Formulary

Algebra

Binomial theorem

$(a+b)^n=\sum _{k=0}^n \binom{n}{k} a^{n-k} b^k$

Trigonometric identities

Transformation

$\sin (-\theta) =-\sin \theta$
$\cos (-\theta) =\cos \theta$
$\tan (-\theta) =-\tan \theta$
$\sin (\pi-\theta)=\sin \theta$
$\cos (\pi-\theta)=-\cos \theta$
$\tan (\pi-\theta)=-\tan \theta$
$\tan (\theta+\pi) =\tan \theta$

$\begin{array}{ll}\cos \theta=\sin \left(\frac{\pi}{2}-\theta\right) & \sin \theta=\cos \left(\frac{\pi}{2}-\theta\right) \ \cot \theta=\tan \left(\frac{\pi}{2}-\theta\right) & \tan \theta=\cot \left(\frac{\pi}{2}-\theta\right) \ \csc \theta=\sec \left(\frac{\pi}{2}-\theta\right) & \sec \theta=\csc \left(\frac{\pi}{2}-\theta\right)\end{array}$

Sum of angles

$\sin (\alpha\pm\beta)=\sin \alpha \cos \beta\pm\cos \alpha \sin \beta$
$\cos (\alpha\pm\beta)=\cos \alpha\cos \beta\mp\sin \alpha \sin \beta$
$\tan (\alpha\pm\beta)=\frac{\tan \alpha\pm\tan \beta}{1\mp\tan \alpha \tan \beta}$

$\cos (2 \theta) =\cos ^2 \theta-\sin ^2 \theta$
$\cos (2 \theta) =2 \cos ^2 \theta-1$
$\cos (2 \theta) =1-2 \sin ^2 \theta$

Product to sum

$\sin \alpha \sin \beta=\frac{1}{2}[\cos (\alpha-\beta)-\cos (\alpha+\beta)]$
$\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha-\beta)+\cos (\alpha+\beta)]$
$\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]$
$\cos \alpha \sin \beta=\frac{1}{2}[\sin (\alpha+\beta)-\sin (\alpha-\beta)]$

Weird ones

$\sec ^2 \theta=1+\tan ^2 \theta$

$\sin 3 \theta =3 \sin \theta-4 \sin ^3 \theta$
$\cos 3 \theta =4 \cos ^3 \theta-3 \cos \theta$
$\tan 3 \theta =\frac{3 \tan \theta-\tan ^3 \theta}{1-3 \tan ^2 \theta}$

Limits

Definition

$\lim _{x \rightarrow a} f(x)=L$ if:
For every $\varepsilon>0$ there exists a $\delta>0$ such that if $0<|x-a|<\delta$ then:
$|f(x)-L|<\varepsilon$

Properties

$\lim _{x \rightarrow a} c=c$
$\lim _{x \rightarrow a} x=a$

Let $\lim _{x \rightarrow a} f(x)=L$ and $\lim _{x \rightarrow a} g(x)=M$ then:

• $\lim _{x \rightarrow a} cf(x) = cL$
  
• $\lim _{x \rightarrow a} [f(x) \pm g(x)]=L \pm M$
  
• $\lim _{x \rightarrow a} [f(x) g(x)]=L M$
  
• $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{L}{M}$ if $M \neq 0$
  
• $\lim _{x \rightarrow a} [f(x)]^n=L^n$
  

Common limits

$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
$\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=0$
$\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$
$\lim _{x \rightarrow 0} \frac{\arcsin x}{x}=1$
$\lim _{x \rightarrow 0} \frac{\arctan x}{x}=1$
$\lim _{x \rightarrow 0} \frac{e^x-1}{x}=1$
$\lim _{x \rightarrow 0} \frac{a^x-1}{x}=\ln a$
$\lim _{x \rightarrow 0} \frac{\sinh x}{x}=1$
$\lim _{x \rightarrow 0} \frac{\cosh x-1}{x}=0$
$\lim _{x \rightarrow 0} \frac{\tanh x}{x}=1$
$\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0$
$\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}=1$
$\lim _{x \rightarrow \infty} e^{-x}=0$
$\lim _{x \rightarrow \infty} (1 + \frac{1}{x})^x=e$
$\lim _{x \rightarrow \infty} (1 + \frac{k}{x})^x=e^k$
$\lim _{x \rightarrow 0} (1 + x)^{\frac{1}{x}}=e$

