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For more info see http://www.lyx.org/ \lyxformat 345 \begin_document \begin_header \textclass scrreprt \begin_preamble \date{} \usepackage{euler} \usepackage{amsopn} \DeclareMathOperator{\arccot}{arccot} \DeclareMathOperator{\arcsec}{arcsec} \DeclareMathOperator{\arccsc}{arccsc} \DeclareMathOperator{\arc}{arc} \DeclareMathOperator{\trig}{trig} \end_preamble \use_default_options true \language english \inputencoding auto \font_roman palatino \font_sans helvet \font_typewriter courier \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize 10 \spacing single \use_hyperref true \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder true \pdf_colorlinks true \pdf_backref false \pdf_pdfusetitle true \papersize letterpaper \use_geometry true \use_amsmath 1 \use_esint 1 \cite_engine basic \use_bibtopic false \paperorientation portrait \leftmargin 3cm \topmargin 3cm \rightmargin 3cm \bottommargin 3cm \secnumdepth -2 \tocdepth 0 \paragraph_separation skip \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \author "" \author "" \end_header \begin_body \begin_layout Title Math Calculus Review \end_layout \begin_layout Author CHSN Review Project \end_layout \begin_layout Publishers \begin_inset Graphics filename ccbysa.png scale 50 \end_inset \end_layout \begin_layout Standard \begin_inset CommandInset toc LatexCommand tableofcontents \end_inset \end_layout \begin_layout Standard \SpecialChar \textcompwordmark{} \end_layout \begin_layout Standard This review guide was written by Dara Adib. Portions of the \begin_inset Quotes eld \end_inset Limits \begin_inset Quotes erd \end_inset and \begin_inset Quotes eld \end_inset Derivatives \begin_inset Quotes erd \end_inset chapters are based off the Calculus Wikibook available on the Internet at \begin_inset Flex URL status collapsed \begin_layout Plain Layout http://en.wikibooks.org/wiki/Calculus \end_layout \end_inset . CHSN Review Project contributors Dara Adib and Paul Sieradzki contributed to the \begin_inset Quotes eld \end_inset Limits \begin_inset Quotes erd \end_inset section of the Calculus Wikibook. \end_layout \begin_layout Standard This is a development version of the text that should be considered a work-in-pr ogress. \end_layout \begin_layout Standard This review guide and other review material are developed by the CHSN Review Project. \end_layout \begin_layout Standard Copyright © 2008-2009 Dara Adib and other contributors to the Calculus Wikibook. This is a freely licensed work, as explained in the Definition of Free Cultural Works ( \begin_inset Flex URL status collapsed \begin_layout Plain Layout freedomdefined.org \end_layout \end_inset ). Except as noted under \begin_inset Quotes eld \end_inset Graphic Credits \begin_inset Quotes erd \end_inset \begin_inset CommandInset ref LatexCommand vpageref reference "sec:graphics" \end_inset , it is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported License. To view a copy of this license, visit \begin_inset Newline linebreak \end_inset \begin_inset Flex URL status collapsed \begin_layout Plain Layout http://creativecommons.org/licenses/by-sa/3.0/ \end_layout \end_inset or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. \end_layout \begin_layout Standard This review guide is provided \begin_inset Quotes eld \end_inset as is \begin_inset Quotes erd \end_inset without warranty of any kind, either expressed or implied. You should not assume that this review guide is error-free or that it will be suitable for the particular purpose which you have in mind when using it. In no event shall the CHSN Review Project be liable for any special, incidental , indirect or consequential damages of any kind, or any damages whatsoever, including, without limitation, those resulting from loss of use, data or profits, whether or not advised of the possibility of damage, and on any theory of liability, arising out of or in connection with the use or performanc e of this review guide or other documents which are referenced by or linked to in this review guide. \end_layout \begin_layout Section* Graphic Credits \begin_inset CommandInset label LatexCommand label name "sec:graphics" \end_inset \end_layout \begin_layout Itemize \begin_inset CommandInset ref LatexCommand prettyref reference "fig:extrema" \end_inset is a public domain graphic by Inductiveload: \begin_inset Newline newline \end_inset \begin_inset Flex URL status collapsed \begin_layout Plain Layout http://commons.wikimedia.org/wiki/File:Maxima_and_Minima.svg \end_layout \end_inset \end_layout \begin_layout Itemize \begin_inset CommandInset ref LatexCommand prettyref reference "fig:inflection" \end_inset is a public domain graphic by Inductiveload: \begin_inset Newline newline \end_inset \begin_inset Flex URL status collapsed \begin_layout Plain Layout http://commons.wikimedia.org/wiki/File:X_cubed_(narrow).svg \end_layout \end_inset \end_layout \begin_layout Chapter Limits \end_layout \begin_layout Standard This chapter was originally designed for a test on limits administered by Jeanine Lennon to her Math 12H (4H/Precalculus) class on April 2, 2008. It was later updated with an \begin_inset Quotes eld \end_inset Addendum \begin_inset Quotes erd \end_inset section (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:addendum" \end_inset ) for a test on limits administered by Jonathan Chernick to his AP \begin_inset Foot status collapsed \begin_layout Plain Layout AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. \end_layout \end_inset Calculus BC class on September 18, 2008. \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard A limit looks at what happens to a function when the input approaches, but does not necessarily reach, a certain value. The general notation for a limit is below. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to c}f(x)=L}$ \end_inset \end_layout \begin_layout Standard This is read as \begin_inset Quotes eld \end_inset the limit of \begin_inset Formula $f(x)$ \end_inset as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $c$ \end_inset is \begin_inset Formula $L$ \end_inset . \begin_inset Quotes erd \end_inset \end_layout \begin_layout Subsection Informal Definition of a Limit \end_layout \begin_layout Standard \begin_inset Formula $L$ \end_inset is the limit of \begin_inset Formula $f(x)$ \end_inset as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $c$ \end_inset . The value of \begin_inset Formula $f(x)$ \end_inset comes close to \begin_inset Formula $L$ \end_inset when \begin_inset Formula $x$ \end_inset is close (but not necessarily equal) to \begin_inset Formula $c$ \end_inset . It can be represented by either of the following forms, with the former being far more common. \end_layout \begin_layout Itemize \begin_inset Formula ${\displaystyle \lim_{x\to c}f(x)=L}$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $f(x)\to L$ \end_inset as \begin_inset Formula $x\to c$ \end_inset \end_layout \begin_layout Section Rules \begin_inset CommandInset label LatexCommand label name "cha:Rules" \end_inset \end_layout \begin_layout Standard Now that a limit has been informally defined, some rules that are useful for manipulating a limit are listed. \end_layout \begin_layout Subsection Identities \end_layout \begin_layout Standard The following identities assume \begin_inset Formula ${\displaystyle \lim_{x\to c}f(x)=L}$ \end_inset and \begin_inset Formula ${\displaystyle \lim_{x\to c}g(x)=M}$ \end_inset . Using these identities, other rules can be deduced. \end_layout \begin_layout Subsubsection Scalar Multiplication \end_layout \begin_layout Standard A scalar is a constant. When a function is multiplied by a constant, scalar \series bold \series default multiplication is performed. \end_layout \begin_layout Itemize \begin_inset Formula ${\displaystyle \lim_{x\to c}kf(x)=k\cdot\lim_{x\to c}f(x)=kL}$ \end_inset \end_layout \begin_layout Subsubsection Addition \end_layout \begin_layout Itemize \begin_inset Formula ${\displaystyle \lim_{x\to c}[f(x)+g(x)]=\lim_{x\to c}f(x)+\lim_{x\to c}g(x)=L+M}$ \end_inset \end_layout \begin_layout Subsubsection Subtraction \end_layout \begin_layout Itemize \begin_inset Formula ${\displaystyle \lim_{x\to c}[f(x)-g(x)]=\lim_{x\to c}f(x)-\lim_{x\to c}g(x)=L-M}$ \end_inset \end_layout \begin_layout Subsubsection Multiplication \end_layout \begin_layout Itemize \begin_inset Formula ${\displaystyle \lim_{x\to c}[f(x)\cdot g(x)]=\lim_{x\to c}f(x)\cdot\lim_{x\to c}g(x)=L\cdot M}$ \end_inset \end_layout \begin_layout Subsubsection Division \end_layout \begin_layout Itemize \begin_inset Formula ${\displaystyle \lim_{x\to c}\frac{f(x)}{g(x)}=\frac{\lim_{x\to c}f(x)}{\lim_{x\to c}g(x)}=\frac{L}{M}}$ \end_inset , where \begin_inset Formula $M\neq0$ \end_inset \end_layout \begin_layout Subsection Constant Rule \end_layout \begin_layout Standard The constant rule states that if \begin_inset Formula $f(x)=k$ \end_inset is constant for all \begin_inset Formula $x$ \end_inset , then the limit as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $c$ \end_inset must be equal to \begin_inset Formula $k$ \end_inset . \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to c}k=k}$ \end_inset \end_layout \begin_layout Subsection Identity Rule \end_layout \begin_layout Standard The identity rule states that if \begin_inset Formula $f(x)=x$ \end_inset , then the limit as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $c$ \end_inset is equal to \begin_inset Formula $c$ \end_inset . \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to c}x=c}$ \end_inset \end_layout \begin_layout Subsection Power Rule \end_layout \begin_layout Standard The rule for products many times results in determining the power rule. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to c}f(x)^{n}=\left(\lim_{x\to c}f(x)\right)^{n}}$ \end_inset \end_layout \begin_layout Section Finding Limits \end_layout \begin_layout Standard If \begin_inset Formula $c$ \end_inset is in the domain of the function and the function can be built out of rational, trigonometric, logarithmic and exponential functions, then the limit is simply the value of the function at \begin_inset Formula $c$ \end_inset . \end_layout \begin_layout Standard If \begin_inset Formula $c$ \end_inset is not in the domain of the function, then in many cases (as with rational functions) the domain of the function includes all of the points near \begin_inset Formula $c$ \end_inset , but not \begin_inset Formula $c$ \end_inset . An example would be if one wanted to find \begin_inset Formula ${\displaystyle \lim_{x\to0}\frac{x}{x}}$ \end_inset , where the domain includes all real numbers except \begin_inset Formula $0$ \end_inset . In that case, one would want to find a similar function, with the hole filled in. The limit of this function at \begin_inset Formula $c$ \end_inset will be the same, while the function is the same at all points not equal to \begin_inset Formula $c$ \end_inset . The limit definition depends on \begin_inset Formula $f(x)$ \end_inset only at the points where \begin_inset Formula $x$ \end_inset is close to \begin_inset Formula $c$ \end_inset but not equal to it. And since the domain of the new function includes \begin_inset Formula $c$ \end_inset , one can now (assuming it's still built out of rational, trigonometric, logarithmic and exponential functions) just evaluate the function at \begin_inset Formula $c$ \end_inset as before. \end_layout \begin_layout Standard In the above example, this is easy; canceling the \begin_inset Formula $x$ \end_inset 's gives 1, which equals \emph on \begin_inset Formula $\nicefrac{x}{x}$ \end_inset \emph default at all points except \begin_inset Formula $0$ \end_inset . Thus, \begin_inset Formula ${\displaystyle \lim_{x\to0}\frac{x}{x}=\lim_{x\to0}1=1}$ \end_inset . In general, when computing limits of rational functions, it's a good idea to look for common factors in the numerator and denominator. \end_layout \begin_layout Subsection Does Not Exist \end_layout \begin_layout Standard Note that the limit might not exist at all. There are a number of ways in which this can occur. \end_layout \begin_layout Subsubsection Not Same from Both Sides \end_layout \begin_layout Standard A left-handed limit is different from the right-handed limit of the same variable, value, and function. Since, the left-handed limit \begin_inset Formula $\neq$ \end_inset right-handed limit, the limit does not exist. This includes cases in which the limit of a certain side does not exist (e.g. \begin_inset Formula ${\displaystyle \lim_{x\to2}\sqrt{x-2}}$ \end_inset , which has no left-handed limit). \end_layout \begin_layout Subsubsection Gap \end_layout \begin_layout Standard There is a gap (more than a point wide) in the function where the function is not defined. As an example, in \begin_inset Formula $f(x)=\sqrt{x^{2}-16}$ \end_inset , \begin_inset Formula $f(x)$ \end_inset does not have any limit when \emph on \begin_inset Formula $-4\leq x\leq4$ \end_inset \emph default . There is no way to \begin_inset Quotes eld \end_inset approach \begin_inset Quotes erd \end_inset the middle of the graph. Note also that the function also has no limit at the endpoints of the two curves generated (at \begin_inset Formula $x=-4$ \end_inset and \begin_inset Formula $x=4$ \end_inset ) since limits from both sides do not exist. \end_layout \begin_layout Subsubsection Jump \end_layout \begin_layout Standard If the graph suddenly jumps to a different level, there is no limit. This is illustrated in the floor function (in which the output value is the greatest integer not greater than the input value). The limit does not exist when the greatest integer function approaches an integer ( \begin_inset Formula ${\displaystyle \lim_{x\to integer}\lfloor x\rfloor}$ \end_inset , also written as int \begin_inset Formula $x$ \end_inset ). \begin_inset Formula $\nicefrac{|x|}{x}$ \end_inset and \begin_inset Formula $\nicefrac{x}{|x|}$ \end_inset are other examples of graphs that contain jumps. \end_layout \begin_layout Subsubsection Infinite Oscillation \end_layout \begin_layout Standard This can be tricky to visualize. A graph continually rises above and below a horizontal line as it approaches a certain \begin_inset Formula $x$ \end_inset -value, for instance infinity. This often means that the limit does not exist, as the graph never approaches a particular value. However, if the height (and depth) of each oscillation diminishes as the graph approaches the \begin_inset Formula $x$ \end_inset -value, so that the oscillations get arbitrarily smaller, then there might actually be a limit. \end_layout \begin_layout Standard The use of oscillation naturally calls to mind the trigonometric functions. An example of a trigonometric function that does not have a limit as \emph on \emph default \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $0$ \end_inset is \begin_inset Formula $f(x)=\sin\frac{1}{x}$ \end_inset . As \begin_inset Formula $x$ \end_inset gets closer to \begin_inset Formula $0$ \end_inset , the function keeps oscillating between \begin_inset Formula $-1$ \end_inset and \begin_inset Formula $1$ \end_inset . \end_layout \begin_layout Subsubsection Incomplete Graph \end_layout \begin_layout Standard Consider the following example. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle g(x)=\left\{ \begin{matrix}2, & \mbox{if }x\mbox{ is rational}\\ 0, & \mbox{if }x\mbox{ is irrational}\end{matrix}\right.