# Important Formulas

## Differentiation Rules

1. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

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

8. 8.

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

9. 9.

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

1. 10.

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

2. 11.

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

3. 12.

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

4. 13.

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

5. 14.

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

6. 15.

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

7. 16.

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

8. 17.

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

9. 18.

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

1. 19.

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

2. 20.

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

3. 21.

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

4. 22.

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

5. 23.

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

6. 24.

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

7. 25.

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

8. 26.

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

9. 27.

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

1. 28.

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

2. 29.

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

3. 30.

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

4. 31.

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

5. 32.

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

6. 33.

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

7. 34.

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

8. 35.

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

9. 36.

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

## Integration Rules

1. 1.

$\displaystyle\int c\cdot f(x)\operatorname{d}\!x=c\int f(x)\operatorname{d}\!x$

2. 2.

$\displaystyle\int f(x)\pm g(x)\operatorname{d}\!x=\lx@parboxnewline[]\hbox{}% \hfill\displaystyle\int f(x)\operatorname{d}\!x\pm\int g(x)\operatorname{d}\!x$

3. 3.

$\displaystyle\int 0\operatorname{d}\!x=C$

4. 4.

$\displaystyle\int 1\operatorname{d}\!x=x+C$

5. 5.

$\displaystyle\int x^{n}\operatorname{d}\!x=\frac{1}{n+1}x^{n+1}+C,% \lx@parboxnewline{\scriptstyle\hbox{}\qquad n\neq-1}$

6. 6.

$\displaystyle\int e^{x}\operatorname{d}\!x=e^{x}+C\vphantom{\frac{1}{1}}$

7. 7.

$\displaystyle\int a^{x}\operatorname{d}\!x=\frac{1}{\ln a}\cdot a^{x}+C$

8. 8.

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

9. 9.

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

10. 10.

$\displaystyle\int\sin x\operatorname{d}\!x=-\cos x+C\vphantom{\frac{1}{x^{2}}}$

1. 11.

$\displaystyle\int\tan x\operatorname{d}\!x=-\ln\left\lvert\cos x\right\rvert+C$

2. 12.

$\displaystyle\int\sec x\operatorname{d}\!x=\ln\left\lvert\sec x+\tan x\right% \rvert+C$

3. 13.

$\displaystyle\int\csc x\operatorname{d}\!x=-\ln\left\lvert\csc x+\cot x\right% \rvert+C$

4. 14.

$\displaystyle\int\cot x\operatorname{d}\!x=\ln\left\lvert\sin x\right\rvert+C$

5. 15.

$\displaystyle\int\sec^{2}x\operatorname{d}\!x=\tan x+C$

6. 16.

$\displaystyle\int\csc^{2}x\operatorname{d}\!x=-\cot x+C\vphantom{\frac{1}{% \sqrt{x^{2}}}}\lx@parboxnewline{\scriptstyle\phantom{1}}$

7. 17.

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

8. 18.

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

9. 19.

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

10. 20.

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

11. 21.

$\displaystyle\int\frac{1}{x^{2}+a^{2}}\operatorname{d}\!x=\frac{1}{a}\tan^{-1}% \left(\frac{x}{a}\right)+C$

12. 22.

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

1. 23.

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

2. 24.

$\displaystyle\int\cosh x\operatorname{d}\!x=\sinh x+C$

3. 25.

$\displaystyle\int\sinh x\operatorname{d}\!x=\cosh x+C$

4. 26.

$\displaystyle\int\tanh x\operatorname{d}\!x=\ln(\cosh x)+C$

5. 27.

$\displaystyle\int\coth x\operatorname{d}\!x=\ln\left\lvert\sinh x\right\rvert+C$

6. 28.

$\displaystyle\int\frac{1}{\sqrt{x^{2}-a^{2}}}\operatorname{d}\!x=\ln\left% \lvert x+\sqrt{x^{2}-a^{2}}\right\rvert+C\lx@parboxnewline{\scriptstyle% \phantom{1}}$

7. 29.

