Memorization List

Trigonometric Identities
$\displaystyle \textrm{sine}=\frac{\textrm{opposite}}{\textrm{hypotenuse}}$$	$\displaystyle \textrm{cosine}=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}$
$\displaystyle \textrm{tangent}=\frac{\textrm{opposite}}{\textrm{adjacent}}$	$\displaystyle \cot x=\frac{\cos x}{\sin x}$
$\displaystyle \cot x=\frac{\cos x}{\sin x}$	$\displaystyle \cot x=\frac{\cos x}{\sin x}$
$\displaystyle \csc x=\frac{1}{\sin x}$	$\displaystyle \cot x=\frac{1}{\tan x}$
$\displaystyle \sin^2 x + \cos^2 = 1$	$\displaystyle 1+\tan^2 x =\sec^2 x$
$\displaystyle 1+\cot^2 x =\csc^2 x$	$\displaystyle \sin(-x)=-\sin x$
$\displaystyle \cos(-x)=\cos x$	$\displaystyle \sin 2x = 2 \sin x \cos x$
$\displaystyle \cos 2x = \cos^2 x - \sin^2 x$	$\cos 2x=2\cos^2 x -1$
$\displaystyle \cos 2x = 1- 2 \sin^2 x$	$\displaystyle \sin (a+b)=\sin a \cos b + \cos a \sin b$

$\displaystyle \sin (a-b)=\sin a \cos b - \cos a \sin b$

$\displaystyle \cos (a+b)=\cos a \cos b - \sin a \sin b$

$\displaystyle \cos (a-b)=\cos a \cos b + \sin a \sin b$

$\displaystyle f \;\textrm{is odd if} \;f(-x)=-f(x)\;\forall x$

$\displaystyle f \;\textrm{is even if} \;f(-x)=f(x)\;\forall x$

Definition of Limit

$$\center \displaystyle \ensuremath \lim_{x\rightarrow a}f(x)=L\;\textrm{if and only if}\;\forall \epsilon > 0\;\exists \;\delta > 0\;\textrm{such that whenever}$\\$|x-a|<\delta\;\rightarrow\;|f(x)-L|<\delta.$

Finding limits as x approaches a number
Result	Limit
c	c
$\frac{0}{0}$$	Use L'Hopital's Rule
$\frac{c}{0}$$	$$\pm \infty$ Limit does not exist. Do left and right hand limits of necessary

$\displaystyle \ensuremath \lim_{x\rightarrow 0}\frac{\sin x}{x}=1$

$\ensurmath \displaystyle \lim_{x\rightarrow 0}\frac{1-\cos x}{x}=0$

Limits at Infinity

$\textrm{Degrees of fraction}$	$\textrm{Limit}$
$\textrm{degree of numerator} $>$ \textrm{degree of denominator}$	$\pm \infty\;\textrm{(limit does not exist)}$
$\textrm{degree of numerator} $<$ \textrm{degree of denominator}$	$0$
$\textrm{degree of numerator} $>$ \textrm{degree of denominator}$	$\textrm{Divide coefficients}$

Continuity at a number

$$f \textrm{is continuous at}\;$x=a$ \textrm{if and only if}$f(a)=\ensuremath \displaystyle \lim_{x\rightarrow a}f(x)$\\ \\ \textrm{(First determine if function value exists, then find the limit.)}$

Intermediate Value Theorm

$\center\texrm{If}\;f is \textrm{continuous on}\;[a,b]\;\textrm{then}\;\forall\;k\;\in\left(f(a),f(b) \right )\;\exists\;c\;\in\;(a,b)\\ \;\textrm{such that}\;f(c)=k.$

$\begin{center}\text{It is often used to show that a function has a zero.}\end{center}$

