math101.md

MATH101

GNU General Public License v3.0 licensed. Source available on github.com/zifeo/EPFL.

Fall 2013: Analysis I

[TOC]

Sets

• subsets
• sets operations

Functions

• vertical line test
• properties
  • even(symetric y-axis)/odd(symetric origine)
    • increasing/decreasing/monotone(in. or de.)
• important functions
  • constant/linear
    • polynomial
    • power
    • rational
    • exponential/logarithm
    • trigonometric : sin, cos, tan, csc, sec, cot
• linear transformation of functions
  • strech vertically : $cf(x)$
    • reflect about y-axis : $-f(x)$
    • shift vertically $f(x)+d$
    • shrink horizontally $f(cx)$
    • reflect about x-axis $f(-x)$
    • shift horizontally $f(x+d)$
• inverse function
  • one-to-one : never takes twice the same value
  • onto : every vale in the codomain is hit at least once
  • bijective : both
• piecewise defined function
• composite function
• some identities
  • $\sin 2x=2\sin x\cos x$
  • $\cos 2x=1-2{\sin }^{2}x$
    • $x^3+y^3=(x+y)(x^2-xy+y^2)$
    • $\sqrt{b}-\sqrt{a}=\frac{b-a}{\sqrt{b}+\sqrt{a}}$
    • $\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots +\frac{1}{2^n}=1-\frac{1}{2^n}$
    • ${\tan }^{2}x+1=\frac{1}{{\cos }^{2}x}$
    • ${\cot }^{2}x+1=\frac{1}{{\sin }^{2}x}$
    • $\sin (a+b)=\sin a\cos b+\cos a\sin b$
    • $\cos (a+b)=\cos a\cos b-\sin a\sin b$

Sequences

• convergence
• properties
  • increasing/decreasing/monotone(in. or de.)

Limits

A limit exists iff both sided limits exist.

• limits rules
• squeeze theorem
• bounds
• sequence definition of limits <=> eplison-delta defintion
• infinite limits
  • $e^x=\underset{n\to \mathrm{\infty }}{lim}(1+\frac{x}{n}{)}^n$
• limits laws for functions

Continuity

Continus if the limit exists at each point.

• intermediate value theorem : suppose $f$ continuous on $[a,b]$ and $f(a)\ne f(b)$, let $N$ be a number between $f(a)$ and $f(b)$, then there exist a $c$ such that $f(c)=N$.

• continuity of an inverse function

• discontinuities

  • removable discontinuities (one point displaced, sided limits equivalent)
    • jump discontinuities (sided limits differ)
    • infinite discontinuities (one sided limit does not exist)
    • be careful of absolute value when simplification
    • simplification does not change the discontinuities

• asymptotes

  • vertical $x\to a,f(x)\to \mathrm{\infty }$
  • horizontal $x\to \mathrm{\infty },f(x)\to a$
    • $\frac{ax-b}{x+c}$ h-asy at $a$, v-asy at $-c$, x-intercept at $x=\frac{b}{a}$
    • slope of the obl-asy $=\underset{x\to \mathrm{\infty }}{lim}\frac{f(x)}{x}$

Derivatives

$f^{\prime }(x)=\underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}$ ou

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$

• tagent line

• differentiable : differentiale except if there is a corner, a discontinuity or a vertical tangent.

• rules of differentiation

  • ${\tan }^{\prime }x={\sec }^{2}x$
    • ${\csc }^{\prime }x=-\csc x\cot x$
    • ${\sec }^{\prime }x=\sec x\tan x$
    • ${\cot }^{\prime }x=-{\csc }^{2}x$
    • ${\sin }^{\prime -1}x=\frac{1}{\sqrt{1-x^2}}$
    • ${\cos }^{\prime -1}x=-\frac{1}{\sqrt{1-x^2}}$
    • ${\tan }^{\prime -1}x=\frac{1}{1+x^2}$
    • ${\csc }^{\prime -1}x=-\frac{1}{x\sqrt{x^2-1}}$
    • ${\sec }^{\prime -1}x=\frac{1}{x\sqrt{x^2-1}}$
    • ${\cot }^{\prime -1}x=-\frac{1}{1+x^2}$
    • ${\ln }^{\prime }g(x)=\frac{g^{\prime }(x)}{g(x)}$
    • ${\log }_a^{\prime }x=\frac{1}{x\ln a}$
    • chain rule

• derivative of inverse function : $(f^{-1}{)}^{\prime }(a)=\frac{1}{f^{\prime }(f^{-1}(a))}$ ou $(f^{-1}{)}^{\prime }(f(a))=\frac{1}{f^{\prime }(a)}$.

• extreme value theorem : if $f$ continuous on $[a,b]$, then $f$ attains an absolute maximum and an absolute minimum on some number in that interval.

• Bolzano-Weierstress theorem : every bounded sequence has a convergent subsequence.

• Fermat's theorem : if $f$ has a local extremum at $c$ and $f^{\prime }(c)$ exists, then $f^{\prime }(c)=0$.

• critical numbers/points : a critcal number of a function $f$ is a number $c$ in the domain of $f$ such that either $f^{\prime }(c)=0$ or $f(c)$ does not exist. Can also be a vertical tangent but it has to be defined in the domain.

• local maximum/minimum near a point

• absolute/global maximum/minimum

• closed interval method : find the values of $f$ at the critical numbers on $[a,b]$ and at the endpoints of the interval, then the largest is the maximum and vice versa.