Continuity

Definition

$f$ is continuous at $x=a$ if:
$\lim _{x \rightarrow a} f(x)=f(a)$

Strict definition

$f$ is continuous at $x=a$ if:
$(\forall \varepsilon>0) (\exists \delta>0) (\forall x \in D(f)):$
$|x-a|<\delta \Rightarrow|f(x)-f(a)|<\varepsilon$

Properties

If $f$ and $g$ are continuous at $x=a$ then:

• $f+g$ is continuous at $x=a$
  
• $f-g$ is continuous at $x=a$
  
• $f \cdot g$ is continuous at $x=a$
  
• $\frac{f}{g}$ is continuous at $x=a$ if $g(a) \neq 0$
  

Intermediate value theorem

If $f$ is continuous on $[a, b]$ and let $M$ be any number between $f(a)$ and $f(b)$ then there exists a number $c$ in $[a, b]$ such that $f(c)=M$

Derivatives

Definition

$\frac{d}{d x} f(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

Critical points

$X = c$ is a critical point of $f$ if:

• $f’(c) = 0$
• $f’(c)$ is undefined

Concavity

$f$ is convex on $I$ if $f’‘(x) > 0$ for all $x \in I$
$f$ is concave on $I$ if $f’‘(x) < 0$ for all $x \in I$

Inflection points

$X = c$ is an inflection point of $f$ if the concavity of $f$ changes at $c$

Common derivatives

$1.\frac{d}{d x} \sin x=\cos x$
$2.\frac{d}{d x} \cos x=-\sin x$
$3.\frac{d}{d x} \tan x=\sec ^2 x$
$4.\frac{d}{d x} \cot x=-\csc ^2 x$
$5.\frac{d}{d x} \sec x=\sec x \tan x$
$6.\frac{d}{d x} \csc x=-\csc x \cot x$
$7.\frac{d}{d x} \arcsin x=\frac{1}{\sqrt{1-x^2}}$
$8.\frac{d}{d x} \arccos x=-\frac{1}{\sqrt{1-x^2}}$
$9.\frac{d}{d x} \arctan x=\frac{1}{1+x^2}$
$10.\frac{d}{d x} \operatorname{arccot} x=-\frac{1}{1+x^2}$
$11.\frac{d}{d x} \sinh x=\cosh x$
$12.\frac{d}{d x} \cosh x=\sinh x$
$13.\frac{d}{d x} \tanh x=\operatorname{sech}^2 x$
$14.\frac{d}{d x} \operatorname{coth} x=-\operatorname{csch}^2 x$
$15.\frac{d}{d x} \log _a x=\frac{1}{x \ln a}$
$16.\frac{d}{d x} a^x=a^x \ln a$

Rolle’s theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ and $f(a)=f(b)$ then there exists a number $c$ in $(a, b)$ such that $f’(c)=0$

Mean value theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ then there exists a number $c$ in $(a, b)$ such that $f’(c)=\frac{f(b)-f(a)}{b-a}$

Extreme value theorem

If $f$ is continuous on $[a, b]$ then $f$ attains an absolute maximum and an absolute minimum on $[a, b]$

To find the absolute maximum and minimum of $f$ on $[a, b]$:

1. Find the critical points of $f$ in $(a, b)$
   
2. Evaluate $f$ at the critical points and at the endpoints of $[a, b]$
   
3. The largest value is the absolute maximum and the smallest value is the absolute minimum
   

L’Hôpital’s rule

If $\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} g(x)=0$ or $\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} g(x)=\pm \infty$ and $\lim _{x \rightarrow a} \frac{f’(x)}{g’(x)}$ exists then:
$\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f’(x)}{g’(x)}$

Taylor series

$f(x)=\sum _{n=0}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^n$

Maclaurin series

The Maclaurin series is a Taylor series centered at $a=0$
$f(x)=\sum _{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} x^n$