$a$

Derivative Theorems
$\displaystyle \ensuremath f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}$		$\displaystyle \ensuremath f'(x)=\lim_{x_1\rightarrow x}\frac{f(x_1)-f(x)}{x_1-x}$
$D_x[c]=0$		$D_x[c\;f(x)]=c\;f'(x)$
$D_x[f(x)g(x)]=f(x)g'(x)+g(x)f'(x)$		$\displaystyle \ensuremath D_x\left[\frac{f(x)}{g(x)} \right ]=\frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}$
$\displaystyle \ensuremath D_x\left[f(x)^n \right ]=n\;f(x)^{n-1}\;f'(x)$		$\displaystyle \ensuremath D_x\left[f(g(x)) \right ]=f'(g(x))\;g'(x)$
$\displaystyle \ensuremath D_x\left[\sin f(x)]=f'(x)\;\cos f(x)$		$\displaystyle \ensuremath D_x\left[\cos f(x)]=-f'(x)\;\sin f(x)$
$\displaystyle \ensuremath D_x\left[\tan f(x)]=f'(x)\;\sec^2 f(x)$		$\displaystyle \ensuremath D_x\left[\cot f(x)]=-f'(x)\;\csc^2 f(x)$
$\displaystyle \ensuremath D_x\left[\sec f(x)]=f'(x)\sec f(x)\tan f(x)$		$\displaystyle \ensuremath D_x\left[\csc f(x)]=-f'(x)\csc f(x)\cot f(x)$
$\displaystyle \ensuremath D_x\left[\sqrt{f(x)}\right]=\frac{f'(x)}{2\sqrt{f(x)}}$		$\displaystyle \ensuremath D_x\left[e^{f(x)})\right]=e^{f(x)}f'(x)$
$\displaystyle \ensuremath D_x\left[a^{f(x)})\right]=a^{f(x)}f'(x)\ln a$		$\displaystyle \ensuremath D_x\left[\ln f(x)\right]=\frac{f'(x)}{f(x)}$
$\displaystyle \ensuremath D_x\left[\log_a f(x)\right]=\frac{f'(x)}{f(x)\ln a}$		$\displaystyle \ensuremath D_x\left[\log_a f(x)\right]=\frac{f'(x)\log_a e}{f(x)}$
$D_x[\sin^{-1}f(x)]=\frac{f'(x)}{\sqrt{1-[f(x)]^2}}$		$D_x[\cos^{-1}f(x)]=-\frac{f'(x)}{\sqrt{1-[f(x)]^2}}$
$D_x[\tan^{-1}f(x)]=\frac{f'(x)}{1+[f(x)]^2}$		$D_x[\cot^{-1}f(x)]=-\frac{f'(x)}{1+[f(x)]^2}$
$D_x[\sec^{-1}f(x)]=\frac{f'(x)}{f(x)\sqrt{[f(x)]^2-1}}$		$D_x[\csc^{-1}f(x)]=-\frac{f'(x)}{f(x)\sqrt{[f(x)]^2-1}}$

Vertical Asymptotes

$\ensuremath f\;\textrm{has a vertical asymptote at}$\;x=a\;\textrm{if}\;f(a)\nexists\;\textrm{and}\;\displaystyle \lim_{x\rightarrow a}f(x)=\pm\infty.$

$\begin{center}\text{One complete sentence for each asymptote.}\end{center}$

Horizontal Asymptotes

$\ensuremath f\;\textrm{has a horizontal asymptote at}$\;y=b\;\textrm{if}\displaystyle \lim_{x\rightarrow \pm \infty}f(x)=b.$

$\begin{center}\text{As always, two sentences\ldots one for} $x\rightarrow \infty$\,\,\, \text{and one for}\,\,\, $x \rightarrow -\infty.$\end{center}$

Linearization

$L(x)=f(a)+f'(a)(x-a)$

$\begin{center} \text{A linearization is just a tangent line!} \end{center}$

$\begin{center} \text{If the curve is concave up, the tangent line approximation will underestimate the actual function value.} \end{center}$

$\begin{center} \text{If the curve is concave down, the tangent line approximation will overestimate the actual function value.} \end{center}$

L'Hopital's Rule

$\textrm{If} \displaystyle \ensuremath \lim_{x \rightarrow a}\frac{f(x)}{g(x)}\;\textrm{is of indeterminant form, then}\;\lim_{x \rightarrow a}\frac{f(x)}{g(x)}=\lim_{x \rightarrow a}\frac{f'(x)}{g'(x)}.$

You can only use this rule if the limit is 0/0.

Inverses

$\center $f\;\textrm{is monotonic and thus has an inverse if and only if}\\ $\;f'(x) \geq 0\;\forall x \;\textrm{or}\;f'(x)\leq 0\;\forall x.$

$\textrm{If}\;(c,d)\;\textrm{is on}\;f\;\textrm{then}\;(f^{-1})'(d))=\frac{1}{f'(c)}.$

Exponential Growth

$\begin{center} \text{The rate of growth in a quantity is directly proportional to the quantity itself.} \end{center}$

$\displaystyle \frac{dy}{dt}=ky$

$y=y_o\;e^{kt}$

Critical Numbers

$\textrm{A number}\;c\;\textrm{is called a critical number if}\;f'(c)=0\;\textrm{or if}\;f'(c)\;\textrm{does not exist.}$

Absolute Extrema

$\begin{center}$f(a)\;\textrm{is called an absolute maximum if}\;f(a)\geq f(x)\;\forall x \in f\\ $f(a)\;\textrm{is called an absolute minimumif}\;f(a)\leq f(x)\;\forall x \in f \end{center}$

Relative Extrema

$\begin{center}$f(a)\;\textrm{is called a relative (local) maximum if}\\\;f(a)\geq f(x)\;\forall x\; \textrm{in some interval containing}\;c.\\ \vspace{3mm}$f(a)\;\textrm{is called a relative (local) minimum if}\\ \;f(a)\leq f(x)\;\forall x\; \textrm{in some interval containing}\;c.\end{center}$

