Important Formulas

Differentiation Rules

1.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(cx)=c$
2.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(u\pm v)=u^{\prime}\pm v^{\prime}$
3.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(u\cdot v)=uv^{\prime}+u^{% \prime}v$
4.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}\left(\dfrac{u}{v}\right)=% \dfrac{vu^{\prime}-uv^{\prime}}{v^{2}}$
5.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(u(v))=u^{\prime}(v)v^{\prime}$
6.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(f^{-1}(x))=\dfrac{1}{f\,^{% \prime}(f^{-1}(x))}$
7.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(c)=0$
8.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(x)=1$
9.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(x^{n})=nx^{n-1}$
10.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}((f(x))^{n})=n(f(x))^{n-1}f\,^{% \prime}(x)$

11.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(e^{x})=e^{x}$
12.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(e^{f(x)})=e^{f(x)}f\,^{\prime}% (x)$
13.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(a^{x})=\ln a\cdot a^{x}$
14.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\ln x)=\frac{1}{x}$
15.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\ln f(x))=\frac{1}{f(x)}f\,^{% \prime}(x)$
16.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\log_{a}x)=\frac{1}{x\ln a}$
17.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\sin x)=\cos x$
18.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\cos x)=-\sin x$
19.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\csc x)=-\csc x\cot x$
20.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\sec x)=\sec x\tan x$

21.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\tan x)=\sec^{2}x$
22.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\cot x)=-\csc^{2}x$
23.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\sin^{-1}x)=\frac{1}{\sqrt{1-x% ^{2}}}$
24.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\cos^{-1}x)=\frac{-1}{\sqrt{1-% x^{2}}}$
25.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\csc^{-1}x)=\frac{-1}{\left% \lvert x\right\rvert\sqrt{x^{2}-1}}$
26.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\sec^{-1}x)=\frac{1}{\left% \lvert x\right\rvert\sqrt{x^{2}-1}}$
27.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\tan^{-1}x)=\frac{1}{1+x^{2}}$
28.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\cot^{-1}x)=\frac{-1}{1+x^{2}}$
29.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\cosh x)=\sinh x$
30.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\sinh x)=\cosh x$

31.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\tanh x)=\operatorname{sech}^{% 2}x$
32.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\operatorname{sech}x)=-% \operatorname{sech}x\tanh x$
33.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\operatorname{csch}x)=-% \operatorname{csch}x\coth x$
34.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\coth x)=-\operatorname{csch}^% {2}x$
35.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\cosh^{-1}x)=\frac{1}{\sqrt{x^% {2}-1}}$
36.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\sinh^{-1}x)=\frac{1}{\sqrt{x^% {2}+1}}$
37.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\operatorname{sech}^{-1}x)=% \frac{-1}{x\sqrt{1-x^{2}}}$
38.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\operatorname{csch}^{-1}x)=% \frac{-1}{\left\lvert x\right\rvert\sqrt{1+x^{2}}}$
39.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\tanh^{-1}x)=\frac{1}{1-x^{2}}$
40.

$\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}(\coth^{-1}x)=\frac{1}{1-x^{2}}$

Integration Rules

1.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int c\cdot f(x)\operatorname% {d}\!x=c\int f(x)\operatorname{d}\!x$
2.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\left(f(x)\pm g(x)\right)% \operatorname{d}\!x=\int f(x)\operatorname{d}\!x\pm\int g(x)\operatorname{d}\!x$
3.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int f(x)g^{\prime}(x)% \operatorname{d}\!x=f(x)g(x)-\int f\,^{\prime}(x)g(x)\operatorname{d}\!x$
4.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int f(g(x))g^{\prime}(x)% \operatorname{d}\!x=\int f(u)\operatorname{d}\!u$ ; $u=g(x)$
5.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int 0\operatorname{d}\!x=C$
6.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int 1\operatorname{d}\!x=x+C$
7.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int x^{n}\operatorname{d}\!x% =\frac{1}{n+1}x^{n+1}+C;\quad n\neq-1$
8.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int e^{x}\operatorname{d}\!x% =e^{x}+C$
9.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int a^{x}\operatorname{d}\!x% =\frac{1}{\ln a}\cdot a^{x}+C$
10.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\ln x\operatorname{d}\!x=% x\ln x-x+C$
11.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{x}\operatorname{% d}\!x=\ln\left\lvert x\right\rvert+C$
12.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\cos x\operatorname{d}\!x% =\sin x+C$
13.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sin x\operatorname{d}\!x% =-\cos x+C$
14.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\tan x\operatorname{d}\!x% =-\ln\left\lvert\cos x\right\rvert+C$
15.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sec x\operatorname{d}\!x% =\ln\left\lvert\sec x+\tan x\right\rvert+C$
16.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\csc x\operatorname{d}\!x% =-\ln\left\lvert\csc x+\cot x\right\rvert+C$

