GNU General Public License v3.0 licensed. Source available on github.com/zifeo/EPFL.
Fall 2013: Analysis I
[TOC]
- subsets
- sets operations
- vertical line test
- properties
- even(symetric y-axis)/odd(symetric origine)
- increasing/decreasing/monotone(in. or de.)
- even(symetric y-axis)/odd(symetric origine)
- important functions
- constant/linear
- polynomial
- power
- rational
- exponential/logarithm
- trigonometric : sin, cos, tan, csc, sec, cot
- constant/linear
- 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
$f(x+d)$
- reflect about y-axis :
- strech vertically :
- 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
$\sin 2x=2\sin x\cos x$ -
$\cos 2x=1-2\sin^2 x$ $x^3+y^3=(x+y)(x^2-xy+y^2)$ $\sqrt{b}-\sqrt{a}=\frac{b-a}{\sqrt{b}+\sqrt{a}}$ $\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots+\frac{1}{2^n}=1-\frac{1}{2^n}$ $\tan^2 x +1=\frac{1}{\cos^2 x}$ $\cot^2 x +1=\frac{1}{\sin^2 x}$ $\sin(a+b)=\sin a\cos b+\cos a\sin b$ $\cos(a+b)=\cos a\cos b-\sin a\sin b$
- convergence
- properties
- increasing/decreasing/monotone(in. or de.)
A limit exists iff both sided limits exist.
- limits rules
- squeeze theorem
- bounds
- sequence definition of limits <=> eplison-delta defintion
- infinite limits
$e^x = \lim_{n\to \infty} (1+\frac{x}{n})^n$
- limits laws for functions
Continus if the limit exists at each point.
-
intermediate value theorem : suppose
$f$ continuous on$[a,b]$ and$f(a)\not =f(b)$ , 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
- removable discontinuities (one point displaced, sided limits equivalent)
-
asymptotes
- vertical
$x \to a, f(x) \to \infty$ - horizontal
$x \to \infty, f(x) \to a$ -
$\frac{ax-b}{x+c}$ h-asy at$a$ , v-asy at$-c$ , x-intercept at$x=\frac{b}{a}$ - slope of the obl-asy
$=\lim_{x\to\infty} \frac{f(x)}{x}$
-
- vertical
-
tagent line
-
differentiable : differentiale except if there is a corner, a discontinuity or a vertical tangent.
-
rules of differentiation
-
$\tan'x=\sec^2x$ $\csc'x = -\csc x\cot x$ $\sec'x = \sec x\tan x$ $\cot'x = -\csc^2x$ $\sin'^{-1}x = \frac{1}{\sqrt{1-x^2}}$ $\cos'^{-1}x = -\frac{1}{\sqrt{1-x^2}}$ $\tan'^{-1}x = \frac{1}{1+x^2}$ $\csc'^{-1}x = -\frac{1}{x\sqrt{x^2-1}}$ $\sec'^{-1}x = \frac{1}{x\sqrt{x^2-1}}$ $\cot'^{-1}x = -\frac{1}{1+x^2}$ $\ln'g(x) = \frac{g'(x)}{g(x)}$ $\log_a'x = \frac{1}{x\ln a}$ - chain rule
-
-
derivative of inverse function :
$(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}$ ou$(f^{-1})'(f(a))=\frac{1}{f'(a)}$ . -
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$f'(c)$ exists, then$f'(c)=0$ . -
critical numbers/points : a critcal number of a function
$f$ is a number$c$ in the domain of$f$ such that either$f'(c)=0$ 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$f'(c)=0$ . -
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$f'(c)=\frac{f(b)-f(a)}{b-a}$ . -
function increasing/decreasing
-
1st derivative test :
- if
$f'$ changes from postive to negative at$c$ , it is a local maximum- if
$f'$ changes from negative to positive at$c$ , it is a local maximum - if
$f'$ does not change sign at$c$ , nothing - if
$\forall x > c, f'(x)>0$ and$\forall x < c, f'(x)<0$ , then$f(c)$ is the absolute maximum - if
$\forall x > c, f'(x)<0$ and$\forall x < c, f'(x)>0$ , then$f(c)$ is the absolute minimum
- if
- if
-
function bendings (inflexion points)
-
concavity test :
- if
$\forall x \in I, f''(x) > 0$ , the function is concave upward on I :)- if
$\forall x \in I, f''(x) < 0$ , the function is concave downward on I :(
- if
- if
-
Cauchy's mean value theorem : let
$f$ ,$g$ be continuous on$[a,b]$ and differentiable in$(a,b)$ , then$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$ . -
De l'Hospital's rule : suppose
$f$ and$g$ are differentiable and$g'(x)\not = 0$ on an open interval$I$ that contains$a$ . If the limit$f$ and the limit$g$ go both to$0$ or$\infty$ as$x\to a$ , then$\lim_{x\to a} \frac{f'(x)}{g'(x)}$ .
-
arithmetic series : converge to
$\sum_{n=1}^\infty n=\frac{n(n+1)}{2}$ -
geometric series : converge to
$\sum_{n=1}^\infty ar^{n-1}=\frac{a}{1-r}$ if$|r|<1$ -
Riemann series :
$\sum_{n=1}^\infty \frac{1}{p^\alpha}$ 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
$\lim_{n\to \infty} \frac{a_n}{b_n}=c$ where$c$ is a finite number bigger than$0$ , then either both converge or diverge.
- limit test : (both positive) if
- comparison test
- alternating series test : if for all postive the next is smaller than the previous one and
$\lim_{n\to \infty} b_n=0$ , then the series converge.
- alternating series test : if for all postive the next is smaller than the previous one and
- ratio test :
$\lim_{n\to \infty} |\frac{next}{previous}|$ gives less than$1$ , it is convergent, divergent if bigger than$1$ , else inconclusive.- root test :
$\lim_{n\to \infty} \sqrt[n]{|a_n|}$ , same conclusion as ratio test. - integral test : for continuous, positive, deacreasing function on
$[1,\infty)$ , the series is convergent iff$\int_1^\infty f(x)dx$ is convergent.
- root test :
- divergence test : if
-
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 :
$\sum_{n=0}^\infty c_n(x-a)^n$ 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
- radius of convergence : series could converge only when
-
Taylor series (called Mclaurin series if centered at
$0$ ) :$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$ -
analyctic functions
-
$\frac{1}{1-ax}=\sum_{n=0}^\infty (ax)^n=1+ax+ax^2+ax^3+\cdots$ $\frac{1}{1+x}=\sum_{n=0}^\infty (-x)^n=1-x+x^2-x^3+\cdots$ $\frac{1}{(1-x)^2}=\sum_{n=0}^\infty (n+1)x^n=1+2x+3x^2+4x^3+\cdots$ $\sin x=\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$ $\cos x=\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$ $\tan^{-1} x=\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots$ $ln(1+x)=\sum_{n=1}^\infty (-1)^n \frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$
-
- Riemann sum :
$\lim_{x\to \infty} \sum_0^n f(x_i^*)\Delta x$ - 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$g'(x)=f(x)$ . if$f$ in continous on$I$ , then$\int_a^b f(x)dx = F(b)-F(a)$ .