Critical Numbers

$\textrm{Values of}\;x\;\textrm{in the domain of}\;f\;\textrm{where}\;f'(x)=0\;\textrm{or}\;f'\;\nexists\;\textrm{are called critical numbers.}$

$\begin{center}\text{If a function has relative extrema, they must occur at critical numbers.\\[1mm]Just because the function has critical numbers does not necessarily mean it has relative extrema.\\[1mm]You must use the first or second derivative tests.} \end{center}$

Extreme Value Theorem

$\begin{center}\textrm{If}\;f\;\textrm{is continuous on}\;$[a,b]\;\textrm{then}\\ $f\;\textrm{has both an absolute maximum and an absolute minimum function value on}\;[a,b].\end{center}$

Rolle's Theorem

$\begin{center}\textrm{If}\;f\;\textrm{is continuous on}\;$[a,b]\;\textrm{and continuous on}\;(a,b)\;\textrm{and}\;f(a)=f(b)\;\textrm{then}\\ \exists \; c \in (a,b)\;\textrm{such that}\;f'(c)=0.\end{center}$

Mean Value Theorem

$\begin{center}\textrm{If}\;f\;\textrm{is continuous on}\;$[a,b]\;\textrm{and continuous on}\;(a,b)\;\textrm{then}\\ \exists \; c \in (a,b)\;\textrm{such that}\;f'(c)=\displaystyle \frac{f(b)-f(a)}{b-a}.\end{center}$

Increasing/Decreasing

$\begin{center}\textrm{If}\;$f'(x)>0\; \forall \;x \in (a,b),\textrm{then}\;f\;\textrm{is increasing on}\;(a,b).\\ \textrm{If}\;$f'(x)<0\; \forall \;x \in (a,b),\textrm{then}\;f\;\textrm{is decreasing on}\;(a,b).\end{center}$

First Derivative Test for Relative Extrema

$\begin{center}\textrm{If}\;$f'(x)\;\textrm{changes from}\;+\;\textrm{to}\;-\;\textrm{at}\;x=c\;\textrm{then}\;f(c)\;\textrm{is a relative maximum.}\\ \textrm{If}\;$f'(x)\;\textrm{changes from}\;-\;\textrm{to}\;+\;\textrm{at}\;x=c\;\textrm{then}\;f(c)\;\textrm{is a relative minimum.}\end{center}$

Concavity

$\begin{center}\textrm{If}\;$f''(x)>0\;\textrm{on}\;(a,b),\;f\;\textrm{is concave up on}\;(a,b).\\ \textrm{If}\;$f''(x)<0\;\textrm{on}\;(a,b),\;f\;\textrm{is concave down on}\;(a,b).\end{center}$

Inflection Points

$f\;\textrm{has an inflection point at}\;x=c\;\textrm{only if}\;f''\;\textrm{changes sign and}\;f''(c)\;\textrm{exists.}$

Second Derivative Test for Relative Extrema

$\begin{center}\textrm{If}\;$f''(c.n.)>0\;f\;\textrm{has a relative minimum at}\;x=c.n.\\ \textrm{If}\;$f''(c.n.)<0\;f\;\textrm{has a relative maximum at}\;x=c.n.\\ \textrm{Test cannot be used if}\;$f''(c.n.)=0\;\textrm{or if}\;f''(c.n.)\;\textrm{does not exist.}\end{center}$

Antiderivatives
$\int u^n\;du=\frac{u^{n+1}}{n+1}+C,\;n \neq -1$	$\int \frac{1}{u}\;du=\ln \|u\|+C$
$\int \sin u\;du=-\cos u +C$	$\int \cos u\;du=\sin u +C$
$\int \tan u\;du=\ln \|\sec u\| + C$	$\int \cot u\;du=\ln \|\sin u\| + C$
$\int \sec u\;du=\ln \|\sec u + \tan u\| + C$	$\int \csc u\;du=\ln \|\csc u - \cot u\| + C$
$\int \sec^2 u\;du=\tan u + C$	$\int \csc^2 u\;du=-\cot u + C$
$\int \sec u \tan u\;du=\sec u + C$	$\int \csc u \cot u\;du=-\csc u + C$
$\int e^u\;du= e^u + C$	$\int a^u\;du= \frac{a^u}{\ln a} + C$
$\int \frac{1}{\sqrt{a^2-u^2}}\;du = \sin^{-1}\frac{u}{a} + C$	$\int \frac{1}{a^2+u^2}\;du = \frac{1}{a}\tan^{-1}\frac{u}{a} + C$
$\int \frac{1}{u\sqrt{u^2-a^2}}\;du = \frac{1}{a}\sec^{-1}\frac{u}{a} + C$

$p$

$x(t_2)=x(t_1)+\int_{t_1}^{t_2}v(t)dt$