17.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\cot x\operatorname{d}\!x% =\ln\left\lvert\sin x\right\rvert+C$
18.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sec^{2}x\operatorname{d}% \!x=\tan x+C$
19.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\csc^{2}x\operatorname{d}% \!x=-\cot x+C$
20.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sec x\tan x\operatorname% {d}\!x=\sec x+C$
21.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\csc x\cot x\operatorname% {d}\!x=-\csc x+C$
22.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\cos^{2}x\operatorname{d}% \!x=\frac{x}{2}+\frac{\sin(2x)}{4}+C$
23.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sin^{2}x\operatorname{d}% \!x=\frac{x}{2}-\frac{\sin(2x)}{4}+C$

24.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{x^{2}+a^{2}}% \operatorname{d}\!x=\frac{1}{a}\tan^{-1}\frac{x}{a}+C$
25.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{\sqrt{a^{2}-x^{2% }}}\operatorname{d}\!x=\sin^{-1}\frac{x}{\left\lvert a\right\rvert}+C$
26.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{x\sqrt{x^{2}-a^{% 2}}}\operatorname{d}\!x=\frac{1}{\left\lvert a\right\rvert}\sec^{-1}\left% \lvert\frac{x}{a}\right\rvert+C$
27.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\cosh x\operatorname{d}\!% x=\sinh x+C$
28.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sinh x\operatorname{d}\!% x=\cosh x+C$
29.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\tanh x\operatorname{d}\!% x=\ln(\cosh x)+C$
30.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\coth x\operatorname{d}\!% x=\ln\left\lvert\sinh x\right\rvert+C$

31.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sec^{3}x\operatorname{d}% \!x=\frac{1}{2}(\sec x\tan x+\ln\left\lvert\sec x+\tan x\right\rvert)+C$
32.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\sqrt{x^{2}+a^{2}}% \operatorname{d}\!x=\frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}\ln\left(x+% \sqrt{x^{2}+a^{2}}\right)+C$
33.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{\sqrt{x^{2}-a^{2% }}}\operatorname{d}\!x=\cosh^{-1}\frac{x}{a}+C=\ln\left(x+\sqrt{x^{2}-a^{2}}% \right)+C$ ; $0<a<x$
34.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{\sqrt{x^{2}+a^{2% }}}\operatorname{d}\!x=\sinh^{-1}\frac{x}{a}+C=\ln\left(x+\sqrt{x^{2}+a^{2}}% \right)+C$ ; $0<a$
35.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{a^{2}-x^{2}}% \operatorname{d}\!x=\begin{cases}\frac{1}{a}\tanh^{-1}\frac{x}{a}+C,&\left% \lvert x\right\rvert<\left\lvert a\right\rvert\\ \frac{1}{a}\coth^{-1}\frac{x}{a}+C,&\left\lvert a\right\rvert<\left\lvert x% \right\rvert\end{cases}\quad=\frac{1}{2a}\ln\left\lvert\frac{a+x}{a-x}\right% \rvert+C$
36.

${\vphantom{\dfrac{1}{\sqrt{x^{2}}}}}\displaystyle\int\frac{1}{x\sqrt{a^{2}-x^{% 2}}}\operatorname{d}\!x=-\frac{1}{a}\operatorname{sech}^{-1}\frac{\left\lvert x% \right\rvert}{a}+C=\frac{1}{a}\ln\left\lvert\frac{x}{a+\sqrt{a^{2}-x^{2}}}% \right\rvert+C$ ; $0<\left\lvert x\right\rvert<a$
37.

