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Calculus
“calculus of infinitesimals”, study of continous change
Differential calculus
study of rates ath which quantities change.
Derivative
Example of a straigth line $y=mx+b$, $m=\Delta y / \Delta x$
“Differentialquotient”:
$m = \lim_{x_1 -> x_0} \frac{f(x_1) - f(x_0)}{x_1 - x_0}$
$f'(x) = \frac{d}{dx} f(x) = \lim_{h -> 0} \frac{f(x + h) - f(x)}{h}$
$f''(x) = \frac{d^2}{dx^2} f(x)$
Geometrically, derivative of f at point x=a is slope of tangent line.
Differential equations
Relation between collection of functions and their derivatives.
Ordinary differential equation:
Differential operator
$y' = 2y + x^2$ ⇒ $f'(x)=2*f(x)+x^2$ ⇒ $\frac{dy}{dx} = 2y + x^2$
$\frac{dy}{dx} = y^2 * x$
$dy = y^2 * x {dx}$
$\frac{1}{y^2} dy = x {dx}$
$\int \frac{1}{y^2} dy = \int x {dx}$