Common Maclaurin series

$1.\sin x=\sum _{n=0}^{\infty} \frac{(-1)^n}{(2 n+1) !} x^{2 n+1}$
$2.\cos x=\sum _{n=0}^{\infty} \frac{(-1)^n}{(2 n) !} x^{2 n}$
$3.e^x=\sum _{n=0}^{\infty} \frac{1}{n !} x^n$
$4.\ln (1+x)=\sum _{n=1}^{\infty} \frac{(-1)^{n+1}}{n} x^n$
$5.\frac{1}{1-x}=\sum _{n=0}^{\infty} x^n$
$6.\frac{1}{1+x}=\sum _{n=0}^{\infty}(-1)^n x^n$
$7.-\ln (1-x)=\sum _{n=1}^{\infty} \frac{x^n}{n}$
$8.\tan ^{-1} x=\sum _{n=0}^{\infty} \frac{(-1)^n}{2 n+1} x^{2 n+1}$
$9.\sinh x=\sum _{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !}$
$10.\cosh x=\sum _{n=0}^{\infty} \frac{x^{2 n}}{(2 n) !}$
$11.(1+x)^p=\sum _{n=0}^{\infty} \frac{p(p-1) \ldots(p-n+1)}{n !} x^n$

Series of real numbers

Definition

A series of real numbers is an expression of the form:
$a_1+a_2+a_3+\ldots=\sum _{n=1}^{\infty} a_n$
where ${a_n}$ is a sequence of real numbers

$\sum _{n=1}^{\infty} a_n = \lim _{n \rightarrow \infty} s_n$
$s_n = \sum _{k=1}^{n} a_k$

Convergence

A series $\sum _{n=1}^{\infty} a_n$ converges if $\lim _{n \rightarrow \infty} s_n$ exists

Necessary condition for convergence

$\lim _{n \rightarrow \infty} a_n = 0$

Convergence tests

Comparison test

If $\sum _{n=1}^{\infty} a_n$ and $\sum _{n=1}^{\infty} b_n$ are series with positive terms and $a_n \leq b_n$ for all $n$ then:

• If $\sum _{n=1}^{\infty} b_n$ converges then $\sum _{n=1}^{\infty} a_n$ converges
  
• If $\sum _{n=1}^{\infty} a_n$ diverges then $\sum _{n=1}^{\infty} b_n$ diverges
  

D’Alembert’s ratio test

If $\lim {n \rightarrow \infty} \frac{a{n+1}}{a_n}=L$ then:

• If $L<1$ then $\sum _{n=1}^{\infty} a_n$ converges
  
• If $L>1$ then $\sum _{n=1}^{\infty} a_n$ diverges
  

Cauchy’s root test

If $\lim _{n \rightarrow \infty} \sqrt[n]{|a_n|}=L$ then:

• If $L<1$ then $\sum _{n=1}^{\infty} a_n$ converges
  
• If $L>1$ then $\sum _{n=1}^{\infty} a_n$ diverges
  

Integral test

If $f$ is a continuous, positive, decreasing function on $[1, \infty)$ and $a_n=f(n)$ then:
$\int _{1}^{\infty} f(x) d x$ converges if and only if $\sum _{n=1}^{\infty} a_n$ converges

Alternating series

An alternating series is a series of the form:
$a_1-a_2+a_3-\ldots=\sum _{n=1}^{\infty} (-1)^{n+1} a_n$
where ${a_n}$ is a sequence of positive real numbers

Alternating series test (Leibniz’s test)

If ${a_n}$ is a sequence of positive real numbers such that:

• $a_{n+1} \leq a_n$ for all $n$
  
• $\lim _{n \rightarrow \infty} a_n = 0$
  

then: $\sum _{n=1}^{\infty} (-1)^{n+1} a_n$ converges

Dirichlet’s test

If ${a_n}$ and ${b_n}$ are sequences such that:

• ${a_n}$ is a decreasing sequence of positive real numbers
  
• $\lim _{n \rightarrow \infty} a_n = 0$
  

• $\sum _{n=1}^{\infty} b_n$ is bounded
  

  then: $\sum _{n=1}^{\infty} a_n b_n$ converges
  

Abel’s test

Suppose the sequence ${a_n}$ is monotonic and the series $\sum _{n=1}^{\infty} b_n$ converges. Then the series $\sum _{n=1}^{\infty} a_n b_n$ converges.