• Rolle's theorem : if $f$ continuous on $[a,b]$ and differentiable on $(a,b)$ and $f(a)=f(b)$, then there is a number $c$ in $(a,b)$ such that $f^{\prime }(c)=0$.

• mean value theorem : if $f$ continuous on $[a,b]$ and differentiable on $(a,b)$, then there is a number $c$ in $(a,b)$ such that $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$.

• function increasing/decreasing

• 1st derivative test :

  • if $f^{\prime }$ changes from postive to negative at $c$, it is a local maximum
    • if $f^{\prime }$ changes from negative to positive at $c$, it is a local maximum
    • if $f^{\prime }$ does not change sign at $c$, nothing
    • if $\mathrm{\forall }x>c,f^{\prime }(x)>0$ and $\mathrm{\forall }x<c,f^{\prime }(x)<0$, then $f(c)$ is the absolute maximum
    • if $\mathrm{\forall }x>c,f^{\prime }(x)<0$ and $\mathrm{\forall }x<c,f^{\prime }(x)>0$, then $f(c)$ is the absolute minimum

• function bendings (inflexion points)

• concavity test :

  • if $\mathrm{\forall }x\in I,f^{″}(x)>0$, the function is concave upward on I :)
    • if $\mathrm{\forall }x\in I,f^{″}(x)<0$, the function is concave downward on I :(

• Cauchy's mean value theorem : let $f$,$g$ be continuous on $[a,b]$ and differentiable in $(a,b)$, then $\frac{f^{\prime }(c)}{g^{\prime }(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$.

• De l'Hospital's rule : suppose $f$ and $g$ are differentiable and $g^{\prime }(x)\ne 0$ on an open interval $I$ that contains $a$. If the limit $f$ and the limit $g$ go both to $0$ or $\mathrm{\infty }$ as $x\to a$, then $\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$.

Series

• arithmetic series : converge to $\sum _{n=1}^{\mathrm{\infty }}n=\frac{n(n+1)}{2}$

• geometric series : converge to $\sum _{n=1}^{\mathrm{\infty }}a{r}^{n-1}=\frac{a}{1-r}$ if $|r|<1$

• Riemann series : $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{p^{\alpha }}$ converges if $\alpha >1$

• laws of sequences

• sequences and series : if the series $\sum a_n$ is convergent, then $lim{a}_n=0$.

• find the terms : $S_n-S_o=a_n$ where $S_o$ is the one before $S_n$.

• convergences tests

  • divergence test : if $lim{a}_n$ does not exist or is not equal to $0$, then the series is divergent.
    • limit test : (both positive) if $\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=c$ where $c$ is a finite number bigger than $0$, then either both converge or diverge.
  • comparison test
    • alternating series test : if for all postive the next is smaller than the previous one and $\underset{n\to \mathrm{\infty }}{lim}{b}_n=0$, then the series converge.
  • ratio test : $\underset{n\to \mathrm{\infty }}{lim}|\frac{next}{previous}|$ gives less than $1$, it is convergent, divergent if bigger than $1$, else inconclusive.
    • root test : $\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{|a_n|}$, same conclusion as ratio test.
    • integral test : for continuous, positive, deacreasing function on $[1,\mathrm{\infty })$, the series is convergent iff ${\int }_1^{\mathrm{\infty }}f(x)dx$ is convergent.

• alternating series

• absolute convergence (conditionnaly) : a series $\sum a_n$ is called absolutely convergent if $\sum |a_n|$ converges, it is conditionnaly convergent.

• power series : $\sum _{n=0}^{\mathrm{\infty }}{c}_n(x-a{)}^n$ is called centered at $a$.

  • radius of convergence : series could converge only when $x=a$, for all $x$ or for a positive number $R$ such that the series converges if $|x-a|<R$. Apply ratio test to find it.
    • infinite many times differentiable
    • derivative/integration

• Taylor series (called Mclaurin series if centered at $0$) : $\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n$

• analyctic functions

  • $\frac{1}{1-ax}=\sum _{n=0}^{\mathrm{\infty }}(ax{)}^n=1+ax+a{x}^2+a{x}^3+\cdots$
    • $\frac{1}{1+x}=\sum _{n=0}^{\mathrm{\infty }}(-x{)}^n=1-x+x^2-x^3+\cdots$
    • $\frac{1}{(1-x{)}^2}=\sum _{n=0}^{\mathrm{\infty }}(n+1)x^n=1+2x+3x^2+4x^3+\cdots$
    • $\sin x=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$
    • $\cos x=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$
    • ${\tan }^{-1}x=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{x^{2n+1}}{2n+1}=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots$
    • $ln(1+x)=\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$

Integral

• Riemann sum : $\underset{x\to \mathrm{\infty }}{lim}\sum _0^{n}f(x_i^{\ast })\mathrm{\Delta }x$
• rules of integration
  • $\int \frac{1}{x}dx=\log x$
  • $\int \frac{1}{x\log x}dx=\log (\log x)$
  • $\int \frac{1}{x{\log }^{2}x}dx=-\frac{1}{\log x}$
• fundamental theorem of calculus : if $f$ is continuous on $I$ then the function $g$ is continuous, differentiable and definded by $g(x)={\int }_a^{x}f(t)dt$ and $g^{\prime }(x)=f(x)$. if $f$ in continous on $I$, then ${\int }_a^{b}f(x)dx=F(b)-F(a)$.