^2)} \\
	\end{eqnarray*}
%\item Log function:
%	$$\frac{d}{dx}(\ln(x)) = \frac{1}{x}.$$
%\item Logs with other bases:
%	$$\frac{d}{dx}(\log_b(x)) = \frac{1}{\ln(b) x}.$$
%	But for this exam, just know the derivative of $\ln(x)$.
%\item Be able to differentiate a piecewise function.  How?  Differentiate
%	each piece.
\item Be able to use multiple rules on a problem when necessary.
\item Know how to use derivatives to find equations of tangent lines (use
	point-slope form).

	\begin{itemize}
	\item Remember point-slope form: $y=m(x-a)+b$.  This is the equation
		of a line with slope $m$ going through the point $(a,b)$.

	\item Example:  $f(x)=x^2$ at $x=3$.  We have $a = 3$ and $b = f(a) =
	3^2 = 9$.  Since $f'(x)=2x$, $f'(3) = 6$ i.e. the slope at $x=3$ is 6.
	(You have to evaluate $f(a)$ to get $b$, \emph{and} you have to
	evaluate $f'(a)$ to get $m$.) Plugging these numbers into point-slope
	form gives
	$$
		y = 6(x-3) + 9 = 6x-18+9
	$$
	which simplifies to
	$$
		y = 6x-9.
	$$

	\item If you prefer, you might wish to memorize the following:
	$$
		y = f(a) + f'(a)(x-a).
	$$
	This works since we use $f'(a)$ for $m$ and $f(a)$ for $b$.
	\item Check:  Graph $y=x^2$ and $y=6x-9$ in your calculator; make sure
	the line is tangent to the curve at $x=3$.
	\end{itemize}

\item Given a formula for $f(x)$, know how to find $f'(c)$ for specific
	$c$, e.g. $f'(5)$.  This takes two steps: (1) find a formula for $f'(x)$;
	(2) evaluate that at $x=5$.

\item Given a graph for $f(x)$, know how to find $f'(c)$ for specific
	$c$, e.g. $f'(5)$.

\item What I may ask you about second derivatives:
	\begin{itemize}
	\item Given a function $f(x)$, find its second derivative.  Just compute
		its first derivative $f'(x)$, then differentiate \emph{that}.
		Example:  $f(x)=x^3$; $f'(x)=3x^2$; $f''(x)=6x$.
	\item Evaluate the second derivative at a point, e.g. $x=5$.
		Differentiate twice to get $f''(x)$, then evaluate that at $x=5$
		to get $f''(5)$.  Example:  if $f(x)=x^3$ then $f''(5)=30$.
	\item Find when the original function $f(x)$ is concave up or down.
		Just find $f''(x)$, then solve the inequality $f''(x)>0$ or $f''(x)<0$,
		respectively.  Example: $f(x)=x^3$.  Then $f'(x)=3x^2$ and $f''(x)=6x$.
		The original function $f(x)$ is concave up when $f''(x)>0$.
		So, solve $6x>0$: $x>0$.  Also graph to check.
	\end{itemize}
\end{itemize}

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