Absolute convergence

A series $\sum _{n=1}^{\infty} a_n$ converges absolutely if $\sum _{n=1}^{\infty} |a_n|$ converges

Theorem

Every absolutely convergent series converges

Conditional convergence

A series $\sum _{n=1}^{\infty} a_n$ converges conditionally if it converges but does not converge absolutely

Power series

$\sum _{n=0}^{\infty} a_n (x-x_0)^n$

Radius of convergence

$R=\frac{1}{\lim _{n \rightarrow \infty} \sqrt[n]{|a_n|}}$

Interval of convergence

$[a-R, a+R]$

Theorems to check for convergence

1. If $\lim _{n \rightarrow \infty} \sqrt[n]{

   a_n

   }=L$ then the radius of convergence is $R=\frac{1}{L}$

2. If $\lim _{n \rightarrow \infty} \frac{

   a_{n+1}

   }{

   a_n

   }=L$ then the radius of convergence is $R=\frac{1}{L}$

Differentiation and integration

A power series can be differentiated and integrated term by term.

$\frac{d}{d x} \sum _{n=0}^{\infty} a_n (x-x_0)^n=\sum _{n=1}^{\infty} n a_n (x-x_0)^{n-1}$

$\int \sum _{n=0}^{\infty} a_n (x-x_0)^n d x=\sum _{n=0}^{\infty} \frac{a_n}{n+1} (x-x_0)^{n+1}+C$

The radius of convergence of the differentiated or integrated series is the same as the original series.

Integrals

Definition

$\int f(x) d x=F(x)+C$

Integral by parts

$\int u \ dv=u v-\int v \ du$

Basic forms

$\int x^n d x=\frac{x^{n+1}}{n+1}+C$
$\int \frac{1}{ax + b} d x=\frac{1}{a} \ln |ax+b|+C$

Differential of an Integral

$\frac{\mathrm{d}}{\mathrm{d} \alpha} \int_{a(\alpha)}^{b(\alpha)} f(x, \alpha) \mathrm{d}x=f(b,\alpha) \frac{\mathrm{d} b}{\mathrm{d} \alpha} - f(a,\alpha) \frac{\mathrm{d} a}{\mathrm{d} \alpha} + \int_{a(\alpha)}^{b(\alpha)} \frac{\partial}{\partial{\alpha}} f(x, \alpha) \mathrm{d}x$

Common integrals

$1.\int x^n d x=\frac{x^{n+1}}{n+1}+C$
$2.\int \frac{1}{x} d x=\ln |x|+C$
$3.\int_{}^{} e^{ax} \mathrm{d}x=\frac{1}{a}e^{ax} + c$
$4.\int a^x d x=\frac{a^x}{\ln a}+C$
$5.\int \ln x d x=x \ln x-x+C$
$6.\int \sin x d x=-\cos x+C$
$7.\int \cos x d x=\sin x+C$
$8.\int \tan x d x=-\ln |\cos x|+C$
$9.\int \cot x d x=\ln |\sin x|+C$
$10.\int \sec x d x=\ln |\sec x+\tan x|+C$
$11.\int \sec ^2 x d x=\tan x+C$
$12.\int \csc ^2 x d x=-\cot x+C$
$13.\int \sec x \tan x d x=\sec x+C$
$14.\int \frac{1}{\sqrt{a^2-x^2}} d x=\arcsin \frac{x}{a}+C$
$15.\int \frac{1}{a^2+x^2} d x=\frac{1}{a} \arctan \frac{x}{a}+C$
$16.\int \frac{1}{x \sqrt{x^2-a^2}} d x=\frac{1}{a} \operatorname{arcsec} \frac{|x|}{a}+C$
$17.\int \frac{1}{\sqrt{x^2+a^2}} d x=\ln \left|x+\sqrt{x^2+a^2}\right|+C$
$18.\int \frac{1}{\sqrt{x^2-a^2}} d x=\ln \left|x+\sqrt{x^2-a^2}\right|+C$
$19.\int \frac{1}{a^2-x^2} d x=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+C$