}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula $g(x)$ \end_inset does not have a limit. For let \begin_inset Formula $x$ \end_inset be a real number, \begin_inset Formula $g(x)$ \end_inset can't have a limit at \begin_inset Formula $x$ \end_inset . No matter how close one gets to \begin_inset Formula $x$ \end_inset , there will be rational numbers (when \begin_inset Formula $g(x)$ \end_inset will be \begin_inset Formula $2$ \end_inset ) and irrational numbers (when \begin_inset Formula $g$ \end_inset will be \begin_inset Formula $0$ \end_inset ). Thus \begin_inset Formula $g(x)$ \end_inset has no limit at any real number. \end_layout \begin_layout Subsection One-Sided Limits \end_layout \begin_layout Standard Sometimes, it is necessary to consider what happens when one approaches an \begin_inset Formula $x$ \end_inset value from one particular direction. To accommodate for this, there are one-sided limits. In a left-handed limit, \begin_inset Formula $x$ \end_inset approaches a from the left hand side (negative). Likewise, in a right-handed limit, \begin_inset Formula $x$ \end_inset approaches a from the right hand side (positive). \end_layout \begin_layout Standard For example, \begin_inset Formula ${\displaystyle \lim_{x\to2}\sqrt{x-2}}$ \end_inset does not exist because there is no left-handed limit. \end_layout \begin_layout Standard The left-handed limit, which does not exist, is expressed as the following. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to2^{-}}\sqrt{x-2}}$ \end_inset \end_layout \begin_layout Standard The right-handed limit, which equals \begin_inset Formula $0$ \end_inset , is expressed as the following. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to2^{+}}\sqrt{x-2}=0}$ \end_inset \end_layout \begin_layout Section Infinite Limits \end_layout \begin_layout Standard Limits can also involve looking at what happens to \begin_inset Formula $f(x)$ \end_inset as \begin_inset Formula $x$ \end_inset gets very big. For example, consider the function \begin_inset Formula $f(x)=\frac{1}{x}$ \end_inset . As \begin_inset Formula $x$ \end_inset becomes very big, \begin_inset Formula $\frac{1}{x}$ \end_inset becomes closer to zero. Without limits it is very difficult to talk about this fact, because \begin_inset Formula $\frac{1}{x}$ \end_inset never actually becomes zero. But the language of limits exists precisely to let one talk about the behavior of a function as it approaches something, without caring about the fact that it will never get there. In this case, however, the same problem as before exists; how big does \begin_inset Formula $x$ \end_inset have to be to be sure that \begin_inset Formula $f(x)$ \end_inset is really going towards \begin_inset Formula $0$ \end_inset ? \end_layout \begin_layout Standard In this case, the bigger \begin_inset Formula $x$ \end_inset gets, the closer \begin_inset Formula $f(x)$ \end_inset should get to \begin_inset Formula $0$ \end_inset . Really, this means that however close one wants \begin_inset Formula $f(x)$ \end_inset to get to \begin_inset Formula $0$ \end_inset , one can find an \begin_inset Formula $x$ \end_inset big enough so \begin_inset Formula $f(x)$ \end_inset is that close. This is written in a similar way to finite limits and is read as \begin_inset Quotes eld \end_inset the limit, as \begin_inset Formula $x$ \end_inset approaches infinity, equals \begin_inset Formula $0$ \end_inset , \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset as \begin_inset Formula $x$ \end_inset approaches infinity, the function approaches \begin_inset Formula $0$ \end_inset . \begin_inset Quotes erd \end_inset \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to\infty}\frac{1}{x}=0}$ \end_inset \end_layout \begin_layout Subsection Rules \end_layout \begin_layout Standard The easiest way to determine limits as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $\pm\infty$ \end_inset is by using the graphing calculator to make observations, or by plugging in high values of positive and negative numbers in a calculator. \end_layout \begin_layout Standard However, there are three rules for determining a limit of a fraction analyticall y as a variable approaches infinity. For each rule, one must look at the variables on both the numerator and denominator of the function. \end_layout \begin_layout Standard Look for the highest term (with the highest exponent) in the numerator. Look for the same in the denominator. These rules are based on that information. \end_layout \begin_layout Itemize If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit is the fractional ratio of the coefficients of the highest terms. \end_layout \begin_layout Itemize If the \emph on numerator \emph default has the highest term, then the fraction is called \begin_inset Quotes eld \end_inset top heavy \begin_inset Quotes erd \end_inset and the limit is infinity. \end_layout \begin_layout Itemize If the \emph on denominator \emph default has the highest term, then the fraction is called \begin_inset Quotes eld \end_inset bottom heavy \begin_inset Quotes erd \end_inset and the limit is zero. \end_layout \begin_layout Standard If there is no denominator stated, it is understood that the denominator is 1 or \begin_inset Formula $1n^{0}$ \end_inset , and the limit will be infinity. \end_layout \begin_layout Section Asymptotes \end_layout \begin_layout Subsection Vertical Asymptotes \end_layout \begin_layout Standard The line \begin_inset Formula $x=a$ \end_inset is a vertical asymptote for the function \begin_inset Formula $f(x)$ \end_inset if at least one of the following statements is true. \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\to a}f(x)=\pm\infty}$ \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\to a^{-}}f(x)=\pm\infty}$ \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\to a^{+}}f(x)=\pm\infty}$ \end_inset \end_layout \begin_layout Standard The limits from both directions do not have to be equal to have an asymptote, but they may be equal. Essentially, a vertical asymptote occurs where the the value of a limit is positive or negative infinity from any direction. \end_layout \begin_layout Standard Recall that this occurs where the fraction of a function is undefined (denominat or equals zero). \end_layout \begin_layout Subsection Horizontal Asymptotes \end_layout \begin_layout Standard The line \begin_inset Formula $y=a$ \end_inset is a horizontal asymptote for the function \begin_inset Formula $f(x)$ \end_inset if \begin_inset Formula ${\displaystyle \lim_{x\to\infty}f(x)=a}$ \end_inset or \begin_inset Formula ${\displaystyle \lim_{x\to-\infty}f(x)=a}$ \end_inset . If \begin_inset Formula ${\displaystyle \lim_{x\to\infty}f(x)=a}$ \end_inset and \begin_inset Formula ${\displaystyle \lim_{x\to-\infty}f(x)=b}$ \end_inset , then the function \begin_inset Formula $f(x)$ \end_inset has two asymptotes at \begin_inset Formula $y=a$ \end_inset and \begin_inset Formula $y=b$ \end_inset . Note that in some functions, the graph may pass through the horizontal asymptote at an \begin_inset Formula $x$ \end_inset value of zero. \end_layout \begin_layout Standard Essentially, a horizontal asymptote occurs at the value of a limit where \begin_inset Formula $x$ \end_inset approaches positive or negative infinity. \end_layout \begin_layout Standard Recall that rules exist for calculating the the value of a limit where \begin_inset Formula $x$ \end_inset approaches positive or negative infinity. \end_layout \begin_layout Section Continuity \end_layout \begin_layout Subsection Definition \end_layout \begin_layout Standard The formal definition of continuity is simple. \end_layout \begin_layout Quote If \begin_inset Formula $f(x)$ \end_inset is defined on an open interval containing \begin_inset Formula $c$ \end_inset , then \begin_inset Formula $f(x)$ \end_inset is said to be continuous at \begin_inset Formula $c$ \end_inset if and only if the limit as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $c$ \end_inset equals \begin_inset Formula $f(c)$ \end_inset . \end_layout \begin_layout Quote \begin_inset Formula ${\displaystyle \lim_{x\rightarrow c}f(x)=f(c)}$ \end_inset \end_layout \begin_layout Standard Note that for \begin_inset Formula $f(x)$ \end_inset to be continuous at \begin_inset Formula $c$ \end_inset , the definition requires three conditions. \end_layout \begin_layout Enumerate \size normal \begin_inset Formula $f(x)$ \end_inset is defined at \begin_inset Formula $c$ \end_inset \end_layout \begin_deeper \begin_layout Enumerate \size normal \begin_inset Formula $f(c)$ \end_inset exists \end_layout \end_deeper \begin_layout Enumerate \size normal The limit as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $c$ \end_inset exists. \end_layout \begin_deeper \begin_layout Enumerate \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula ${\displaystyle \lim_{x\rightarrow c}f(x)}$ \end_inset exists \end_layout \end_deeper \begin_layout Enumerate \size normal The limit and \begin_inset Formula $f(c)$ \end_inset are equal. \end_layout \begin_deeper \begin_layout Enumerate \size normal \begin_inset Formula ${\displaystyle f(c)=\lim_{x\rightarrow c}f(x)}$ \end_inset \end_layout \end_deeper \begin_layout Standard If any of these do not hold then \begin_inset Formula $f(x)$ \end_inset is not continuous at \begin_inset Formula $c$ \end_inset . \end_layout \begin_layout Standard Notice how this relates to the idea of continuity. To be continuous, the function must be uniformly \begin_inset Quotes eld \end_inset smooth \begin_inset Quotes erd \end_inset (e.g. no \begin_inset Quotes eld \end_inset gaps, \begin_inset Quotes erd \end_inset breaks, or sharp turns/corners) within an interval. \end_layout \begin_layout Quote A function is said to be continuous if it is continuous at every point \begin_inset Formula $c$ \end_inset in its domain. \end_layout \begin_layout Standard A function may be continuous at a certain point, but not a continuous function (throughout). Likewise, a discontinuous function may be continuous at a certain point. \end_layout \begin_layout Subsection Removable Discontinuities \end_layout \begin_layout Description discontinuity point where a function is not continuous \end_layout \begin_layout Standard If there is a \begin_inset Quotes eld \end_inset gap \begin_inset Quotes erd \end_inset one point wide on a graph \size normal ( \begin_inset Formula $f(c)$ \end_inset \size default does not exist) or if there is a \begin_inset Quotes eld \end_inset jump \begin_inset Quotes erd \end_inset one point wide on a graph ( \size normal \begin_inset Formula ${\displaystyle f(c)\neq\lim_{x\rightarrow c}f(x)}$ \end_inset ), the discontinuity is removable. Gap discontinuities ( \family roman \series medium \shape up \emph off \bar no \noun off \color none \begin_inset Formula ${\displaystyle \lim_{x\rightarrow c}f(x)}$ \end_inset does not exist), j \family default \series default \shape default \emph default \bar default \noun default \color inherit ump discontinuities \size default ( \size normal \begin_inset Formula ${\displaystyle f(c)\neq\lim_{x\rightarrow c}f(x)}$ \end_inset ), and infinite oscillation discontinuities are non-removable. \end_layout \begin_layout Standard The function \begin_inset Formula $f(x)=\frac{x^{2}-9}{x-3}$ \end_inset is considered to have a removable discontinuity at \begin_inset Formula $x=3$ \end_inset . It is discontinuous at that point because the fraction then becomes \begin_inset Formula $\frac{0}{0}$ \end_inset which is undefined. Therefore the function fails the very first condition of continuity. \end_layout \begin_layout Standard If the function is slightly modified, the discontinuity can be removed and the function becomes continuous. Standard algebraic techniques for simplifying fractions and algebraic expressio ns (e.g. factoring, multiplying by conjugates) can be used. \end_layout \begin_layout Standard To make the function \begin_inset Formula $f(x)$ \end_inset continuous, \begin_inset Formula $f(x)$ \end_inset must be simplified. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle f(x)=\frac{x^{2}-9}{x-3}=\frac{(x+3)(x-3)}{(x-3)}=\frac{x+3}{1}\times\frac{x-3}{x-3}=\frac{x+3}{1}\times1=x+3}$ \end_inset \end_layout \begin_layout Standard As long as \begin_inset Formula $x\neq3$ \end_inset , the function \begin_inset Formula $f(x)$ \end_inset can be simplified to get a new function \begin_inset Formula $g(x)$ \end_inset . \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle g(x)=\left\{ \begin{matrix}x+3, & \mbox{if }x\ne3\\ 6, & \mbox{if }x=3\end{matrix}\right.}$ \end_inset \end_layout \begin_layout Standard Note that the function \begin_inset Formula $g(x)$ \end_inset is not the same as the original function \begin_inset Formula $f(x)$ \end_inset , because \begin_inset Formula $g(x)$ \end_inset has the extra point \begin_inset Formula $(3,6)$ \end_inset . \begin_inset Formula $g(x)$ \end_inset is now defined for \begin_inset Formula $x=3$ \end_inset , and therefore continuous. \end_layout \begin_layout Subsection Properties \end_layout \begin_layout Quote If \begin_inset Formula $f(x)$ \end_inset and \begin_inset Formula $g(x)$ \end_inset are continuous, then the following are also continuous. \end_layout \begin_deeper \begin_layout Itemize \begin_inset Formula $f(x)+g(x)$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $f(x)\cdot g(x)$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $f(x)-g(x)$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula ${\displaystyle \frac{f(x)}{g(x)}}$ \end_inset , \begin_inset Formula $g\neq0$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $k\times f(x)$ \end_inset , where \begin_inset Formula $k$ \end_inset is a constant \end_layout \end_deeper \begin_layout Subsection Intermediate Value Theorem \end_layout \begin_layout Standard A graph of a continuous function has no breaks, so a point between two \begin_inset Formula $x$ \end_inset -values has a \begin_inset Formula $y$ \end_inset -value between the \begin_inset Formula $y$ \end_inset -values of the respective \begin_inset Formula $x$ \end_inset -values. \end_layout \begin_layout Quote If a function is continuous on the closed interval \begin_inset Formula $[a,b]$ \end_inset , then for every value \begin_inset Formula $k$ \end_inset between \begin_inset Formula $f(a)$ \end_inset and \begin_inset Formula $f(b)$ \end_inset there is a value \begin_inset Formula $c$ \end_inset on \begin_inset Formula $[a,b]$ \end_inset such that \begin_inset Formula $f(c)=k$ \end_inset . \end_layout \begin_layout Standard This can be used to approximate when the \begin_inset Formula $y$ \end_inset -value of a function will be a certain value (e.g. the \begin_inset Formula $x$ \end_inset \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none -value \family default \series default \shape default \size default \emph default \bar default \noun default \color inherit when \begin_inset Formula $y=4$ \end_inset ). \end_layout \begin_layout Subsection Calculating Continuities \end_layout \begin_layout Standard One should be able to determine where a function is discontinuous. In some cases, one may be required to determine the value(s) of variable(s) in rule(s) of a function so that the function will be continuous. A system of equations is required when there are multiple variables. \end_layout \begin_layout Section Trigonometric Functions \end_layout \begin_layout Standard In most cases, limits with trigonometric functions can be treated the same way as other limits. \end_layout \begin_layout Standard One can substitute into the expression if possible, or use the graphing calculator. \end_layout \begin_layout Standard If divide by zero occurs, one may eliminate removable discontinuities if they exist or use the graphing calculator. In some cases, factoring to eliminate removable discontinuities can only be done if trigonometric identities are used first. \end_layout \begin_layout Labeling \labelwidthstring 00.00.0000 Note When graphing, stay in radian mode as the limits are provided in radian mode unless stated otherwise. \end_layout \begin_layout Subsection Trigonometric Identities \end_layout \begin_layout Subsubsection Pythagorean Identities \end_layout \begin_layout Enumerate \begin_inset Formula $\sin^{2}\theta+\cos^{2}\theta=1$ \end_inset \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula $1+\tan^{2}\theta=\sec^{2}\theta$ \end_inset \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula $1+\cot^{2}\theta=\csc^{2}\theta$ \end_inset \end_layout \begin_layout Subsubsection Quotient Identities \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}}$ \end_inset \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \cot\theta=\frac{\cos\theta}{\sin\theta}}$ \end_inset \end_layout \begin_layout Section Addendum \begin_inset CommandInset label LatexCommand label name "sec:addendum" \end_inset \end_layout \begin_layout Standard This section was designed for a test on limits administered by Jonathan Chernick to his AP \begin_inset Foot status collapsed \begin_layout Plain Layout AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. \end_layout \end_inset Calculus BC class on September 18, 2008. It is not covered in Math 12H/4H. \end_layout \begin_layout Subsection Further Trigonometric Identities \end_layout \begin_layout Standard These identities can be used for the same purpose as the other trigonometric identities. To use these identities, the limits may need to be multiplied by a certain factor or separated based on the rules \begin_inset CommandInset ref LatexCommand vpageref reference "cha:Rules" \end_inset . \end_layout \begin_layout Subsubsection Sine \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\rightarrow0}\frac{\sin x}{x}=1}$ \end_inset \end_layout \begin_layout Subsubsection Cosine \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\rightarrow0}\frac{1-\cos x}{x}=0}$ \end_inset \end_layout \begin_layout Subsection Squeeze (Sandwich) Theorem \end_layout \begin_layout Standard The squeeze theorem, also known as the sandwich theorem, is used to find the limit of a function by comparison with two other functions whose limits are known or easily computed. It refers to a function \begin_inset Formula $f(x)$ \end_inset whose values are squeezed between the values of two other functions \begin_inset Formula $g(x)$ \end_inset and \begin_inset Formula $h(x)$ \end_inset , both of which have the same limit \begin_inset Formula $L$ \end_inset . If the value of \begin_inset Formula $f(x)$ \end_inset is trapped between the values of the two functions \begin_inset Formula $g(x)$ \end_inset and \begin_inset Formula $h(x)$ \end_inset , the values of \begin_inset Formula $f(x)$ \end_inset must also approach \begin_inset Formula $L$ \end_inset . \end_layout \begin_layout Quote If the following are true: \end_layout \begin_deeper \begin_layout Enumerate \begin_inset Formula $g(x)\le f(x)\le h(x)$ \end_inset for all \begin_inset Formula $x$ \end_inset not equal to \begin_inset Formula $c$ \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\rightarrow c}g(x)=\lim_{x\rightarrow c}h(x)=L}$ \end_inset \end_layout \end_deeper \begin_layout Quote Then \begin_inset Formula ${\displaystyle \lim_{x\rightarrow c}f(x)=L}$ \end_inset . \end_layout \begin_layout Subsubsection Example \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\to0}x\sin\frac{1}{x}}$ \end_inset \end_layout \begin_layout Standard Note that the sine of anything is in the interval \begin_inset Formula $[-1,1]$ \end_inset . In other words, \begin_inset Formula $-1\le\sin x\le1$ \end_inset for all \begin_inset Formula $x$ \end_inset ). As a result, for all nonzero \begin_inset Formula $x$ \end_inset , \begin_inset Formula $-1\times\left|x\right|\le x\sin\frac{1}{x}\le1\times\left|x\right|$ \end_inset . Simplified, this means \begin_inset Formula ${\displaystyle -\left|x\right|\le x\sin\frac{1}{x}\le\left|x\right|}$ \end_inset . Since \begin_inset Formula ${\displaystyle \lim_{x\to0}-\left|x\right|=\lim_{x\to0}\left|x\right|=0}$ \end_inset , \begin_inset Formula ${\displaystyle \lim_{x\to0}x\sin\frac{1}{x}=0}$ \end_inset . \end_layout \begin_layout Subsection End Behavior \end_layout \begin_layout Standard The end behavior of a graph describes how it appears as \begin_inset Formula $x$ \end_inset approaches infinity to the right ( \begin_inset Formula $x$ \end_inset increases) or to the left ( \begin_inset Formula $x$ \end_inset decreases). End behavior is expressed as a behavior model. The behavior model of a graph depends on the highest order term in the equation. In rational expressions (fractions), this would be the division of the highest order term in the numerator by the highest order term in the denominato r. \end_layout \begin_layout Standard For example, the behavior model of \begin_inset Formula ${\displaystyle \frac{2x^{5}+x^{4}-x^{2}+1}{3x^{2}-5x+7}}$ \end_inset is \begin_inset Formula ${\displaystyle \frac{2x^{5}}{3x^{2}}}$ \end_inset . The limit as \begin_inset Formula $x$ \end_inset approaches both positive and negative infinity would be positive infinity. \end_layout \begin_layout Subsubsection Differing Behavior \end_layout \begin_layout Standard Sometimes, right-hand and left-hand behavior differ. \end_layout \begin_layout Standard If the function is \begin_inset Formula $f(x)$ \end_inset and its left-hand behavior model is \begin_inset Formula $g(x)$ \end_inset , \begin_inset Formula ${\displaystyle \lim_{x\rightarrow\infty^{-}}\frac{f(x)}{g(x)}=1}$ \end_inset . Likewise, if the function is \begin_inset Formula $f(x)$ \end_inset and its right-hand behavior model is \begin_inset Formula $h(x)$ \end_inset , \begin_inset Formula ${\displaystyle \lim_{x\rightarrow\infty^{+}}\frac{f(x)}{h(x)}=1}$ \end_inset . \end_layout \begin_layout Paragraph Example \end_layout \begin_layout Standard \begin_inset Formula $f(x)=x+e^{-x}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\rightarrow\infty^{-}}\frac{x+e^{-x}}{e^{-x}}=\lim_{x\rightarrow\infty^{-}}\frac{x}{e^{-x}}+\lim_{x\rightarrow\infty^{-}}\frac{e^{-x}}{e^{-x}}=0+1=1}$ \end_inset . Therefore, \begin_inset Formula $y=e^{-x}$ \end_inset is the left-hand behavior model. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\rightarrow\infty^{+}}\frac{x+e^{-x}}{x}=\lim_{x\rightarrow\infty^{+}}\frac{x}{x}=\lim_{x\rightarrow\infty^{+}}\frac{e^{-x}}{x}=1+0=1}$ \end_inset . Therefore, \begin_inset Formula $y=x$ \end_inset is the right-hand behavior model. \end_layout \begin_layout Chapter Derivatives \end_layout \begin_layout Standard This chapter was originally designed for a test on derivatives administered by Jeanine Lennon to her Math 12H (4H/Precalculus) class on April 18, 2008. It was updated for a quiz on the derivatives of trigonometric functions on April 29, 2008, and later updated with an \begin_inset Quotes eld \end_inset Addendum \begin_inset Quotes erd \end_inset section (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:addendum2" \end_inset ) for a test on derivatives administered by Jonathan Chernick to his AP \begin_inset Foot status collapsed \begin_layout Plain Layout AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. \end_layout \end_inset Calculus BC class on October 14, 2008. \end_layout \begin_layout Section Introduction \end_layout \begin_layout Standard The slope of a curve cannot be determined by using the formula \begin_inset Formula $m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ \end_inset , but the slopes of tangent lines drawn to a curve can be determined. To create an infinite number of tangent lines, two points on the curve must be \begin_inset Quotes eld \end_inset pushed \begin_inset Quotes erd \end_inset together so that their distance, \begin_inset Formula $h$ \end_inset , approaches zero. \end_layout \begin_layout Standard The concept of a limit is used to find a derivative. The derivative is the \begin_inset Formula $m_{tan}$ \end_inset (slope of tangent line) on a curve at a specific point. \end_layout \begin_layout Description derivative slope of a curve at a given point on the curve \end_layout \begin_layout Description normal \begin_inset space ~ \end_inset line line perpendicular to a tangent line at the point of tangency \end_layout \begin_layout Subsection Definition \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle f^{\prime}(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Tangent Lines \end_layout \begin_layout Standard The derivative can be used to calculate the equation of a line tangent to a curve at a certain point. The derivative is the slope of the tangent line, and when the coordinates of the certain point on the curve are known, the calculated slope and the coordinates of the certain point on the curve (values can be calculated by plugging into equation of curve) can be plugged into \begin_inset Formula $y=mx+b$ \end_inset or the point-slope formula to determine the equation of the tangent line. \end_layout \begin_layout Standard If the slope of a curve at a given point (derivative) is equal to the slope of another curve at a given point, then the two curves have parallel tangent lines at the indicated points. \end_layout \begin_layout Section Notation \end_layout \begin_layout Standard The derivative notation is special and unique in mathematics. There are two kinds of notations --- Leibniz notation and Newtonian notation. \end_layout \begin_layout Subsection Leibniz Notation \end_layout \begin_layout Standard The Leibniz notation is expressed as \begin_inset Formula $\frac{dy}{dx}$ \end_inset , meaning \begin_inset Quotes eld \end_inset rate of change in \begin_inset Formula $y$ \end_inset with respect to \begin_inset Formula $x$ \end_inset \begin_inset Quotes erd \end_inset or as \begin_inset Formula $\frac{d}{dx}$ \end_inset , which literally means \begin_inset Quotes eld \end_inset derivative with respect to \begin_inset Formula $x$ \end_inset . \begin_inset Quotes erd \end_inset Because the derivative of function \begin_inset Formula $y$ \end_inset is defined as a function representing the slope of function \begin_inset Formula $y$ \end_inset , the second (or double) derivative is the function representing the slope of the first derivative function. In Leibniz notation, this is written as: \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d^{2}y}{dx^{2}}}$ \end_inset \end_layout \begin_layout Subsection Newtonian Notation \end_layout \begin_layout Standard With the Newtonian notation, the derivative of the function \begin_inset Formula $f(x)$ \end_inset is denoted by \begin_inset Formula $f^{\prime}(x)$ \end_inset , and its second (or double) derivative is denoted by \begin_inset Formula $f^{\prime\prime}(x)$ \end_inset . This is read as \begin_inset Quotes eld \end_inset \begin_inset Formula $f$ \end_inset double prime of \begin_inset Formula $x$ \end_inset , \begin_inset Quotes erd \end_inset or \begin_inset Quotes eld \end_inset the second derivative of \begin_inset Formula $f(x)$ \end_inset . \begin_inset Quotes erd \end_inset \end_layout \begin_layout Section Higher Order Derivatives \end_layout \begin_layout Standard The second derivative is the derivative of the derivative of a function. Subsequent derivatives can be calculated by calculating the derivative of the previous derivative. The following are notations for derivatives of different orders. \end_layout \begin_layout Standard \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Order \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Newtonian Notation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Leibniz Notation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Leibniz Notation \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout First Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $f^{\prime}(x)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{dy}{dx}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[f(x)\right]}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Second Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $f^{\prime\prime}(x)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{2}y}{dx^{2}}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{2}}{dx^{2}}\left[f(x)\right]}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Third Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $f^{\prime\prime\prime}(x)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{3}y}{dx^{3}}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{3}}{dx^{3}}\left[f(x)\right]}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Fourth Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $f^{(4)}(x)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{4}y}{dx^{4}}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{4}}{dx^{4}}\left[f(x)\right]}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout N \begin_inset Formula $^{\text{th}}$ \end_inset Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $f^{(n)}(x)$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{n}y}{dx^{n}}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d^{n}}{dx^{n}}\left[f(x)\right]}$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Standard One should not write \begin_inset Formula $f^{n}(x)$ \end_inset to indicate the \begin_inset Formula $n^{\textrm{th}}$ \end_inset derivative, as this is easily confused with the quantity \begin_inset Formula $f(x)$ \end_inset all raised to the \begin_inset Formula $n^{\textrm{th}}$ \end_inset power. \end_layout \begin_layout Section Rules \end_layout \begin_layout Standard Rules for calculating the derivatives of general functions have been developed. As a result, it is possible to calculate the derivative of a wide variety of functions. In many cases the use of multiple rules are required. \end_layout \begin_layout Subsection Constant Function \begin_inset CommandInset label LatexCommand label name "sec:Constant-Function" \end_inset \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout For any constant \begin_inset Formula $c$ \end_inset , \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[c]=0}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The function \begin_inset Formula $f(x)=c$ \end_inset is a horizontal line, which has a constant slope of zero. Therefore, it should be expected that the derivative of this function is zero, regardless of the value of \begin_inset Formula $x$ \end_inset . It is important to understand that \begin_inset Formula $e$ \end_inset and \begin_inset Formula $\pi$ \end_inset are constants, and that their derivative is therefore zero. \end_layout \begin_layout Subsection Linear Function \begin_inset CommandInset label LatexCommand label name "sec:Linear-Function" \end_inset \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout For any constants \begin_inset Formula $m$ \end_inset and \begin_inset Formula $c$ \end_inset , \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[mx+c]=m}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The function \begin_inset Formula $f(x)=mx+c$ \end_inset is a line with a slope of \begin_inset Formula $m$ \end_inset . \end_layout \begin_layout Subsection Constant Multiplier Rule \begin_inset CommandInset label LatexCommand label name "sec:Constant-Multiplier-Rule" \end_inset \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout For any constant \begin_inset Formula $c$ \end_inset , \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard In the definition of a derivative, one can factor \begin_inset Formula $c$ \end_inset out of the numerator and then out of the entire limit. \end_layout \begin_layout Subsection Addition/Subtraction Rule \begin_inset CommandInset label LatexCommand label name "sec:Addition/Subtraction-Rule" \end_inset \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout For the given functions \begin_inset Formula $f(x)$ \end_inset and \begin_inset Formula $g(x)$ \end_inset , \end_layout \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula ${\displaystyle \frac{d}{dx}[f(x)\pm g(x)]=\frac{d}{dx}[f(x)]\pm\frac{d}{dx}[g(x)]}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard As a result, one can take an equation, break it up into terms, figure out the derivative individually, and build the answer back up. \end_layout \begin_layout Subsection Power Rule \begin_inset CommandInset label LatexCommand label name "sec:Power-Rule" \end_inset \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout For any constant exponent \begin_inset Formula $n$ \end_inset , \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[x^{n}\right]=nx^{n-1},x\ne0}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard This rule is actually in effect in linear equations too, since \begin_inset Formula $x^{n-1}=x^{0}$ \end_inset when \begin_inset Formula $n=1$ \end_inset , and any real number or variable to the zero power is one. \end_layout \begin_layout Standard This rule also applies to fractional and negative powers. Therefore, \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[\sqrt{x}\,\right]=\frac{d}{dx}\left[x^{1/2}\right]=\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt{x}}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Since polynomials are sums of monomials, using this rule and the addition/subtra ction rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Addition/Subtraction-Rule" \end_inset ) lets one calculate the derivative of any polynomial. \end_layout \begin_layout Subsubsection Simple Fractions \end_layout \begin_layout Standard When taking the derivative of simple fractions, one can use the following shortcut to quickly do so. The calculations of derivatives of more complex fractions require use of the quotient rule. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[\frac{c}{x^{b}}\right]=\frac{d}{dx}\left[cx^{-b}\right]=-cbx^{-b-1}=-cbx^{-(b+1)}=\frac{-cb}{x^{b+1}}}$ \end_inset , where \begin_inset Formula $c$ \end_inset is a constant \end_layout \end_inset \end_layout \begin_layout Subsection Chain Rule \begin_inset CommandInset label LatexCommand label name "sec:Chain-Rule" \end_inset \end_layout \begin_layout Standard The chain rule allows one to calculate the derivative of an unexpanded expressio n without expanding the expression. This is done by calculating the derivative of the composite of two functions. \end_layout \begin_layout Standard For example, see the function \begin_inset Formula $f(x)=(a+b)^{c}$ \end_inset . To make this the composite of two functions, \begin_inset Formula $g(x)=a+b$ \end_inset and \begin_inset Formula $f(x)=g(x)^{c}$ \end_inset . This function can be rewritten as the composite function \begin_inset Formula $f(g(x))$ \end_inset , where \begin_inset Formula $g(x)$ \end_inset is the polynomial ( \begin_inset Formula $a+b$ \end_inset ) and \begin_inset Formula $f(x)$ \end_inset is \begin_inset Formula $g(x)$ \end_inset to the \begin_inset Formula $c^{\textrm{th}}$ \end_inset power. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout According to the chain rule, \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[f(g(x))]=f^{\prime}(g(x))\times g^{\prime}(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard An example of this situation is \begin_inset Formula $f(x)=(3x+4)^{3}$ \end_inset . In this case, \begin_inset Formula $g(x)=3x+4$ \end_inset and \begin_inset Formula $f(x)=g(x)^{3}$ \end_inset . According to the chain rule, \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[(3x+4)^{3}\right]=3(3x+4)^{2}\times\frac{d}{dx}\left[3x+4\right]=3(3x+4)^{2}\times(3+0)=9(3x+4)^{2}}$ \end_inset \end_layout \begin_layout Subsection Product Rule \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none The derivative of the function \begin_inset Formula $f(x)=A\times B$ \end_inset would \family default \series default \shape default \size default \emph on \bar default \noun default \color inherit not \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none be \begin_inset Formula $f^{\prime}(a)\times f^{\prime}(b)$ \end_inset . The product rule allows one to correctly calculate the derivative of the product of two functions. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout According to the product rule, \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[f(x)\times g(x)]=f(x)\times g^{\prime}(x)+g(x)\times f^{\prime}(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The derivative of the product of two functions is the first function multiplied by the derivative of the other function, added to the first function multiplied by the derivative of the second function. \end_layout \begin_layout Standard The \size normal mnemonic device \begin_inset Quotes eld \end_inset one-D-two plus two-D-one \begin_inset Quotes erd \end_inset can be used to remember this rule. \end_layout \begin_layout Standard \end_layout \begin_layout Subsection Quotient Rule \begin_inset CommandInset label LatexCommand label name "sec:Quotient-Rule" \end_inset \end_layout \begin_layout Standard As with multiplying, \size normal the derivative of a quotient \size default is not \size normal the quotient of the derivatives. \family roman \series medium \shape up \emph off \bar no \noun off \color none The quotient rule allows one to correctly calculate the derivative of the quotient of two functions. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout According to the quotient rule, \end_layout \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)\times f^{\prime}(x)-f(x)\times g^{\prime}(x)}{g(x)^{2}}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The mnemonic device \size normal \begin_inset Quotes eld \end_inset low-D-high minus high-D-low over the square of what's below \begin_inset Quotes erd \end_inset can be used to remember this rule. \end_layout \begin_layout Section Basic Polynomials \end_layout \begin_layout Standard With these rules, the derivative of any polynomial can be determined. Here is a step-by-step example of the process of calculating the derivative of a fairly simple polynomial. The chain, product, and quotient rules are not covered. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[6x^{5}+3x^{2}+3x+1\right]}$ \end_inset \end_layout \begin_layout Standard The addition/subtraction rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Addition/Subtraction-Rule" \end_inset ) splits the equation into several terms. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[6x^{5}\right]+\frac{d}{dx}\left[3x^{2}\right]+\frac{d}{dx}[3x]+\frac{d}{dx}[1]}$ \end_inset \end_layout \begin_layout Standard The constant (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Constant-Function" \end_inset ) and linear (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Linear-Function" \end_inset ) rules get rid of some terms. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \frac{d}{dx}\left[6x^{5}\right]+\frac{d}{dx}\left[3x^{2}\right]+3+0}$ \end_inset \end_layout \begin_layout Standard The constant multiplier rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Constant-Multiplier-Rule" \end_inset ) moves the constants outside of the derivatives. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle 6\frac{d}{dx}\left[x^{5}\right]+3\frac{d}{dx}[x]+3}$ \end_inset \end_layout \begin_layout Standard The power rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Power-Rule" \end_inset ) works on the individual monomials. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle 6\left(5x^{4}\right)+3(2x)+3}$ \end_inset \end_layout \begin_layout Standard Simplifying obtains the final answer. \end_layout \begin_layout Standard \begin_inset Formula $30x^{4}+6x+3$ \end_inset \end_layout \begin_layout Section Graphing Calculator \end_layout \begin_layout Standard In some cases it may be easier or required to calculate derivatives using the graphing calculator. It can also be used to check one's answer. \end_layout \begin_layout Standard There are two methods of calculating a derivative of a graph with a Texas Instruments graphing calculator. These instructions are designed for a TI-84 Plus calculator, but they may used on other Texas Instruments graphing calculators, though slight modificatio n may be necessary. \end_layout \begin_layout Standard Unless otherwise specified, the graphing calculator should be in radian mode. \end_layout \begin_layout Enumerate Math \begin_inset Formula $\longrightarrow$ \end_inset 8 (8. nDeriv) \begin_inset Formula $\longrightarrow$ \end_inset enter with form \emph on function, \begin_inset Formula $x$ \end_inset , \begin_inset Formula $x$ \end_inset value \emph default \begin_inset Formula $\longrightarrow$ \end_inset Enter \end_layout \begin_deeper \begin_layout Enumerate replace \emph on function \emph default with the appropriate function \end_layout \begin_layout Enumerate replace \emph on \begin_inset Formula $x$ \end_inset value \emph default with the appropriate value \end_layout \end_deeper \begin_layout Enumerate Graph function \begin_inset Formula $\longrightarrow$ \end_inset 2nd \begin_inset Formula $\longrightarrow$ \end_inset Trace (Calc) \begin_inset Formula $\longrightarrow$ \end_inset enter \begin_inset Formula $x$ \end_inset value \begin_inset Formula $\longrightarrow$ \end_inset Enter \end_layout \begin_deeper \begin_layout Enumerate use the appropriate \begin_inset Formula $x$ \end_inset value \end_layout \end_deeper \begin_layout Subsection Does Not Exist \end_layout \begin_layout Standard The graphing calculator may display an incorrect answer when calculating derivatives that do not exist (e.g. at a corner). Graphing calculators like the TI-84 Plus calculate derivatives by using the symmetric difference quotient. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}}$ \end_inset \end_layout \begin_layout Standard The problem with this method is that the calculator will actually calculate the average slope over a very small area instead of the true derivative (instantaneous slope). At a corner, the average slope over a very small area will be zero, but the correct answer is that the derivative does not exist. \end_layout \begin_layout Section Differentiability \end_layout \begin_layout Subsection Definition \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout For \begin_inset Formula $f(x)$ \end_inset to be differentiable at point \begin_inset Formula $c$ \end_inset , the following must be true: \end_layout \begin_layout Enumerate \begin_inset Formula $f(x)$ \end_inset must be continuous at point \begin_inset Formula $c$ \end_inset \end_layout \begin_deeper \begin_layout Enumerate \size normal \begin_inset Formula $f(x)$ \end_inset is defined at \begin_inset Formula $c$ \end_inset \end_layout \begin_deeper \begin_layout Enumerate \size normal \begin_inset Formula $f(c)$ \end_inset exists \end_layout \end_deeper \begin_layout Enumerate \size normal The limit as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $c$ \end_inset exists. \end_layout \begin_deeper \begin_layout Enumerate \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula ${\displaystyle \lim_{x\rightarrow c}f(x)}$ \end_inset exists \end_layout \end_deeper \begin_layout Enumerate \size normal The limit and \begin_inset Formula $f(c)$ \end_inset are equal. \end_layout \begin_deeper \begin_layout Enumerate \size normal \begin_inset Formula ${\displaystyle f(c)=\lim_{x\rightarrow c}f(x)}$ \end_inset \end_layout \end_deeper \end_deeper \begin_layout Enumerate The derivative from both sides must be equal \end_layout \begin_deeper \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\rightarrow c^{-}}f^{\prime}(x)=\lim_{x\rightarrow c^{+}}f^{\prime}(x)}$ \end_inset \end_layout \end_deeper \begin_layout Plain Layout If any of these do not hold then \begin_inset Formula $f(x)$ \end_inset is not differentiable at \begin_inset Formula $c$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard Notice how this relates to the idea of differentiability. To be differentiable, the function must have a uniform rate of change (e.g. no corners, cusps, or vertical tangents) within an interval. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout A function is said to be differentiable if it is differentiable at every point \begin_inset Formula $c$ \end_inset in its domain. \end_layout \end_inset \end_layout \begin_layout Standard A function may be differentiable at a certain point, but not a differentiable function (throughout). Likewise, a non-differentiable function may be differentiable at a certain point. \end_layout \begin_layout Subsection Not Differentiable \end_layout \begin_layout Subsubsection Corner \end_layout \begin_layout Standard A function does not have a derivative at a corner. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\rightarrow a^{-}}f^{\prime}(x)\neq\lim_{x\rightarrow a^{+}}f^{\prime}(x)}$ \end_inset \end_layout \begin_layout Subsubsection Cusp \end_layout \begin_layout Standard A cusp occurs when the limit of the slope from one side of a curve goes to \begin_inset Formula $-\infty$ \end_inset and the other side of the curve goes to \begin_inset Formula $+\infty$ \end_inset . As a result, a function does not have a derivative at a cusp. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\rightarrow a^{-}}f^{\prime}(x)\neq\lim_{x\rightarrow a^{+}}f^{\prime}(x)}$ \end_inset \end_layout \begin_layout Subsubsection Vertical Tangent \end_layout \begin_layout Standard A function does not have a derivative at a vertical tangent. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \lim_{x\rightarrow a}f^{\prime}(x)=\infty}$ \end_inset , therefore the limit does not exist \end_layout \begin_layout Subsubsection Endpoint \end_layout \begin_layout Standard A function is not differentiable at an endpoint because the derivative can only be calculated from one side. However, since an endpoint has a one-sided derivative, the endpoints on the graph of the derivative of a function are filled in. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout Endpoints are a source of a lot of inconsistency in calculus. \end_layout \end_inset \end_layout \begin_layout Section Trigonometric Functions \end_layout \begin_layout Standard Trigonometric identities can be used to simplify expressions before or after finding a derivative. \end_layout \begin_layout Subsection Trigonometric Identities \end_layout \begin_layout Subsubsection Pythagorean Identities \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Enumerate \begin_inset Formula $\sin^{2}\theta+\cos^{2}\theta=1$ \end_inset \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula $1+\tan^{2}\theta=\sec^{2}\theta$ \end_inset \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula $1+\cot^{2}\theta=\csc^{2}\theta$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Quotient Identities \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Enumerate \begin_inset Formula ${\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}}$ \end_inset \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \cot\theta=\frac{\cos\theta}{\sin\theta}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Sum of Two Angles \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula $\sin(A+B)=\sin A\cos B+\cos A\sin B$ \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula $\cos(A+B)=\cos A\cos B-\sin A\sin B$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Derivation \end_layout \begin_layout Standard Sine, cosine, tangent, cotangent, secant, and cosecant are trigonometric functions. Each trigonometric function has a derivative. \end_layout \begin_layout Standard \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Trigonometric Function \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sin x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\cos x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\cos x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-\sin x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tan x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sec^{2}x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\cot x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-\csc^{2}x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sec x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sec x\times\tan x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\csc x$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $-\csc x\times\cot x$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Subsubsection Sine \end_layout \begin_layout Standard The derivative of sine is cosine. \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula ${\displaystyle \frac{d}{dx}[\sin(x)]=\cos(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Cosine \end_layout \begin_layout Standard The derivative of cosine is negative sine. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[\cos(x)]=-\sin(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Tangent \end_layout \begin_layout Standard Using the quotient rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Quotient-Rule" \end_inset ) and the Pythagorean identity \begin_inset Formula $\cos^{2}(x)+\sin^{2}(x)=1$ \end_inset , the derivative of tangent can be derived. \end_layout \begin_layout Standard \begin_inset Formula $ $ \end_inset \begin_inset Formula \begin{eqnarray*} \tan(x) & = & \frac{\sin(x)}{\cos(x)}\\ \frac{d}{dx}[\tan(x)] & = & \frac{\cos^{2}(x)+\sin^{2}(x)}{\cos^{2}(x)}\\ \frac{d}{dx}[\tan(x)] & = & \frac{1}{\cos^{2}(x)}\\ \frac{d}{dx}[\tan(x)] & = & \sec^{2}(x)\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Therefore, the derivative of tangent is the square of secant. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}\tan(x)=\sec^{2}(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Cotangent \end_layout \begin_layout Standard Using the quotient rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Quotient-Rule" \end_inset ) and the Pythagorean identity \begin_inset Formula $\cos^{2}(x)+\sin^{2}(x)=1$ \end_inset , the derivative of cotangent can be derived. \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \cot(x) & = & \frac{\cos(x)}{\sin(x)}\\ \frac{d}{dx}[\cot(x)] & = & \frac{-\sin^{2}(x)-\cos^{2}(x)}{\sin^{2}(x)}\\ \frac{d}{dx}[\cot(x)] & = & \frac{-1}{\sin^{2}(x)}\\ \frac{d}{dx}[\cot(x)] & = & -\csc^{2}(x)\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Therefore, the derivative of cotangent is the negative of the square of cosecant. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[\cot(x)]=-\csc^{2}(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Secant \end_layout \begin_layout Standard Using the quotient rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Quotient-Rule" \end_inset ), the derivative of secant can be derived. \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula \begin{eqnarray*} \sec(x) & = & \frac{1}{\cos(x)}\\ \frac{d}{dx}[\sec(x)] & = & \frac{\sin(x)}{\cos^{2}(x)}\\ \frac{d}{dx}[\sec(x)] & = & \frac{1}{\cos(x)}\times\frac{\sin(x)}{\cos(x)}\\ \frac{d}{dx}[\sec(x)] & = & \sec(x)\times\tan(x)\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Therefore, the derivative of secant is secant multiplied by tangent. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[\sec(x)]=\sec(x)\times\tan(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Cosecant \end_layout \begin_layout Standard Using the quotient rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Quotient-Rule" \end_inset ), the derivative of cosecant can be derived. \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \csc(x) & = & \frac{1}{-\sin(x)}\\ \frac{d}{dx}[\csc(x)] & = & -\frac{\cos(x)}{sin^{2}(x)}\\ \frac{d}{dx}[\csc(x)] & = & -\frac{1}{\sin(x)}\times\frac{\cos(x)}{sin(x)}\\ \frac{d}{dx}[\csc(x)] & = & -\csc(x)\times\cot(x)\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Therefore, the derivative of cosecant is the negative of cosecant multiplied by cotangent. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[\csc(x)]=-\csc(x)\times\cot(x)}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Combining with Derivative Rules \end_layout \begin_layout Standard In most cases, one must determine the derivative of an an example that requires the use of derivative rules in addition to the knowledge of the derivatives of trigonometric function. One may apply the form \begin_inset Formula $\mathrm{trig}\,(a)$ \end_inset to many examples, where \begin_inset Formula $\mathrm{trig}$ \end_inset is the trigonometric function and \begin_inset Formula $a$ \end_inset is the angle. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout Based on the chain rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Chain-Rule" \end_inset ), the derivative of \begin_inset Formula $\mathrm{trig}\,(a)$ \end_inset would be \begin_inset Formula $(\frac{d}{dx}[\mathrm{trig}])(a)\times\frac{d}{dx}[a]$ \end_inset . \begin_inset VSpace medskip \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[\mathrm{trig}\,(a)]=\left(\frac{d}{dx}[\mathrm{trig}](a)\right)\times\frac{d}{dx}[a]}$ \end_inset , \end_layout \begin_layout Plain Layout where \begin_inset Formula $\frac{d}{dx}[\mathrm{trig}]$ \end_inset is the derivative of the trigonometric function, and \begin_inset Formula $\frac{d}{dx}[a]$ \end_inset is the derivative of the angle. \end_layout \end_inset \end_layout \begin_layout Subsubsection Example \end_layout \begin_layout Description Original \begin_inset space ~ \end_inset Function \begin_inset Formula $\sin(2x)$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \frac{d}{dx}[\sin(2x)] & = & \left(\frac{d}{dx}\left[\sin\right](2x)\right)\times\frac{d}{dx}[2x]\\ \frac{d}{dx}[\sin(2x)] & = & (\cos(2x))\times2\\ \frac{d}{dx}[\sin(2x)] & = & 2\cos(2x)\end{eqnarray*} \end_inset \end_layout \begin_layout Section Asymptotes \end_layout \begin_layout Standard A linear asymptote is a straight line that a graph approaches, but does not become identical to. Asymptotes are formally defined using limits. For more information on limits, see the \emph on CHSN Math Calculus/12H Limits Review Report \emph default . \end_layout \begin_layout Subsection Vertical Asymptotes \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout The line \begin_inset Formula $x=a$ \end_inset is a vertical asymptote for the function \begin_inset Formula $f(x)$ \end_inset if at least one of the following statements is true. \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\to a}f(x)=\pm\infty}$ \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\to a^{-}}f(x)=\pm\infty}$ \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula ${\displaystyle \lim_{x\to a^{+}}f(x)=\pm\infty}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard The limits from both directions do not have to be equal to have an asymptote, but they may be equal. Essentially, a vertical asymptote occurs where the the value of a limit is positive or negative infinity from any direction. \end_layout \begin_layout Standard Recall that this occurs where the fraction of a function is undefined (denominat or equals zero). \end_layout \begin_layout Subsubsection Removable Discontinuities \end_layout \begin_layout Standard The function \begin_inset Formula $f(x)=\frac{x^{2}-9}{x-3}$ \end_inset is considered to have a removable discontinuity at \begin_inset Formula $x=3$ \end_inset . It is discontinuous at that point because the fraction then becomes \begin_inset Formula $\frac{0}{0}$ \end_inset which is undefined. \end_layout \begin_layout Standard Standard algebraic techniques for simplifying fractions and algebraic expression s (i.e. factoring, multiplying by conjugates) can be used to eliminate the discontinuit y. \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle f(x)=\frac{x^{2}-9}{x-3}=\frac{(x+3)(x-3)}{(x-3)}=\frac{x+3}{1}\cdot\frac{x-3}{x-3}=\frac{x+3}{1}\cdot1=x+3}$ \end_inset \end_layout \begin_layout Standard However, the function is not really continuous, and an open circle must be left in the graph at the removable discontinuity. \end_layout \begin_layout Subsection Horizontal Asymptotes \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout The line \begin_inset Formula $y=a$ \end_inset is a horizontal asymptote for the function \begin_inset Formula $f(x)$ \end_inset if \begin_inset Formula ${\displaystyle \lim_{x\to\infty}f(x)=a}$ \end_inset or \begin_inset Formula ${\displaystyle \lim_{x\to-\infty}f(x)=a}$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard If \begin_inset Formula ${\displaystyle \lim_{x\to\infty}f(x)=a}$ \end_inset and \begin_inset Formula ${\displaystyle \lim_{x\to-\infty}f(x)=b}$ \end_inset , then the function \begin_inset Formula $f(x)$ \end_inset has two asymptotes at \begin_inset Formula $y=a$ \end_inset and \begin_inset Formula $y=b$ \end_inset . Note that in some functions, the graph may pass through the horizontal asymptote at an \begin_inset Formula $x$ \end_inset value of zero. \end_layout \begin_layout Standard Essentially, a horizontal asymptote occurs at the value of a limit where \begin_inset Formula $x$ \end_inset approaches positive or negative infinity. \end_layout \begin_layout Standard Recall that rules exist for calculating the the value of a limit where \begin_inset Formula $x$ \end_inset approaches positive or negative infinity. \end_layout \begin_layout Subsubsection Rules \end_layout \begin_layout Standard The easiest way to determine limits as \begin_inset Formula $x$ \end_inset approaches \begin_inset Formula $\pm\infty$ \end_inset is by using the graphing calculator to make observations, or by plugging in high values of positive and negative numbers in a calculator. \end_layout \begin_layout Standard However, there are three rules for determining a limit of a fraction analyticall y as a variable approaches infinity. For each rule, one must look at the variables on both the numerator and denominator of the function. \end_layout \begin_layout Standard Look for the highest term (with the highest exponent) in the numerator. Look for the same in the denominator. These rules are based on that information. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Itemize If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit is the fractional ratio of the coefficients of the highest terms. \end_layout \begin_layout Itemize If the \emph on numerator \emph default has the highest term, then the fraction is called \begin_inset Quotes eld \end_inset top heavy \begin_inset Quotes erd \end_inset and the limit is infinity. \end_layout \begin_layout Itemize If the \emph on denominator \emph default has the highest term, then the fraction is called \begin_inset Quotes eld \end_inset bottom heavy \begin_inset Quotes erd \end_inset and the limit is zero. \end_layout \end_inset \end_layout \begin_layout Standard If there is no denominator stated, it is understood that the denominator is 1 or \begin_inset Formula $1n^{0}$ \end_inset , and the limit will be infinity. \end_layout \begin_layout Section Sketching with Asymptotes \end_layout \begin_layout Standard A series of steps can be taken to sketch with asymptotes. As a result, curves may be sketched without a graphing calculator. \end_layout \begin_layout Enumerate Find the \begin_inset Formula $x$ \end_inset -intercept by setting \begin_inset Formula $y$ \end_inset equal to zero. \end_layout \begin_layout Enumerate Find the \begin_inset Formula $y$ \end_inset -intercept by setting \begin_inset Formula $x$ \end_inset equal to zero. \end_layout \begin_layout Enumerate Find the horizontal asymptote(s). \end_layout \begin_layout Enumerate Find the vertical asymptotes(s). \end_layout \begin_layout Enumerate Plot the \begin_inset Formula $x$ \end_inset -intercept and \begin_inset Formula $y$ \end_inset -intercept. \end_layout \begin_layout Enumerate Sketch the asymptote(s). \end_layout \begin_layout Enumerate Find the limits of both sides of the vertical asymptote by using test points. \end_layout \begin_layout Enumerate Sketch the curve using the determined information and the sketched asymptotes. \end_layout \begin_layout Standard In some problems only limits will be provided. From these limits horizontal and vertical asymptotes can be determined. While the \begin_inset Formula $x$ \end_inset -intercept and \begin_inset Formula $y$ \end_inset -intercept are not provided, it is still possible to sketch the graph. The sketch will be less accurate, but that is acceptable when provided with \emph on limit \emph default ed information. \end_layout \begin_layout Section Stationary Points \end_layout \begin_layout Subsection Extrema \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \begin_inset Graphics filename Maxima_and_Minima.svg scale 40 \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:extrema" \end_inset Graph demonstrating extrema on a curve \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Standard Maxima and minima are points where a function reaches a highest or lowest value, respectively. A maximum occurs when positive slope changes to negative slope and a minimum occurs when negative slope changes to a positive slope. There are two kinds of extrema (a word meaning maximum or minimum): global and local, sometimes referred to as \begin_inset Quotes eld \end_inset absolute \begin_inset Quotes erd \end_inset and \begin_inset Quotes eld \end_inset relative, \begin_inset Quotes erd \end_inset respectively. A global maximum is a point that takes the largest value on the entire range of the function, while a global minimum is the point that takes the smallest value on the range of the function. Local extrema are the largest or smallest values of the function in the immediate vicinity. See Figure \begin_inset CommandInset ref LatexCommand vref reference "fig:extrema" \end_inset . \end_layout \begin_layout Standard All extrema look like the crest of a hill or the bottom of a bowl on a graph of the function. A global extremum is always a local extremum too, because it is the largest or smallest value on the entire range of the function, and therefore also in its vicinity. It is also possible to have a function with no extrema, global or local (e.g. \begin_inset Formula $y=x$ \end_inset ) \begin_inset Note Comment status open \begin_layout Plain Layout or is it all extrema? \end_layout \end_inset . \end_layout \begin_layout Standard At an extremum, the \begin_inset Formula $y$ \end_inset -value is the value of the extremum and the \begin_inset Formula $x$ \end_inset -value is where the extremum occurs. \end_layout \begin_layout Subsection \begin_inset Quotes eld \end_inset Flatpoints \begin_inset Quotes erd \end_inset \end_layout \begin_layout Standard It is important to note that not all cases in which the first derivative of a function is equal to zero are turning points or extrema, though the first derivative of a function is equal to zero or does not exist at all turning points and extrema. \begin_inset Quotes eld \end_inset Flatpoints \begin_inset Quotes erd \end_inset (e.g. triple roots) may also occur when the first derivative of a function is equal to zero, but they are not turning points nor extrema because no slope change occurs. \end_layout \begin_layout Subsection Classification \end_layout \begin_layout Standard At any extremum, the slope of the graph is zero or undefined, as the graph must stop rising or falling at an extremum, and begin to fall or rise. Because of this, extrema are also commonly called stationary points or turning points. If the graph has one or more of these stationary points, these may be found by setting the first derivative equal to zero and finding the roots of the resulting equation as well as values where the function is undefined. These values are referred to as critical points. Note that if the domain is restricted, the endpoints of the domain must also be checked to see if they are global extrema. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Description critical \begin_inset space ~ \end_inset point point in domain of \begin_inset Formula $f$ \end_inset where \begin_inset Formula $f^{\prime}=0$ \end_inset or \begin_inset Formula $f^{\prime}$ \end_inset does not exist \end_layout \begin_layout Plain Layout Extrema can only occur at critical points and endpoints. \end_layout \end_inset \end_layout \begin_layout Standard True extrema require a sign change in the first derivative. This makes sense --- the graph must rise (positive first derivative) and fall (negative first derivative) to form a maximum. In between rising and falling, on a smooth curve, there will ideally be a point of zero slope --- the maximum. A minimum would exhibit similar properties, but in reverse. \end_layout \begin_layout Subsubsection First Derivative Test \end_layout \begin_layout Standard This leads to a simple method to classify a stationary point --- plug \begin_inset Formula $x$ \end_inset values (test points) slightly left and right into the derivative of the function. If the results have opposite signs then it is a true extremum. To calculate the coordinates of the minimum or maximum point, one would plug the determined \begin_inset Formula $x$ \end_inset value into the original function to find its \begin_inset Formula $y$ \end_inset value. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Itemize If \begin_inset Formula $f^{\prime}(x)<0$ \end_inset for \begin_inset Formula $x0$ \end_inset for \begin_inset Formula $x>c$ \end_inset , then \begin_inset Formula $f(c)$ \end_inset is a local minimum. \end_layout \begin_layout Itemize If \begin_inset Formula $f^{\prime}(x)>0$ \end_inset for \begin_inset Formula $xc$ \end_inset , then \begin_inset Formula $f(c)$ \end_inset is a local maximum. \end_layout \end_inset \end_layout \begin_layout Standard Caution must be exercised with this method, as, if a point too far from the extremum is picked, one could take it on the far side of another extremum and incorrectly classify the point. A more rigorous method to classify a stationary point is called the extremum test that uses the second derivative, but this simple method is acceptable. \end_layout \begin_layout Subsubsection Second Derivative Test \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Itemize If \begin_inset Formula $f^{\prime}(c)=0$ \end_inset and \begin_inset Formula $f^{\prime\prime}(c)>0$ \end_inset , then \begin_inset Formula $c$ \end_inset is a local minimum. \end_layout \begin_layout Itemize If \begin_inset Formula $f^{\prime}(c)=0$ \end_inset and \begin_inset Formula $f^{\prime\prime}(c)<0$ \end_inset , then \begin_inset Formula $c$ \end_inset is a local maximum. \end_layout \end_inset \end_layout \begin_layout Standard Note that the second derivative test cannot be used to verify an extrema if the first or second derivative does not exist. \end_layout \begin_layout Subsubsection Information \end_layout \begin_layout Standard \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Stationary Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout First Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Second Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Minimum Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout zero or undefined \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout positive or undefined \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Maximum Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout zero or undefined \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout negative or undefined \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Quotes eld \end_inset Flatpoint \begin_inset Quotes erd \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout zero \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout zero \end_layout \end_inset \end_inset \end_layout \begin_layout Standard \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Stationary Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout First Derivative Sign Before \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout First Derivative Sign After \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Minimum Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout negative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout positive \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Maximum Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout positive \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout negative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Quotes eld \end_inset Flatpoint \begin_inset Quotes erd \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout same sign \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout same sign \end_layout \end_inset \end_inset \end_layout \begin_layout Section Inflection Points \end_layout \begin_layout Standard \begin_inset Float figure wide false sideways false status open \begin_layout Plain Layout \noindent \align center \begin_inset Graphics filename X_cubed_(narrow).svg scale 40 \end_inset \end_layout \begin_layout Plain Layout \begin_inset Caption \begin_layout Plain Layout \begin_inset CommandInset label LatexCommand label name "fig:inflection" \end_inset Graph containing an inflection point \end_layout \end_inset \end_layout \begin_layout Plain Layout \end_layout \end_inset \end_layout \begin_layout Standard Inflection points occur when the second derivative of a function is equal to zero. The curve changes from being concave up (positive second derivative) to concave down (negative second derivative), or vice versa. See Figure \begin_inset CommandInset ref LatexCommand vref reference "fig:inflection" \end_inset . \begin_inset Quotes eld \end_inset Flatpoints \begin_inset Quotes erd \end_inset are a specific type of inflection point where the graph flattens out (first derivative is zero), but the sign of the slope does not change. These points are called stationary points of inflection. Other inflection points are not \begin_inset Quotes eld \end_inset flatpoints, \begin_inset Quotes erd \end_inset and there is no flattening out (i.e. sine curve); these points are known as non-stationary points of inflection. \end_layout \begin_layout Standard \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Curvature \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Second Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout First Derivative Graph \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Concave Up ( \begin_inset Quotes eld \end_inset smile \begin_inset Quotes erd \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout positive \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout increasing \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Concave Down ( \begin_inset Quotes eld \end_inset frown \begin_inset Quotes erd \end_inset ) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout negative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout decreasing \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Inflection Point \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout zero or undefined \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout extrema \end_layout \end_inset \end_inset \end_layout \begin_layout Section Optimization \end_layout \begin_layout Standard Optimization is the use of Calculus in the real world. Calculus is a useful tool for maximizing or minimizing (also known as \begin_inset Quotes eld \end_inset optimizing \begin_inset Quotes erd \end_inset ) a situation. \end_layout \begin_layout Subsection Formulas \end_layout \begin_layout Subsubsection Volume \end_layout \begin_layout Description cube \begin_inset Formula $A=a^{3}$ \end_inset , where \begin_inset Formula $a$ \end_inset is the length of the side of each edge of the cube \end_layout \begin_layout Description rectangular \begin_inset space ~ \end_inset prism \begin_inset Formula $V=abc$ \end_inset , where \begin_inset Formula $a$ \end_inset , \begin_inset Formula $b$ \end_inset , and \begin_inset Formula $c$ \end_inset are the lengths of the 3 sides of the prism \end_layout \begin_layout Description cylinder \begin_inset Formula $V=\pi r^{2}h$ \end_inset , where \begin_inset Formula $r$ \end_inset is the radius and \begin_inset Formula $h$ \end_inset is the height of the cylinder \end_layout \begin_layout Description sphere \begin_inset Formula ${\displaystyle V=\frac{4}{3}\pi r^{3}}$ \end_inset , where \begin_inset Formula $r$ \end_inset represents the radius of the sphere \end_layout \begin_layout Subsubsection Surface Area \end_layout \begin_layout Description cube \begin_inset Formula $A=6a^{2}$ \end_inset , where \begin_inset Formula $a$ \end_inset is the length of the side of each edge of the cube \end_layout \begin_layout Description rectangular \begin_inset space ~ \end_inset prism \begin_inset Formula $A=2ab+2bc+2ac$ \end_inset , where \begin_inset Formula $a$ \end_inset , \begin_inset Formula $b$ \end_inset , and \begin_inset Formula $c$ \end_inset are the lengths of the 3 sides of the prism \end_layout \begin_layout Description sphere \begin_inset Formula $A=4\pi r^{2}$ \end_inset , where \begin_inset Formula $r$ \end_inset is radius of the sphere \end_layout \begin_layout Description cylinder \begin_inset Formula $A=2\pi r^{2}+2\pi rh$ \end_inset , where \begin_inset Formula $r$ \end_inset is the radius and \begin_inset Formula $h$ \end_inset is the height of the cylinder \end_layout \begin_layout Section Addendum \begin_inset CommandInset label LatexCommand label name "sec:addendum2" \end_inset \end_layout \begin_layout Standard This section was designed for a test on derivatives administered by Jonathan Chernick to his AP \begin_inset Foot status collapsed \begin_layout Plain Layout AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. \end_layout \end_inset Calculus BC class on October 14, 2008. It is not covered in Math 12H/4H. \end_layout \begin_layout Subsection Alternative Definition of Derivative \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle f^{\prime}(x)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Parametric Equations \end_layout \begin_layout Standard Parametric equations are typically defined by two equations that specify both the \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset coordinates of a graph using a parameter. They are graphed using the parameter (usually \begin_inset Formula $t$ \end_inset ) to figure out both the \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset coordinates. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout The derivative of the parametrized curve \begin_inset Formula $x(t),y(t)$ \end_inset is: \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}}$ \end_inset , \begin_inset Formula ${\displaystyle \frac{dx}{dt}\neq0}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Example \end_layout \begin_layout Description parametrized \begin_inset space ~ \end_inset equation \begin_inset Formula $x=t,y=t^{2}$ \end_inset \end_layout \begin_layout Standard \begin_inset Formula ${\displaystyle \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{2t}{1}=2t}$ \end_inset \end_layout \begin_layout Subsection Implicit Differentiation \end_layout \begin_layout Description explicit \begin_inset space ~ \end_inset relationship function in which \begin_inset Formula $f(x)$ \end_inset is given in terms of \begin_inset Formula $x$ \end_inset and constants; for every \begin_inset Formula $x$ \end_inset -value there is one \begin_inset Formula $y$ \end_inset -value \end_layout \begin_layout Description implicit \begin_inset space ~ \end_inset relationship relationship between two or more variables; two or more functions put together \end_layout \begin_layout Standard Ordinary differentiation is explicit differentiation. Implicit differentiation is useful when differentiating an equation that cannot be explicitly differentiated because it is impossible or hard to isolate variables (e.g. \begin_inset Formula $x^{2}+xy+y^{2}=16$ \end_inset ). \end_layout \begin_layout Standard In many difficult problems involving implicit differentiation (e.g. multiple choice), it is necessary to substitute the dependent variable (e.g. \begin_inset Formula $y$ \end_inset ) and its derivatives (e.g. \begin_inset Formula $\frac{dy}{dx}$ \end_inset , \begin_inset Formula $\frac{d^{2}y}{dx^{2}}$ \end_inset ) based on the original equation or previous determined derivative expressions. \end_layout \begin_layout Subsubsection Example \end_layout \begin_layout Description function \begin_inset Formula $x^{2}+y^{2}=1$ \end_inset \end_layout \begin_layout Paragraph Explicit Differentiation \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} x^{2}+y^{2} & = & 1\\ y^{2} & = & 1-x^{2}\\ y & = & \pm\sqrt{1-x^{2}}\\ y & = & \pm(1-x^{2})^{\frac{1}{2}}\\ \frac{dy}{dx} & = & -\frac{x}{y}\end{eqnarray*} \end_inset \end_layout \begin_layout Paragraph Implicit Differentiation \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} x^{2}+y^{2} & = & 1\\ 2x+2y\frac{dy}{dx} & = & 0\\ 2y\frac{dy}{dx} & = & -2x\\ \frac{dy}{dx} & = & \frac{-2x}{2y}\\ \frac{dy}{dx} & = & \frac{-x}{y}\end{eqnarray*} \end_inset \end_layout \begin_layout Subsection Inverse Functions \end_layout \begin_layout Description inverse \begin_inset space ~ \end_inset function \begin_inset Quotes eld \end_inset opposite \begin_inset Quotes erd \end_inset of a function; if \begin_inset Formula $f(x)=a$ \end_inset , \begin_inset Formula $f^{-1}(a)=f(x)$ \end_inset ; reflected over line \begin_inset Formula $y=x$ \end_inset \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout The composition of a function and its inverse is \begin_inset Formula $x$ \end_inset because the two functions \begin_inset Quotes eld \end_inset undo \begin_inset Quotes erd \end_inset each other. \end_layout \begin_layout Plain Layout \begin_inset Formula $f\left(f^{-1}(x)\right)=x$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard With use of the chain rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Chain-Rule" \end_inset ), the relationship between the derivative of a function and the derivative of its inverse can be determined. \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} f\left(f^{-1}(x)\right) & = & x\\ f^{\prime}\left[f^{-1}(a)\right]\times\left[f^{-1}\right]^{\prime}(a) & = & 1\\ \left[f^{-1}\right]^{\prime}(a) & = & \frac{1}{f^{\prime}\left[f^{-1}(a)\right]}\end{eqnarray*} \end_inset \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout A function and its inverse have reciprocal slopes with reversed \begin_inset Formula $(x,y)$ \end_inset values. \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \left[f^{-1}\right]^{\prime}(a)=\frac{1}{f^{\prime}\left[f^{-1}(a)\right]}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Paragraph Example \end_layout \begin_layout Standard \begin_inset Formula $f(x)=x^{3}+x-2$ \end_inset , find \begin_inset Formula $\left[f^{-1}\right]^{\prime}(0)$ \end_inset \end_layout \begin_layout Standard \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula \begin{eqnarray*} 0 & = & x^{3}+x-2\\ x & = & 1\end{eqnarray*} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} f^{\prime}(x) & = & 3x^{2}+1\\ f^{\prime}(1) & = & 4\end{eqnarray*} \end_inset \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \left[f^{-1}\right]^{\prime}(a) & = & \frac{1}{f^{\prime}\left[f^{-1}(a)\right]}\\ \left[f^{-1}\right]^{\prime}(0) & = & \frac{1}{f^{\prime}\left[f^{-1}(0)\right]}\\ \left[f^{-1}\right]^{\prime}(0) & = & \frac{1}{f^{\prime}(1)}\\ \left[f^{-1}\right]^{\prime}(0) & = & \frac{1}{4}\end{eqnarray*} \end_inset \end_layout \begin_layout Subsubsection Inverse Trigonometric Functions \end_layout \begin_layout Standard The inverse trigonometric functions are the inverse functions of the trigonometr ic functions. The inverse of the trigonometric functions \begin_inset Formula $\sin$ \end_inset , \begin_inset Formula $\cos$ \end_inset , \begin_inset Formula $\tan$ \end_inset , \begin_inset Formula $\cot$ \end_inset , \begin_inset Formula $\sec$ \end_inset , and \begin_inset Formula $\csc$ \end_inset is \begin_inset Formula $\arcsin$ \end_inset , \begin_inset Formula $\arccos$ \end_inset , \begin_inset Formula $\arctan$ \end_inset , \begin_inset Formula $\arccot$ \end_inset , \begin_inset Formula $\arcsec$ \end_inset , and \begin_inset Formula $\arccsc$ \end_inset , respectively. \end_layout \begin_layout Standard The notations \begin_inset Formula $\sin^{-1}$ \end_inset , \begin_inset Formula $\cos^{-1}$ \end_inset , etc. are often used for \begin_inset Formula $\arcsin$ \end_inset , \begin_inset Formula $\arccos$ \end_inset , etc., respectively, but this convention may result in confusion between multiplicative inverse and compositional inverse since this logically conflicts