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\centerline{\textbf{Exam \#3 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 3 covers sections 3.6 through 4.7.  (Remember we skipped sections
3.9, 3.10, and 4.4).  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 3.6 through 4.7.  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} \\
2008-11-02

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

Topics:

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\begin{itemize}
\item Even though sections 3.1 through 3.5 were on the last exam, we're still
using everything from chapter 3.  You very much still need to know the product
rule, quotient rule, chain rule, derivatives of $\sin$, $\cos$,
$\tan$, etc.

\item Inverse-trig rules:
	\begin{eqnarray*}
		\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 Implicit differentiation:  Given an equation relating two variables
(say, $y$ and $x$), differentiate both sides and solve for $dy/dx$.
Remember to use the chain rule since $y$ depends on $x$.

	Example:
	\begin{eqnarray*}
		x^3 + y^2 &=& 3 \\
		3x^2 + 2y \frac{dy}{dx} &=& 0 \\
		\frac{dy}{dx} &=& \frac{-3x^2}{2y}.
	\end{eqnarray*}

\item Hyperbolic functions:
	\begin{itemize}
	\item Definitions:
		\begin{eqnarray*}
			\sinh(x) &=& \frac{1}{2}(e^x-e^{-x}), \\
			\cosh(x) &=& \frac{1}{2}(e^x+e^{-x}), \\
			\tanh(x) &=& \sinh(x)/\cosh(x).
		\end{eqnarray*}
	\item Be able to differentiate all three (use definitions and the chain
	rule, or memorize):
		\begin{eqnarray*}
			d/dx(\sinh(x)) &=& \cosh(x), \\
			d/dx(\cosh(x)) &=& \sinh(x), \\
			d/dx(\tanh(x)) &=& 1/\cosh^2(x).
		\end{eqnarray*}
	\item My primary interest in these functions for calculus per se is to add
	some variety to chain-rule problems.  Also, having you prove hyperbolic
	identities such as $\cosh^2(x)-\sinh^2(x)=1$ is an algebra exercise.  (It
	checks your proficiency with exponent rules.)

	\end{itemize}

\item Linear approximation:
	\begin{itemize}
	\item Know how to write an equation for the tangent line $\ell(x)$
		to a function $f(x)$.  Remember: point-slope form
		$y=m(x-a)+b$ with $b = f(a)$ and $m=f'(a)$.
	\item Graph and check for tangency at $x=a$.
	\item Remember that if $f''(a)>0$ (concave up there), the function
		will curve up away from the tangent line;
		if $f''(a)<0$ (concave down there), the function
		will curve down away from the tangent line.
	\item Example: $e^x$ near $x=1$.  $f(1) = e$; and $f'(x)=e^x$ so
		$f'(1)=e$.  Then $\ell(x)=e+e(x-1)=ex$.  Graph both $e^x$ and $ex$ and
		observe that the tangent line is below the original function so the
		linear approximation is an underapproximation.  We expect this since
		$e^x$ is concave up everywhere, so certainly concave up at $1$.
	\end{itemize}

\item Using first and second derivatives:
	\begin{itemize}
	\item Critical points: derivative is zero or undefined.

	\item Local maxima and minima:  occur at critical points or boundary points.

	\item How to tell if a critical point is a local maximum, local minimum, or
	a false alarm: graph, first derivative test, second derivative test.
	Graphing usually suffices unless there are parameters involved.

	\item Inflection points:  where the graph changes concavity.  Be able to
	compute a second derivative and solve for $x$.  Use the graph to determine
	if the point you find is an inflection point or a false alarm.
	(Examples: $x^4$ and $x^3$ at $x=0$.  Both have zero second derivatives
	there; only the latter function actually changes concavity there.)

	\end{itemize}

\item Families of curves:
	\begin{itemize}
	\item Be able to experimentally find out what effect a parameter has.
	E.g. for $f(x)=e^{-ax}$, plot it for bigger and smaller values of $a$ and
	tell me what they do.  (The more you remember from transformations of
	functions in college algebra / precalculus, the easier this will be.)

	\item If the first derivative is positive/negative at a point, the original
	function is increasing/decreasing there, respectively.

	\item If the second derivative is positive/negative at a point, the
	original function is concave up/down there, respectively.

	\item Find parameters to make a function satisfy certain constraints,
	including horizontal/vertical asymptotes, $x$- and $y$-intercepts, critical
	points, inflection points.

	Example: Let $f(x)=x^2+bx+c$.  Find $b$ and $c$ such that $f(x)$ has a
	$y$-intercept of 2 and a critical point at $x=3/2$.  The $y$-intercept is
	$f(0) = c = 2$.  Critical point at $x=3/2$ means $f'(3/2) = 0$.  Compute
	$f'(x)=2x+b$; then $f'(3/2) = 3+b=0$ so $b=-3$.