$\displaystyle\int\frac{1}{x\sqrt{x^{2}+a^{2}}}\operatorname{d}\!x=-\frac{1}{a}% \operatorname{csch}^{-1}\frac{\left\lvert x\right\rvert}{a}+C=\frac{1}{a}\ln% \left\lvert\frac{x}{a+\sqrt{a^{2}+x^{2}}}\right\rvert+C$ ; $x\neq 0$ , $a>0$

The Unit Circle

Definitions of the Trigonometric Functions

Unit Circle Definition

$\displaystyle\sin\theta$	$\displaystyle=y$	$\displaystyle\cos\theta$	$\displaystyle=x$
$\displaystyle\csc\theta$	$\displaystyle=\dfrac{1}{y}$	$\displaystyle\sec\theta$	$\displaystyle=\dfrac{1}{x}$
$\displaystyle\tan\theta$	$\displaystyle=\frac{y}{x}$	$\displaystyle\cot\theta$	$\displaystyle=\frac{x}{y}$

Right Triangle Definition

$\displaystyle\sin\theta$	$\displaystyle=\frac{\text{O}}{\text{H}}$	$\displaystyle\csc\theta$	$\displaystyle=\frac{\text{H}}{\text{O}}$
$\displaystyle\cos\theta$	$\displaystyle=\frac{\text{A}}{\text{H}}$	$\displaystyle\sec\theta$	$\displaystyle=\frac{\text{H}}{\text{A}}$
$\displaystyle\tan\theta$	$\displaystyle=\frac{\text{O}}{\text{A}}$	$\displaystyle\cot\theta$	$\displaystyle=\frac{\text{A}}{\text{O}}$

Common Trigonometric Identities

Pythagorean Identities

	$\displaystyle\sin^{2}x+\cos^{2}x=1$
	$\displaystyle\tan^{2}x+1=\sec^{2}x$
	$\displaystyle 1+\cot^{2}x=\csc^{2}x$

Cofunction Identities

$\displaystyle\sin(\tfrac{\pi}{2}-x)$	$\displaystyle=\cos x$	$\displaystyle\csc(\tfrac{\pi}{2}-x)$	$\displaystyle=\sec x$
$\displaystyle\cos(\tfrac{\pi}{2}-x)$	$\displaystyle=\sin x$	$\displaystyle\sec(\tfrac{\pi}{2}-x)$	$\displaystyle=\csc x$
$\displaystyle\tan(\tfrac{\pi}{2}-x)$	$\displaystyle=\cot x$	$\displaystyle\cot(\tfrac{\pi}{2}-x)$	$\displaystyle=\tan x$

Even/Odd Identities

$\displaystyle\sin(-x)$	$\displaystyle=-\sin x$	$\displaystyle\csc(-x)$	$\displaystyle=-\csc x$
$\displaystyle\cos(-x)$	$\displaystyle=\phantom{-}\cos x$	$\displaystyle\sec(-x)$	$\displaystyle=\phantom{-}\sec x$
$\displaystyle\tan(-x)$	$\displaystyle=-\tan x$	$\displaystyle\cot(-x)$	$\displaystyle=-\cot x$

Sum to Product Formulas

	$\displaystyle\sin x+\sin y$	$\displaystyle=2\sin(\tfrac{x+y}{2})\cos(\tfrac{x-y}{2})$
	$\displaystyle\sin x-\sin y$	$\displaystyle=2\sin(\tfrac{x-y}{2})\cos(\tfrac{x+y}{2})$
	$\displaystyle\cos x+\cos y$	$\displaystyle=2\cos(\tfrac{x+y}{2})\cos(\tfrac{x-y}{2})$
	$\displaystyle\cos x-\cos y$	$\displaystyle=2\sin(\tfrac{x+y}{2})\sin(\tfrac{y-x}{2})$

Power-Reducing Formulas

	$\displaystyle\sin^{2}x$	$\displaystyle=\frac{1-\cos 2x}{2}$
	$\displaystyle\cos^{2}x$	$\displaystyle=\frac{1+\cos 2x}{2}$
	$\displaystyle\tan^{2}x$	$\displaystyle=\frac{1-\cos 2x}{1+\cos 2x}$

Double Angle Formulas

	$\displaystyle\sin 2x$	$\displaystyle=2\sin x\cos x$
	$\displaystyle\cos 2x$	$\displaystyle=\cos^{2}x-\sin^{2}x$
		$\displaystyle=2\cos^{2}x-1$
		$\displaystyle=1-2\sin^{2}x$
	$\displaystyle\tan 2x$	$\displaystyle=\frac{2\tan x}{1-\tan^{2}x}$

Product to Sum Formulas

	$\displaystyle\sin x\sin y$	$\displaystyle=\tfrac{1}{2}\bigl(\cos(x-y)-\cos(x+y)\bigr)$
	$\displaystyle\cos x\cos y$	$\displaystyle=\tfrac{1}{2}\bigl(\cos(x-y)+\cos(x+y)\bigr)$
	$\displaystyle\sin x\cos y$	$\displaystyle=\tfrac{1}{2}\bigl(\sin(x+y)+\sin(x-y)\bigr)$