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

8. 30.

$\displaystyle\int\frac{1}{a^{2}-x^{2}}\operatorname{d}\!x=\frac{1}{2a}\ln\left% \lvert\frac{a+x}{a-x}\right\rvert+C$

9. 31.

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

10. 32.

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

11. 33.

$\displaystyle\int\sqrt{x^{2}+a^{2}}\operatorname{d}\!x=\vphantom{\frac{1}{x^{2% }}}\lx@parboxnewline\hbox{}\hfill\displaystyle\frac{x}{2}\sqrt{x^{2}+a^{2}}+% \frac{a^{2}}{2}\ln\left(x+\sqrt{x^{2}+a^{2}}\right)+C$

## 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

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

### Double Angle Formulas

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

### Sum to Product Formulas

 $\displaystyle\sin x+\sin y$ $\displaystyle=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$ $\displaystyle\sin x-\sin y$ $\displaystyle=2\sin\left(\frac{x-y}{2}\right)\cos\left(\frac{x+y}{2}\right)$ $\displaystyle\cos x+\cos y$ $\displaystyle=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$ $\displaystyle\cos x-\cos y$ $\displaystyle=2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{y-x}{2}\right)$

### Power–Reducing Formulas

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

### Even/Odd Identities

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

### Product to Sum Formulas

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

### Angle Sum/Difference Formulas

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

## Areas and Volumes

#### Triangles

$\displaystyle h=a\sin\theta$ $\displaystyle\text{Area}=\frac{1}{2}bh$ Law of Cosines: $\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos\theta$

#### Right Circular Cone

$\displaystyle\text{Volume}=\frac{1}{3}\pi r^{2}h$ $\displaystyle\text{Surface Area}=$ $\displaystyle\pi r\sqrt{r^{2}+h^{2}}+\pi r^{2}$

#### Parallelograms

$\displaystyle\text{Area}=bh$

#### Right Circular Cylinder

$\displaystyle\text{Volume}=\pi r^{2}h$ $\displaystyle\text{Surface Area}=$ $\displaystyle 2\pi rh+2\pi r^{2}$

#### Trapezoids

$\displaystyle\text{Area}=\frac{1}{2}(a+b)h$

#### Sphere

$\displaystyle\text{Volume}=\frac{4}{3}\pi r^{3}$ $\displaystyle\text{Surface Area}=4\pi r^{2}$

#### Circles

$\displaystyle\text{Area}=\pi r^{2}$ $\displaystyle\text{Circumference}=2\pi r$

#### General Cone

$\displaystyle\text{Area of Base}=A$ $\displaystyle\text{Volume}=\frac{1}{3}Ah$

#### Sectors of Circles

$\displaystyle\theta\text{ in radians}$ $\displaystyle\text{Area}=\frac{1}{2}\theta r^{2}$ $\displaystyle s=r\theta$

#### General Right Cylinder

$\displaystyle\text{Area of Base}=A$ $\displaystyle\text{Volume}=Ah$

## 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 $zero$ of the polynomial and a solution of the equation $p(x)=0$. Furthermore, $(x-a)$ is a $factor$ 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.

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 $rational$ $zero$ 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

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

 $\displaystyle a^{0}=1,\;\;a\neq 0$ $\displaystyle(ab)^{x}$ $\displaystyle=a^{x}b^{x}$ $\displaystyle a^{x}a^{y}$ $\displaystyle=a^{x+y}$ $\displaystyle\sqrt{a}$ $\displaystyle=a^{1/2}$ $\displaystyle\frac{a^{x}}{a^{y}}$ $\displaystyle=a^{x-y}$ $\displaystyle\sqrt[n]{a}$ $\displaystyle=a^{1/n}$ $\displaystyle\left(\frac{a}{b}\right)^{x}=\frac{a^{x}}{b^{x}}$ $\displaystyle\sqrt[n]{a^{m}}$ $\displaystyle=a^{m/n}$ $\displaystyle a^{-x}$ $\displaystyle=\frac{1}{a^{x}}$ $\displaystyle\sqrt[n]{ab}$ $\displaystyle=\sqrt[n]{a}\sqrt[n]{b}$ $\displaystyle(a^{x})^{y}$ $\displaystyle=a^{xy}$ $\displaystyle\sqrt[n]{\frac{a}{b}}$ $\displaystyle=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