with the structure of expressions like \begin_inset Formula $\sin^{2}x$ \end_inset , which do not refer to function composition but rather multiplication. \end_layout \begin_layout Standard Each inverse trigonometric function has a derivative. \end_layout \begin_layout Standard \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Trigonometric Function \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Inverse ( \begin_inset Formula $\arc$ \end_inset notation) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Inverse ( \begin_inset Formula $^{-1}$ \end_inset notation) \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sin$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arcsin$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sin^{-1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\cos$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arccos$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\cos^{-1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tan$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arctan$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\tan^{-1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\cot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arccot$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\cot^{-1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sec$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arcsec$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\sec^{-1}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\csc$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arccsc$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\csc^{-1}$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Standard In the table below, \begin_inset Formula $u$ \end_inset can represent any differentiable expression, using the chain rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Chain-Rule" \end_inset ). \end_layout \begin_layout Standard \noindent \align center \begin_inset Tabular \begin_inset Text \begin_layout Plain Layout Inverse Trigonometric Function \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout Derivative \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arcsin u$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle }$ \end_inset \begin_inset Formula ${\displaystyle \frac{1}{\sqrt{1-u^{2}}}\times\frac{du}{dx}}$ \end_inset , \begin_inset Formula $|u|<1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arccos u$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{-1}{\sqrt{1-u^{2}}}\times\frac{du}{dx}}$ \end_inset , \begin_inset Formula $|u|<1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arctan u$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{1}{1+u^{2}}\times\frac{du}{dx}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arccot u$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{-1}{1+u^{2}}\times\frac{du}{dx}}$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arcsec u$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{1}{|u|\sqrt{u^{2}-1}}\times\frac{du}{dx}}$ \end_inset , \begin_inset Formula $|u|>1$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula $\arccsc u$ \end_inset \end_layout \end_inset \begin_inset Text \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{-1}{|u|\sqrt{u^{2}-1}}\times\frac{du}{dx}}$ \end_inset , \begin_inset Formula $|u|>1$ \end_inset \begin_inset Formula ${\displaystyle }$ \end_inset \end_layout \end_inset \end_inset \end_layout \begin_layout Paragraph Strategies for Simplifying \end_layout \begin_layout Standard In many difficult problems (e.g. multiple choice) where simplifying is necessary, there are some strategies for doing so. If simplifying is not required, these strategies are not necessary. \end_layout \begin_layout Itemize If an expression under an absolute value is always positive, the absolute value symbols can be removed. \end_layout \begin_layout Itemize Combine terms into terms with a common denominator. \end_layout \begin_layout Itemize Factor out variables from square roots. \end_layout \begin_layout Subsection More Rules \end_layout \begin_layout Standard If the original expression is a constant raised to a variable power, use the \begin_inset Formula $c^{x}$ \end_inset rule ( \begin_inset CommandInset ref LatexCommand prettyref reference "sub:cx" \end_inset ). If the original expression contains a variable in the base and exponent, logarithmic differentiation (page \begin_inset CommandInset ref LatexCommand pageref reference "sub:Logarithmic-Differentiation" \end_inset ) must be used. \end_layout \begin_layout Subsubsection \begin_inset Formula $e^{x}$ \end_inset \end_layout \begin_layout Standard The derivative of \begin_inset Formula $e^{x}$ \end_inset is itself. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout Based on the chain rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Chain-Rule" \end_inset ), where \begin_inset Formula $u$ \end_inset is any differentiable expression, \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[e^{u}]=e^{u}\times\frac{du}{dx}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection \begin_inset Formula $c^{x}$ \end_inset \begin_inset CommandInset label LatexCommand label name "sub:cx" \end_inset \end_layout \begin_layout Standard \begin_inset Formula $c$ \end_inset represents a constant. The derivative of \begin_inset Formula $c^{x}$ \end_inset is \begin_inset Formula $ $ \end_inset \begin_inset Formula $c^{x}\times\ln c$ \end_inset , \begin_inset Formula $c>0$ \end_inset and \begin_inset Formula $c\neq1$ \end_inset . \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout Based on the chain rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Chain-Rule" \end_inset ), where \begin_inset Formula $c$ \end_inset is a constant, \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[c^{u}]=\ln c\times c^{u}\times\frac{du}{dx}}$ \end_inset , \begin_inset Formula $c>0$ \end_inset and \begin_inset Formula $c\neq1$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection \begin_inset Formula $\ln x$ \end_inset \end_layout \begin_layout Standard The derivative of \begin_inset Formula $\ln x$ \end_inset is \begin_inset Formula $\frac{1}{x}$ \end_inset , \begin_inset Formula $x>0$ \end_inset . \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout Based on the chain rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Chain-Rule" \end_inset ), where \begin_inset Formula $u$ \end_inset is any differentiable expression, \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[\ln u]=\frac{1}{u}\times\frac{du}{dx}}$ \end_inset , \begin_inset Formula $u>0$ \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Logarithms \end_layout \begin_layout Paragraph Properties \end_layout \begin_layout Standard These properties hold true for both \begin_inset Formula $\log$ \end_inset and \begin_inset Formula $\ln$ \end_inset . \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Itemize \begin_inset Formula $\log(xy)=\log x+\log y$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $\log(x/y)=\log x-\log y$ \end_inset \end_layout \begin_layout Itemize \begin_inset Formula $\log x^{a}=a\ln x$ \end_inset \end_layout \end_inset \end_layout \begin_layout Paragraph Change of Base \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \log_{a}x=\frac{\log x}{\log a}=\frac{\ln x}{\ln a}}$ \end_inset \end_layout \end_inset \end_layout \begin_layout Paragraph \begin_inset Formula $\log_{b}x$ \end_inset \end_layout \begin_layout Standard The derivative of \begin_inset Formula $\log_{b}x$ \end_inset is \begin_inset Formula ${\displaystyle \frac{1}{x\ln(b)}}$ \end_inset . \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout Based on the chain rule (page \begin_inset CommandInset ref LatexCommand pageref reference "sec:Chain-Rule" \end_inset ), where \begin_inset Formula $u$ \end_inset is any differentiable expression, \end_layout \begin_layout Plain Layout \begin_inset Formula ${\displaystyle \frac{d}{dx}[\log_{b}u]=\frac{1}{u\ln(b)}\times\frac{du}{dx}}$ \end_inset ; \begin_inset Formula $b>0$ \end_inset , \begin_inset Formula $b\neq1$ \end_inset , and \begin_inset Formula $u>0$ \end_inset \end_layout \end_inset \end_layout \begin_layout Paragraph Logarithmic Differentiation \begin_inset CommandInset label LatexCommand label name "sub:Logarithmic-Differentiation" \end_inset \end_layout \begin_layout Standard Logarithmic differentiation is a differentiation process used to take the derivative of a variable raised to a variable or other complex situations. The natural log ( \begin_inset Formula $\ln$ \end_inset ) of both sides of an equation are taken, and the result is implicitly different iated. \end_layout \begin_layout Subsection Extreme Value Theorem \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f$ \end_inset is continuous on the interval \begin_inset Formula $[a,b]$ \end_inset , \begin_inset Formula $f$ \end_inset has both a maximum and a minimum value in the interval. \end_layout \end_inset \end_layout \begin_layout Standard Note that brackets \begin_inset Formula $[\,]$ \end_inset refer to a closed interval including the endpoints while parentheses \begin_inset Formula $(\,)$ \end_inset refer to an interval not including the endpoints. \end_layout \begin_layout Subsection Mean Value Theorem \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f$ \end_inset is continuous on the interval \begin_inset Formula $[a,b]$ \end_inset and differentiable on the interval \begin_inset Formula $(a,b)$ \end_inset , there exists a point \begin_inset Formula $c$ \end_inset on \begin_inset Formula $(a,b)$ \end_inset such that \begin_inset Formula ${\displaystyle f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}}$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard In other words, somewhere on the interval the slope of the tangent line equals (at least once) the slope of the secant line connecting the two endpoints. \end_layout \begin_layout Subsection Rolle's Theorem \end_layout \begin_layout Standard Rolle's Theorem is a special case of the Mean Value Theorem. \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f$ \end_inset is continuous on the interval \begin_inset Formula $[a,b]$ \end_inset , differentiable on the interval \begin_inset Formula $(a,b)$ \end_inset , and \begin_inset Formula $f(a)=f(b)$ \end_inset , then there exists a point \begin_inset Formula $c$ \end_inset on \begin_inset Formula $(a,b)$ \end_inset such that \begin_inset Formula $f^{\prime}(c)=0$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Chapter Integrals \end_layout \begin_layout Standard This chapter was designed for a test on integrals administered by Jonathan Chernick to his AP \begin_inset Foot status collapsed \begin_layout Plain Layout AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this product. \end_layout \end_inset Calculus BC class on November 26, 2008. It is not covered in Math 12H/4H. \end_layout \begin_layout Section Definite Integrals \end_layout \begin_layout Subsection Definition \end_layout \begin_layout Description definite \begin_inset space ~ \end_inset integral area between a curve and the \begin_inset Formula $x$ \end_inset -axis (area underneath the \begin_inset Formula $x$ \end_inset -axis is negative) \end_layout \begin_layout Standard A finite number of rectangles can be used to estimate this area. A larger number of rectangles will give a more accurate estimate, and an infinite number of rectangles can give an exact answer. \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[f(x)dx]\approx A_{k}=\sum_{k=1}^{n}a_{k}=a_{1}+a_{2}+\cdots+a_{n-1}+a_{n}\] \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Riemann Sums \end_layout \begin_layout Standard This area can be expressed as the infinite limit of Riemann sums. As \begin_inset Formula $n$ \end_inset gets larger the width of the rectangles gets smaller and when \begin_inset Formula $n$ \end_inset approaches infinity, the exact area is calculated. \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f(x)$ \end_inset is a continuous on the closed interval \begin_inset Formula $[a,b]$ \end_inset , the definite integral of \begin_inset Formula $f(x)$ \end_inset between \begin_inset Formula $a$ \end_inset and \begin_inset Formula $b$ \end_inset is: \begin_inset Formula \[ \int_{a}^{b}[f(x)]dx=\lim_{n\to\infty}\left(\sum_{k=1}^{n}f(c_{k})\right)\left(\frac{b-a}{n}\right)\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $c_{k}$ \end_inset are sample points in the interval. \end_layout \end_inset \end_layout \begin_layout Subsection Notation \end_layout \begin_layout Standard When considering the expression \begin_inset Formula $\int_{a}^{b}[f(x)]dx$ \end_inset , the function \begin_inset Formula $f(x)$ \end_inset is called the integrand and the interval \begin_inset Formula $[a,b]$ \end_inset is the interval of integration. \begin_inset Formula $a$ \end_inset and \begin_inset Formula $b$ \end_inset are the lower and upper limits of integration, respectively. \end_layout \begin_layout Section Rectangular Approximation Method \end_layout \begin_layout Standard Rectangular Approximation Method (RAM) is a method of estimating definite integrals by calculating the area of a certain number of rectangles. A larger number of rectangles will give a more accurate estimate. \end_layout \begin_layout Subsection Left Rectangular Approximation Method (LRAM) \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[f(x)dx]\approx\Delta x(f(a)+f(a+\Delta x)+\cdots+f(b-2\Delta x)+f(b-\Delta x))\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $\Delta x$ \end_inset is the width of the rectangles ( \begin_inset Formula $\frac{b-a}{n}$ \end_inset ) and \begin_inset Formula $n$ \end_inset is the number of rectangles. \end_layout \end_inset \end_layout \begin_layout Subsection Right Rectangular Approximation Method (RRAM) \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[f(x)dx]\approx\Delta x(f(a+\Delta x)+f(a+2\Delta x)+\cdots+f(b-\Delta x)+f(b))\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $\Delta x$ \end_inset is the width of the rectangles ( \begin_inset Formula $\frac{b-a}{n}$ \end_inset ) and \begin_inset Formula $n$ \end_inset is the number of rectangles. \end_layout \end_inset \end_layout \begin_layout Subsection Midpoint Rectangular Approximation Method (MRAM) \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[f(x)dx]\approx\Delta x(f(a+\frac{\Delta x}{2})+f(a+\Delta x)+\cdots+f(b-\Delta x)+f(b-\frac{\Delta x}{2}))\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $\Delta x$ \end_inset is the width of the rectangles ( \begin_inset Formula $\frac{b-a}{n}$ \end_inset ) and \begin_inset Formula $n$ \end_inset is the number of rectangles. \end_layout \end_inset \end_layout \begin_layout Section Trapezoidal Approximation Method \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[f(x)dx]\approx\left(\frac{1}{2}\right)(\Delta x)\left(f(a)+2f(a+\Delta x)+\cdots+2f(b-\Delta x)+f(b)\right)\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $\Delta x$ \end_inset is the width of the trapezoids ( \begin_inset Formula $\frac{b-a}{n}$ \end_inset ) and \begin_inset Formula $n$ \end_inset is the number of trapezoids. \end_layout \end_inset \end_layout \begin_layout Standard An integral approximated with this rule on a concave-up function will be an overestimate because the trapezoids include all of the area under the curve and extend over it. Using this method on a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. \end_layout \begin_layout Section Graphing Calculator \end_layout \begin_layout Standard These instructions are designed for a TI-84 Plus calculator, but they may used on other Texas Instruments graphing calculators, though slight modificatio n may be necessary. Unless otherwise specified, the graphing calculator should be in radian mode. \end_layout \begin_layout Subsection Definite Integral Rectangular Approximations \end_layout \begin_layout Standard In some cases it may be easier or required to calculate