Angle Sum/Difference Formulas

	$\displaystyle\sin(x\pm y)$	$\displaystyle=\sin x\cos y\pm\cos x\sin y$
	$\displaystyle\cos(x\pm y)$	$\displaystyle=\cos x\cos y\mp\sin x\sin y$
	$\displaystyle\tan(x\pm y)$	$\displaystyle=\frac{\tan x\pm\tan y}{1\mp\tan x\tan y}$

Domains and ranges of inverse trigonometric functions

Inverse Function	Domain	Range	Inverse Function	Domain	Range
$\sin^{-1}x$	$[-1,1]$	$[-\pi/2,\pi/2]$	$\csc^{-1}x$	$(-\infty,-1]\cup[1,\infty)$	$[-\pi/2,0)\cup(0,\pi/2]$
$\cos^{-1}x$	$[-1,1]$	$[0,\pi]$	$\sec^{-1}x$	$(-\infty,-1]\cup[1,\infty)$	$[0,\pi/2)\cup(\pi/2,\pi]$
$\tan^{-1}x$	$(-\infty,\infty)$	$(-\pi/2,\pi/2)$	$\cot^{-1}x$	$(-\infty,\infty)$	$(0,\pi)$

Areas and Volumes

Triangles $h=a\sin\theta$ Area ${}=\frac{1}{2}bh$ Law of Cosines: $c^{2}=a^{2}+b^{2}-2ab\cos\theta$		Right Circular Cone Volume ${}=\frac{1}{3}\pi r^{2}h$ Surface Area = $\pi r\sqrt{r^{2}+h^{2}}+\pi r^{2}$
Parallelograms Area ${}=bh$		Right Circular Cylinder Volume ${}=\pi r^{2}h$ Surface Area = $2\pi rh+2\pi r^{2}$
Trapezoids Area ${}=\frac{1}{2}(a+b)h$		Sphere Volume ${}=\frac{4}{3}\pi r^{3}$ Surface Area ${}=4\pi r^{2}$
Circles Area ${}=\pi r^{2}$ Circumference ${}=2\pi r$		General Cone Area of Base ${}=A$ Volume ${}=\frac{1}{3}Ah$
Sectors of Circles $\theta$ in radians Area ${}=\frac{1}{2}\theta r^{2}$ $s=r\theta$		General Right Cylinder Area of Base ${}=A$ Volume ${}=Ah$

Algebra

Factors and Zeros of Polynomials

Let $p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{1}x+a_{0}$ be a polynomial. If $p(a)=0$ , then $a$ is a $z e r o$ of the polynomial and a solution of the equation $p(x)=0$ . Furthermore, $(x-a)$ is a $f a c t o r$ of the polynomial.

Fundamental Theorem of Algebra

An $n$ th degree polynomial has $n$ (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.

Quadratic Formula

If $p(x)=ax^{2}+bx+c$ , then the zeros of $p$ are $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Special Factoring

\displaystyle x^{2}-a^{2}

\displaystyle=(x-a)(x+a)

\displaystyle x^{3}\pm a^{3}

\displaystyle=(x\pm a)(x^{2}\mp ax+a^{2})

\displaystyle x^{4}-a^{4}

\displaystyle=(x^{2}-a^{2})(x^{2}+a^{2})

Binomial Theorem

	$\displaystyle(x+y)^{2}$	$\displaystyle=x^{2}+2xy+y^{2}$	$\displaystyle(x+y)^{3}$	$\displaystyle=x^{3}+3x^{2}y+3xy^{2}+y^{3}$
	$\displaystyle(x+y)^{4}$	$\displaystyle=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}$	$\displaystyle(x+y)^{n}$	$\displaystyle=\sum_{k=0}^{n}\binom{n}{k}x\mkern 1.35mu^{n-k}y\mkern 1.35mu^{k}$

Rational Zero Theorem
If $p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb+a_{1}x+a_{0}$ has integer coefficients, then every $r a t i o n a l$ $z e r o$ of $p$ is of the form $x=r/s$ , where $r$ is a factor of $a_{0}$ and $s$ is a factor of $a_{n}$ .

