Change in variables

For any function f(x) . Over the interval of ${\displaystyle x}$ to ${\displaystyle x+\Delta x}$

Change in variable x

${\displaystyle \Delta x=(x+\Delta x)-x}$

Change in function f(x)

${\displaystyle \Delta f(x)=f(x+\Delta x)-f(x)}$

Rate of change

Rate of change

${\displaystyle {\frac {\Delta f(x)}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

Limit

${\displaystyle \lim _{x\to a}f(x)}$

Finite Limit We call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle c}$ if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently close (and not equal) to ${\displaystyle c}$ .

When this holds we write

${\displaystyle \lim _{x\to c}f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to c}$

Infinite Limit We call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches infinity if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently large.

When this holds we write

${\displaystyle \lim _{x\to \infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to \infty }$

Similarly, we call ${\displaystyle L}$ the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches negative infinity if ${\displaystyle f(x)}$ becomes arbitrarily close to ${\displaystyle L}$ whenever ${\displaystyle x}$ is sufficiently negative.

When this holds we write

${\displaystyle \lim _{x\to -\infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to -\infty }$

Differentiation

Let ${\displaystyle f(x)}$ be a function. Then

${\displaystyle {\frac {d}{dt}}f(t)=f'(x)=\sum \lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$ wherever this limit exists.

In this case we say that ${\displaystyle f}$ is differentiable at ${\displaystyle x}$ and its derivative at ${\displaystyle x}$ is ${\displaystyle f'(x)}$ .

Integration

Mathematics operation on a continuous function to find its area under graph . There are 2 types of integration

Indefinite Integral

${\displaystyle \int f(x)dx=F(x)+C}$

Where ${\displaystyle F}$ satisfies ${\displaystyle F'(x)=f(x)}$

Definite Integral

Suppose ${\displaystyle f}$ is a continuous function on ${\displaystyle [a,b]}$ and ${\displaystyle \Delta x={\frac {b-a}{n}}}$ . Then the definite integral of ${\displaystyle f}$ between ${\displaystyle a}$ and ${\displaystyle b}$ is

${\displaystyle \int \limits _{a}^{b}f(x)dx=\lim _{n\to \infty }A_{n}=\lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i}^{*})\Delta x}$

Where ${\displaystyle x_{i}^{*}}$ are any sample points in the interval ${\displaystyle [x_{i-1},x_{i}]}$ and ${\displaystyle x_{k}=a+k\cdot \Delta x}$ for ${\displaystyle k=0,\dots ,n}$ .}}

Solving differential equations

Given

${\displaystyle {\frac {d}{dt}}f(t)=-sf(t)}$

${\displaystyle {\frac {df(t)}{f(t)}}=-sdt}$

${\displaystyle \int {\frac {df(t)}{f(t)}}=\int -sdt}$

${\displaystyle Lnf(t)=-st+c}$

${\displaystyle f(t)=e^{-st+c}=Ae^{-st}}$

${\displaystyle A=e^{c}}$

Partial Differential Equations

${\displaystyle {\frac {\partial }{\partial t}}f(t)=-sf(t)}$

Integral Transformation

Any function f(t) can be transform into Laplace function or Fourier function by using Laplace transform or Fourier transform

${\displaystyle f(t)}$	${\displaystyle F(s)}$	${\displaystyle F(j\omega )}$
${\displaystyle f(t)}$	${\displaystyle \int f(t)e^{-st}dt}$	${\displaystyle \int f(t)e^{-j\omega t}dt}$
${\displaystyle {\frac {d}{dt}}}$	${\displaystyle s}$	${\displaystyle j\omega }$
${\displaystyle \int dt}$	${\displaystyle {\frac {1}{s}}}$	${\displaystyle {\frac {1}{j\omega }}}$

Example

${\displaystyle f(t)}$	${\displaystyle F(s)}$	${\displaystyle F(j\omega )}$
${\displaystyle L{\frac {d}{dt}}}$	${\displaystyle sL}$	${\displaystyle j\omega L}$
${\displaystyle {\frac {1}{L}}\int dt}$	${\displaystyle {\frac {1}{sL}}}$	${\displaystyle {\frac {1}{j\omega L}}}$