	Example:  Find $a$ so that $f(x)=x^3+ax^2+bx+c$ has an inflection
	point at $x=4$.  Compute $f'(x)=3x^2+2ax+b$ and $f''(x)=6x+2a$.
	Then we want $f''(4)=0$ so $24+2a=0$ so $a=-12$.  Then $b$ and $c$
	can be anything, say 0.  Graph $x^3-12x^2$ to check.  Does the
	concavity change at $x=4$?
	\end{itemize}

\item Optimization:
	\begin{itemize}
	\item Global maxima/minima of continuous functions occur only at critical
	points and boundary points.
	\item Upper/lower bounds are either global max/min (function
	reaches its extreme output values) or asymptotes (function approaches
	its extreme output values).
	\end{itemize}

\item Optimization and modeling:
	\begin{itemize}
	\item Draw a picture.  Label quantities and their relationships (these
	will become equations).
	\item Know what you want to optimize.
	\item Know what you can vary.
	\item Often, you will have an equation with more than one input.
		Eliminate a variable until the quantity you want to optimize is a
		function of one variable.
	\item Solve the 4.3 problem.
	\item Example:  Rectangle of maximum area with perimeter 200.
		Want to optimize $A=xy$.  This has two inputs, $x$ and $y$.  Since
		$2x+2y=200$, eliminate $y=100-x$.  Maximize $A=x(100-x)$.
	\item You will often need to know some geometry/trig facts to set these
	problems up, such as:
		\begin{itemize}
		\item Definitions of sine, cosine, and tangent in terms of sides
			of a triangle (SOHCAHTOA).
		\item Pythagorean theorem.
		\item Equation for a circle of radius $r$.
		\item Area of a circle, rectangle, triangle.
		\item Volume of a rectangular box or circular cylinder.
		\end{itemize}
	\item Also remember that areas of complicated objects (e.g. surface area of
	box with no lid --- I didn't memorize that formula!?!) are built up of
	simpler pieces (areas of the four sides and the bottom, all five of which
	are rectangles).
	\end{itemize}

\item Related rates:
	\begin{itemize}
	\item You have two quantities which are related by an equation.
	\item They both vary with respect to some variable (often, time).
	\item You know the rate of change of one quantity, either as a formula
	or for a single data point.
	\item Differentiate both sides (use the chain rule) and solve for the
	unknown rate of change.
	\item As with optimization problems, you may need to eliminate a
	variable.
	\item Example:  $A=\pi r^2$ and $dr/dt = 0.4$.  Find $dA/dt$ at $r=6$.
	Differentiating both sides and using the chain rule:
		$\frac{dA}{dt} = 2 \pi r \frac{dr}{dt}$.
	Then
		$\frac{dA}{dt}|_{r=6} = 2 \pi \cdot 6 \cdot 0.4 = 4.8 \pi$.

	\end{itemize}

\item L'H\^opital's rule:
	\begin{itemize}
	\item If $f(x)$ and $g(x)$ are differentiable at a point $a$,
		$f(a)=g(a)=0$, and $g'(a)=0$, then
		$\lim_{x\to a} f(x)/g(x) = f'(a)/g'(a)$.
	\item Example:  $h(x)=\sin(x)/x)$.  Graph it and see that it looks like
	$h(0)$ should be 1.  Using l'H\^opital's rule, $\lim_{x\to 0} h(x)
	= \cos(0)/1 = 1$ as expected.
	\item Also works for limits at infinity (horizontal asymptotes).
		In particular, this gives us yet another way to know that the
		horizontal asymptote of a balanced rational function is the ratio
		of leading coefficients (use l'H\^opital's rule repeatedly).
	\item Use it in the following situations:
		\begin{eqnarray*}
			\lim_{x\to a} \frac{f(x)}{g(x)} &=& \frac{0}{0} \\
			\lim_{x\to a} \frac{f(x)}{g(x)} &=& \frac{\infty}{\infty} \\
			\lim_{x\to \pm\infty} \frac{f(x)}{g(x)} &=& \frac{0}{0} \\
			\lim_{x\to \pm\infty} \frac{f(x)}{g(x)} &=& \frac{\infty}{\infty} \\
		\end{eqnarray*}
	\item Know when this rule applies and when it does not.
	Example: $$h(x)=f(x)/g(x)=(x+1)/(x+2).$$ Then $f'(x)/g'(x)=1/1=1$ everywhere,
	but $h(3) \ne 1$.  What went wrong?
	\end{itemize}
\end{itemize}

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