Factoring by Grouping

$acx^{3}+adx^{2}+bcx+bd=ax^{2}(cx+d)+b(cx+d)=(ax^{2}+b)(cx+d)$

Arithmetic Operations

$\displaystyle ab+ac=a(b+c)$	$\displaystyle\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$	$\displaystyle\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$
$\displaystyle\frac{\left(\dfrac{a\vphantom{d}}{b}\right)}{\left(\dfrac{c}{d}% \right)}=\left(\frac{a}{b}\right)\left(\frac{d}{c}\right)=\frac{ad}{bc}$	$\displaystyle\frac{\left(\dfrac{a}{b}\right)}{c}=\frac{a}{bc}$	$\displaystyle\frac{a}{\left(\dfrac{b}{c}\right)}=\frac{ac}{b}$
$\displaystyle a\left(\frac{b}{c}\right)=\frac{ab}{c}$	$\displaystyle\frac{a-b}{c-d}=\frac{b-a}{d-c}$	$\displaystyle\frac{ab+ac}{a}=b+c$

Exponents and Radicals

\displaystyle a^{0}=1,\;\;a\neq 0

\displaystyle(ab)^{x}

\displaystyle=a^{x}b^{x}

\displaystyle a^{x}a^{y}

\displaystyle=a^{x+y}

\displaystyle\sqrt{a}

\displaystyle=a^{1/2}

\displaystyle\frac{a^{x}}{a^{y}}

\displaystyle=a^{x-y}

\displaystyle\sqrt[n]{a}

\displaystyle=a^{1/n}

\displaystyle\left(\frac{a}{b}\right)^{x}=\frac{a^{x}}{b^{x}}

\displaystyle\sqrt[n]{a^{m}}

\displaystyle=a^{m/n}

\displaystyle a^{-x}

\displaystyle=\frac{1}{a^{x}}

\displaystyle\sqrt[n]{ab}

\displaystyle=\sqrt[n]{a}\sqrt[n]{b}

\displaystyle(a^{x})^{y}

\displaystyle=a^{xy}

\displaystyle\sqrt[n]{\frac{a}{b}}

\displaystyle=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}

Additional Formulas

Summation Formulas

\displaystyle\sum^{n}_{i=1}{c}

\displaystyle=cn

\displaystyle\sum^{n}_{i=1}{i}

\displaystyle=\frac{n(n+1)}{2}

\displaystyle\sum^{n}_{i=1}{i^{2}}

\displaystyle=\frac{n(n+1)(2n+1)}{6}

\displaystyle\sum^{n}_{i=1}{i^{3}}

\displaystyle=\left(\frac{n(n+1)}{2}\right)^{2}

Trapezoidal Rule

$\displaystyle\int_{a}^{b}{f(x)}\operatorname{d}\!x\approx\frac{\Delta x}{2}% \left[f(x_{1})+2f(x_{2})+2f(x_{3})+\dotsb+2f(x_{n})+f(x_{n+1})\right]$

with $\text{Error}\leq\dfrac{(b-a)^{3}}{12n^{2}}\left[\max\left\lvert f\,^{\prime% \prime}(x)\right\rvert\right]$

Simpson’s Rule

$\displaystyle\int_{a}^{b}{f(x)}\operatorname{d}\!x\approx\frac{\Delta x}{3}% \left[f(x_{1})+4f(x_{2})+2f(x_{3})+4f(x_{4})+\dotsb+2f(x_{n-1})+4f(x_{n})+f(x_% {n+1})\right]$

with $\text{Error}\leq\dfrac{(b-a)^{5}}{180n^{4}}\left[\max\left\lvert f\mkern 1.35% mu^{(4)}(x)\right\rvert\right]$

Arc Length

$\displaystyle L=\int_{a}^{b}{\sqrt{1+f\,^{\prime}(x)^{2}}}\operatorname{d}\!x$

Work Done by a Variable Force

$\displaystyle W=\int_{a}^{b}{F(x)}\operatorname{d}\!x$

Force Exerted by a Fluid

$\displaystyle F=\int_{a}^{b}{w\,d(y)\,\ell(y)}\operatorname{d}\!y$

Taylor Series Expansion for $f(x)$

$\displaystyle p_{n}(x)=f(c)+f\,^{\prime}(c)(x-c)+\frac{f\,^{\prime\prime}(c)}{% 2!}(x-c)^{2}+\frac{f\,^{\prime\prime\prime}(c)}{3!}(x-c)^{3}+\dotsb+\frac{f\,^% {(n)}(c)}{n!}(x-c)^{n}+\dotsb$