### 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 Vertical axis Horizontal axis Foci and vertices Foci and vertices 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

Test Series

Condition(s) of Convergence

Condition(s) of Divergence

Comment

$n^{\text{th}}$-Term

Test for

Divergence

$\displaystyle\sum_{n=1}^{\infty}a_{n}$ $\displaystyle{\lim_{n\to\infty}a_{n}\neq 0}$

cannot show convergence.

Alternating

Series

$\displaystyle\sum_{n=1}^{\infty}(-1)^{n}b_{n}$ $\displaystyle\lim_{n\to\infty}b_{n}=0$

$b_{n}$ must be positive and decreasing

Geometric

Series

$\displaystyle\sum_{n=0}^{\infty}ar\mkern 1.35mu ^{n}$ $\left\lvert r\right\rvert<1$ $\left\lvert r\right\rvert\geq 1$ Sum $=\dfrac{a}{1-r}$

Telescoping

Series

$\displaystyle\sum_{n=1}^{\infty}b_{n}-b_{n+m}$ $\displaystyle{\lim_{n\to\infty}b_{n}=L}$

Sum $=$

$\displaystyle\left(\sum_{n=1}^{m}b_{n}\right)-L$

$p$-Series $\displaystyle\sum_{n=1}^{\infty}\frac{1}{(an+b)^{p}}$ $p>1$ $p\leq 1$

$p$-Series For

Logarithms

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(an+b)(\log n)^{p}}$ $p>1$ $p\leq 1$

logarithm’s base doesn’t affect convergence.

Integral

Test

$\displaystyle\sum_{n=1}^{\infty}a_{n}$

$\displaystyle\int_{1}^{\infty}a(n)\operatorname{d}\!n$

convergesd

$\displaystyle\int_{1}^{\infty}a(n)\operatorname{d}\!n$

diverges

$a_{n}=a(n)$ must be positive and decreasing

Direct

Comparison

$\displaystyle\sum_{n=1}^{\infty}a_{n}$

$\displaystyle\sum_{n=0}^{\infty}b_{n}$

converges and

$0\leq a_{n}\leq b_{n}$

$\displaystyle\sum_{n=0}^{\infty}b_{n}$

diverges and

$0\leq b_{n}\leq a_{n}$

Limit

Comparison

$\displaystyle\sum_{n=1}^{\infty}a_{n}$

$\displaystyle\sum_{n=0}^{\infty}b_{n}$

converges and

$\displaystyle\lim_{n\to\infty}a_{n}/b_{n}<\infty$

$\displaystyle\sum_{n=0}^{\infty}b_{n}$

diverges and

\begin{aligned} \lim_{n\to\infty}a_{n}/b_{n}&>0\\[-4.3pt] \text{or }&=\infty\end{aligned}

$a_{n},b_{n}>0$
Ratio Test $\displaystyle\sum_{n=1}^{\infty}a_{n}$

$\displaystyle\lim_{n\to\infty}\left\lvert\frac{a_{n+1}}{a_{n}}\right\rvert<1$

\begin{aligned} \lim_{n\to\infty}\left\lvert\frac{a_{n+1}}{a_{n}}\right\rvert&% >1\\ \text{or }&=\infty\end{aligned}

limit of 1 is indeterminate

Root Test $\displaystyle\sum_{n=1}^{\infty}a_{n}$

$\displaystyle\lim_{n\to\infty}\left\lvert a_{n}\right\rvert^{1/n}<1$

\begin{aligned} \lim_{n\to\infty}\left\lvert a_{n}\right\rvert^{1/n}&>1\\[-2.1% 5pt] \text{or }&=\infty\end{aligned}

limit of 1 is indeterminate 