rectangular approximatio ns of definite integrals using the graphing calculator, especially when using a large number of rectangles. \end_layout \begin_layout Standard The program \family typewriter RAM \family default must be added to the calculator's memory. Once installed, set the \begin_inset Formula $y_{1}$ \end_inset of the calculator's graph to the function being integrated and run the program with \family typewriter PRGM \family default \begin_inset Formula $\longrightarrow$ \end_inset \family typewriter RAM \family default . \end_layout \begin_layout Subsection Definite Integral Calculations \end_layout \begin_layout Standard In some cases it may be easier or required to calculate definite integrals using the graphing calculator, especially when the function is too complex. It can also be used to check one's answer. \end_layout \begin_layout Standard \begin_inset Note Comment status open \begin_layout Plain Layout todo \end_layout \end_inset \end_layout \begin_layout Section Fundamental Theorem of Calculus \end_layout \begin_layout Standard Every continuous function has an antiderivative. \end_layout \begin_layout Subsection Part I \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f$ \end_inset is continuous on the closed interval \begin_inset Formula $[a,b]$ \end_inset and \begin_inset Formula $F(x)=\int_{a}^{x}[f(t)dt]$ \end_inset on the closed interval \begin_inset Formula $[a,b]$ \end_inset , then \begin_inset Formula $F$ \end_inset is differentiable on the open interval \begin_inset Formula $(a,b)$ \end_inset and \begin_inset Formula $F^{\prime}(x)=f(x)$ \end_inset for all \begin_inset Formula $x$ \end_inset in the open interval \begin_inset Formula $(a,b)$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard By definition \begin_inset Formula $F(x)$ \end_inset is the antiderivative of \begin_inset Formula $f(x)$ \end_inset in the open interval \begin_inset Formula $(a,b)$ \end_inset . \end_layout \begin_layout Subsection Part II \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f$ \end_inset is continuous on the closed interval \begin_inset Formula $[a,b]$ \end_inset and \begin_inset Formula $F$ \end_inset is an antiderivative of \begin_inset Formula $f$ \end_inset , then: \begin_inset Formula \[ \int_{a}^{b}[f(x)dx]=F(b)-F(a)\] \end_inset \end_layout \end_inset \end_layout \begin_layout Standard It is therefore possible to calculate a definite integral using rules for antiderivatives (indefinite integrals). \end_layout \begin_layout Subsection Corollary \end_layout \begin_layout Standard Integration and differentiation are inverses of each other. \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f$ \end_inset is continuous on the closed interval \begin_inset Formula $[a,b]$ \end_inset then: \begin_inset Formula \[ \frac{d}{dx}\left[\int_{a}^{x}[f(t)dt]\right]=f(x)\] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \frac{d}{du}\left[\int_{a}^{u}[f(t)\ dt]\right]=f(u)du\] \end_inset \end_layout \end_inset \end_layout \begin_layout Section Integral Rules \end_layout \begin_layout Standard Rules for calculating the integrals of general functions have been developed. As a result, it is possible to calculate the integrals of a wide variety of functions. In many cases the use of multiple rules are required. \end_layout \begin_layout Standard In the following rules, \begin_inset Formula $C$ \end_inset represents the constant of integration. \end_layout \begin_layout Subsection Constant Function \end_layout \begin_layout Standard The definite integral of a constant function is a rectangle with the height being the constant and the width being the interval of integration. \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int[cdx]=cx+C\] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[cdx]=c(b-a)\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $c$ \end_inset is a constant. \end_layout \end_inset \end_layout \begin_layout Subsection Addition/Subtraction Rule \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f(x)$ \end_inset and \begin_inset Formula $g(x)$ \end_inset are continuous on the closed interval \begin_inset Formula $[a,b]$ \end_inset , then: \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \int[(f(x)\pm g(x))dx]=\int[f(x)dx]\pm\int[g(x)dx]+C\] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[(f(x)\pm g(x))dx]=\int_{a}^{b}[f(x)dx]\pm\int_{a}^{b}[g(x)dx]\] \end_inset \end_layout \end_inset \end_layout \begin_layout Standard As a result, one can take an equation, break it up into terms, figure out the definite integrals individually, and build the answer back up. \end_layout \begin_layout Subsection Constant Multiplier Rule \begin_inset CommandInset label LatexCommand label name "sec:Constant-Multiplier-Rule-1" \end_inset \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ {\displaystyle \int[c\times f(x)dx]=c\int[f(x)dx]}\] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ {\displaystyle \int_{a}^{b}[c\times f(x)dx]=c\int_{a}^{b}[f(x)dx]}\] \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Power Rule \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int[x^{n}dx]=\frac{x^{n+1}}{n+1}+C\] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[x^{n}dx]=\frac{b^{n+1}-a^{n+1}}{n+1}\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $n$ \end_inset is a constant exponent not equal to \begin_inset Formula $-1$ \end_inset and \begin_inset Formula $x\ne0$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Standard Expressions containing roots (i.e. square roots) can be intregrated by using a fractional value for \begin_inset Formula $n$ \end_inset ( \begin_inset Formula $\sqrt[b]{x^{a}}=x^{\nicefrac{a}{b}}$ \end_inset ). Expressions containing algebraic monomials in the denominator of a fraction can be integrated by inverting the sign of \begin_inset Formula $n$ \end_inset ( \begin_inset Formula $\frac{1}{x^{n}}=x^{-n}$ \end_inset ). \end_layout \begin_layout Subsection Logarithms \end_layout \begin_layout Subsubsection \begin_inset Formula $\frac{1}{x}$ \end_inset Rule \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int\left[\frac{dx}{x}\right]=\ln|x|+C\] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}\left[\frac{dx}{x}\right]=\ln|b|-\ln|a|\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $x\ne0$ \end_inset . \end_layout \end_inset \end_layout \begin_layout Subsubsection \begin_inset Formula $e^{x}$ \end_inset Rule \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int\left[e^{kx}dx\right]=\frac{e^{kx}}{k}+C\] \end_inset \end_layout \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}\left[e^{kx}dx\right]=\frac{e^{kb}}{k}-\frac{e^{ka}}{k}\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $k$ \end_inset is a constant. \end_layout \end_inset \end_layout \begin_layout Subsubsection \begin_inset Formula $a^{x}$ \end_inset Rule \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int\left[a^{x}dx\right]=\frac{a^{x}}{\ln a}+C\] \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Trigonometry \end_layout \begin_layout Itemize integrating the derivatives of the six trigonometric functions \end_layout \begin_layout Itemize integrating the derivatives of the inverse trigonometric functions \end_layout \begin_layout Standard See the \emph on Math Calculus/12H Derivatives Review Report \emph default for more information. \end_layout \begin_layout Subsubsection Constant \end_layout \begin_layout Standard If the constant is outside the trigonometric function, use the constant multiplier rule ( \begin_inset CommandInset ref LatexCommand prettyref reference "sec:Constant-Multiplier-Rule-1" \end_inset ). If the constant is inside the trigonometric function, use the following rule. \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int[(\trig kx)dx]=\frac{(\int[trig]kx)}{k}+C\] \end_inset \end_layout \begin_layout Plain Layout where \begin_inset Formula $k$ \end_inset is a constant. \end_layout \end_inset \end_layout \begin_layout Subsection Definite Integrals \end_layout \begin_layout Subsubsection Additivity Rule \end_layout \begin_layout Standard The area under the graph of \begin_inset Formula $f(x)$ \end_inset between \begin_inset Formula $a$ \end_inset and \begin_inset Formula $b$ \end_inset is the area between \begin_inset Formula $a$ \end_inset and \begin_inset Formula $c$ \end_inset plus the area between \begin_inset Formula $c$ \end_inset and \begin_inset Formula $b$ \end_inset . \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{b}[f(x)dx]=\int_{a}^{c}[f(x)dx]+\int_{c}^{b}[f(x)dx]\] \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Zero Rule \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{a}^{a}[f(x)dx]=0\] \end_inset \end_layout \end_inset \end_layout \begin_layout Subsubsection Order of Integration Rule \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ \int_{b}^{a}[f(x)dx]=-\int_{a}^{b}[f(x)dx]\] \end_inset \end_layout \end_inset \end_layout \begin_layout Section Mean Value of Definite Integrals \end_layout \begin_layout Subsection Mean Value \end_layout \begin_layout Standard The average (arithmetic mean) \begin_inset Formula $y$ \end_inset -value of a function over an interval is the integral over the interval divided by the length of the interval. \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout \begin_inset Formula \[ f_{avg}=\frac{\int_{a}^{b}[f(x)dx]}{b-a}\] \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Mean Value Theorem \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $f$ \end_inset is continuous on the closed interval \begin_inset Formula $[a,b]$ \end_inset , then at some point \begin_inset Formula $c$ \end_inset in \begin_inset Formula $[a,b]$ \end_inset there exists the following: \end_layout \begin_layout Plain Layout \begin_inset Formula \[ f(c)=\frac{\int_{a}^{b}[f(x)dx]}{b-a}\] \end_inset \end_layout \end_inset \end_layout \begin_layout Section Initial Value Problems \end_layout \begin_layout Subsection Introduction \end_layout \begin_layout Standard An equation that contains a derivative is called a differential equation. For example, \begin_inset Formula $\frac{dy}{dx}=2x$ \end_inset is a differential equation. Every differential equation of a function corresponds to a specific equation at a particular point (referred to as a particular solution), assuming the point is in the function's domain. \end_layout \begin_layout Standard An initial value problem provides a differential equation and a particular point through which the function passes through. The specific equation is determined by calculating the value of \begin_inset Formula $C$ \end_inset . \end_layout \begin_layout Subsection Example \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \frac{dy}{dx} & = & 2x,\quad y(1)=6\\ \int\left[\frac{dy}{dx}\right] & = & \int[2xdx]\\ y & = & x^{2}+C\\ 6 & = & (1)^{2}+C\\ 6 & = & 1+C\\ C & = & 5\\ y & = & x^{2}+5\end{eqnarray*} \end_inset \end_layout \begin_layout Section Slope Fields \end_layout \begin_layout Standard Slope fields (also known as direction fields) are a logical extension to initial value problems as they provide a sketch of the differential equation for any value of \begin_inset Formula $C$ \end_inset . \end_layout \begin_layout Standard A table containing the value of \begin_inset Formula $\frac{dy}{dx}$ \end_inset (the function's slope) at different \begin_inset Formula $x$ \end_inset and \begin_inset Formula $y$ \end_inset values is used to create a slope field. \end_layout \begin_layout Subsection Approaches \end_layout \begin_layout Standard These approaches reduce the time required to make or analyze slope fields and the possibility of making errors. \end_layout \begin_layout Subsubsection Patterns \end_layout \begin_layout Paragraph Horizontal Pattern \end_layout \begin_layout Standard When the differential equation only contains the letter \begin_inset Formula $y$ \end_inset (e.g. \begin_inset Formula $\frac{dy}{dx}=y$ \end_inset ), there is a horizontal pattern. \end_layout \begin_layout Paragraph Vertical Pattern \end_layout \begin_layout Standard When the differential equation only contains the letter \begin_inset Formula $x$ \end_inset (e.g. \begin_inset Formula $\frac{dy}{dx}=x$ \end_inset ), there is a vertical pattern. \end_layout \begin_layout Subsubsection Direction of Slope \end_layout \begin_layout Standard Determining whether the slopes of points in a certain vicinity are positive or negative is useful for comparing slope fields. \end_layout \begin_layout Subsubsection Zero/No Slope \end_layout \begin_layout Standard Determining where the slopes of points are infinity (vertical and undefined) and where they are zero is useful for comparing slope fields. \end_layout \begin_layout Section Separation of Variables \end_layout \begin_layout Standard Separation of variables is one method to isolate variables in a differentiable equation. The separated variables can then be integrated. \end_layout \begin_layout Standard \begin_inset Box Boxed position "t" hor_pos "c" has_inner_box 1 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Plain Layout If \begin_inset Formula $\frac{dy}{dx}=g(x)h(y)$ \end_inset , then \begin_inset Formula $\frac{dy}{h(y)}=g(x)dx$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Basically, \begin_inset Formula $\frac{dy}{dx}$ \end_inset is being treated as a fraction, which can be can be separated. \end_layout \begin_layout Subsection Example \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} \frac{dy}{dx} & = & yx\\ \frac{dy}{y} & = & xdx\\ \ln|y| & = & \frac{1}{2}x^{2}\\ e^{\ln|y|} & = & e^{\frac{1}{2}x^{2}}\\ y & = & e^{\frac{1}{2}x^{2}+C}\end{eqnarray*} \end_inset \end_layout \begin_layout Section Integration By Substitution \end_layout \begin_layout Standard Integration by substitution is a method for integrating a composition of function, when the entire integral can be expressed in terms of constants, \begin_inset Formula $u$ \end_inset , and \begin_inset Formula $du$ \end_inset . \end_layout \begin_layout Standard \begin_inset Note Comment status open \begin_layout Plain Layout todo: more detail including definite integrals ( \begin_inset Formula $u(a)$ \end_inset to \begin_inset Formula $u(b)$ \end_inset ), \begin_inset Formula $du=\_dt$ \end_inset \end_layout \end_inset \end_layout \begin_layout Standard Integration by substitution may be used in combination with rules for inverse trigonometric functions. \end_layout \begin_layout Section Trigonometric Identities \end_layout \begin_layout Standard Using trigonometric identities may be necessary to simplify expressions before integrating. \end_layout \begin_layout Subsection Pythagorean Identities \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Enumerate \begin_inset Formula \[ \sin^{2}\theta+\cos^{2}\theta=1\] \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula \[ 1+\tan^{2}\theta=\sec^{2}\theta\] \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula \[ 1+\cot^{2}\theta=\csc^{2}\theta\] \end_inset \end_layout \end_inset \end_layout \begin_layout Subsection Quotient Identities \end_layout \begin_layout Standard \begin_inset Box Framed position "t" hor_pos "c" has_inner_box 0 inner_pos "t" use_parbox 0 width "100col%" special "none" height "1in" height_special "totalheight" status open \begin_layout Enumerate \begin_inset Formula \[ \tan\theta=\frac{\sin\theta}{\cos\theta}\] \end_inset \end_layout \begin_layout Enumerate \begin_inset Formula \[ \cot\theta=\frac{\cos\theta}{\sin\theta}\] \end_inset \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset Note Comment status open \begin_layout Itemize graphing calculator todo \end_layout \begin_layout Itemize integration by substitution \end_layout \begin_deeper \begin_layout Itemize definite integrals ( \begin_inset Formula $u(a)$ \end_inset to \begin_inset Formula $u(b)$ \end_inset ) \end_layout \begin_layout Itemize \begin_inset Formula $du=\_dt$ \end_inset \end_layout \end_deeper \begin_layout Itemize long division \end_layout \end_inset \end_layout \end_body \end_document