Standard Form of Conic Sections

Parabola		Ellipse	Hyperbola with foci and vertices
Vertical axis	Horizontal axis	Ellipse	on $x$ -axis	on $y$ -axis
$y=\dfrac{x^{2}}{4p}$	$x=\dfrac{y^{2}}{4p}$	$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$	$\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$	$\dfrac{y^{2}}{b^{2}}-\dfrac{x^{2}}{a^{2}}=1$

Summary of Tests for Series

Notation: Infinite series $\displaystyle\sum_{n=1}^{\infty}a_{n}$ with sequence of partial sums $\{S_{n}\}=\{a_{1}+a_{2}+a_{3}+\dots+a_{n}\}$

Test	Series	Convergence or Divergence	Comment
Definition of Series	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	series converges if and only if $\{S_{n}\}$ converges	used when a formula for $S_{n}$ can be found
Divergence Test	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	diverges if $\displaystyle\lim_{n\to\infty}a_{n}\neq 0$	no conclusion if $\displaystyle\lim_{n\to\infty}a_{n}=0$
Alternating Series	$\displaystyle\pm\sum_{n=1}^{\infty}(-1)^{n}b_{n}$	converges if $b_{n}>0$ , $\{b_{n}\}$ is decreasing, and $\displaystyle\lim_{n\to\infty}b_{n}=0$	check that conditions hold eventually; no information about divergence
Geometric Series	$\displaystyle\sum_{n=0}^{\infty}ar\mkern 1.35mu^{n}$	converges if and only if $\left\lvert r\right\rvert<1$	Sum $=\dfrac{a}{1-r}$
Telescoping Series	$\displaystyle\sum_{n=1}^{\infty}b_{n}-b_{n+m}$	converges if and only if $\{S_{n}\}$ converges	most terms of $S_{n}$ subtract away
$p$ -Series	$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(an+b)^{p}}$	converges if and only if $p>1$	assumes $an+b\neq 0$
$p$ -Series For Logarithms	$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(an+b)(\log n)^{p}}$	converges if and only if $p>1$	logarithm’s base doesn’t affect convergence.
Integral Test	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	converges if and only if $\displaystyle\int_{k}^{\infty}a(n)\operatorname{d}\!n$ converges	$a_{n}=a(n)$ must be positive and decreasing eventually
Direct Comparison	$\displaystyle\sum_{n=1}^{\infty}a_{n}$ , $\displaystyle\sum_{n=1}^{\infty}b_{n}$ $0<a_{n}\leq b_{n}$	$\sum b_{n}\text{ converges }\Rightarrow\sum a_{n}$ converges $\sum a_{n}\text{ diverges }\Rightarrow\sum b_{n}$ diverges	consider geometric or $p$ -series
Limit Comparison	$\displaystyle\sum_{n=1}^{\infty}a_{n}$ , $\displaystyle\sum_{n=1}^{\infty}b_{n}$ $0<a_{n},b_{n}$	if $\displaystyle\lim_{n\to\infty}a_{n}/b_{n}=L$ $L>0$ : both converge or diverge together $L=0$ : $\sum b_{n}\text{ converges }\Rightarrow\sum a_{n}$ converges $L=\infty$ : $\sum b_{n}\text{ diverges }\Rightarrow\sum a_{n}$ diverges	consider geometric or $p$ -series
Ratio/Root Test	$\displaystyle\sum_{n=1}^{\infty}a_{n}$	$L=\begin{cases}\displaystyle\lim_{n\to\infty}\left\lvert a_{n+1}/a_{n}\right% \rvert&\text{Ratio Test}\\ \displaystyle\lim_{n\to\infty}\left\lvert a_{n}\right\rvert^{1/n}&\text{Root % Test}\end{cases}$ $L<1$ : converges $L>1$ or $L=\infty$ : diverges $L=1$ : test indeterminate	use Ratio Test for products, factorials, or powers in terms use Root Test for series of the form $a_{n}=(b_{n})^{n}$

Absolute convergence: $\displaystyle\sum_{n=1}^{\infty}\left\lvert a_{n}\right\rvert$ converges (and by Absolute Convergence Theorem, $\displaystyle\sum_{n=1}^{\infty}a_{n}$ converges)

Conditional convergence: $\displaystyle\sum_{n=1}^{\infty}a_{n}$ converges but $\displaystyle\sum_{n=1}^{\infty}\left\lvert a_{n}\right\rvert$ diverges