Calculus

Break the complicated problem (shape) into small pieces (slices), solve them separately and put them back together (Azad, 2015).
- Azad’s Site
- slope measures rate of change
- are measures accumulation of change

division (average step) -> $y/x$ -> split the whole into equal parts
differentiation (actual step) -> $\frac{d}{dx} y$ -> split the whole into possibly different parts
multiplication -> $z * x$ -> accumulate average steps.
integration -> $\int z * dx$ -> accumulate possibly different steps
- $\int_a^b step(x)\ \mathrm{d}x$, where $x$ is in the range of interpretation from $a$ to $b$.
- $\int_a^b step(x)\ \mathrm{d}x$ = $Original(b) - Original(a)$ where the former is $f(x)$ and the latter $F(x)$

derivatives with respect to radius -> rings : $\frac{d}{dr} Area$ -> $\int_0^r 2\pi r\ \mathrm{d}r$
derivatives with respect to perimeter -> pizza slice : $\frac{d}{dp} Area$ -> $\int_{p_{min}}^{p_{max}} (pizza\ slice)\ \mathrm{d}p$
derivatives with respect to x-axis -> boards : $\frac{d}{dx} Area$ -> $\int_{x_{min}}^{x_{max}} (board)\ \mathrm{d}x$

Integrals ore often written as unspecified slices, and only then we worry about the exact slice formula.
Often, when there is only 1 variable, it is skipped: $\int_0^r 2\pi r\ \mathrm{d}r$ is the same as $\int_0^r 2\pi r$
$\mathrm{d}r$ implies moving along $r$.
- $\int_0^r 2\pi r\ \mathrm{d}r$ -> in the $2\pi r$ part $r$ changes with each step, ranging from $r_{min}$ to $r_{max}$
$f`(x)$, $\dot f$, $\frac{d}{dx}f$, and $\frac{\mathrm{d}f}{\mathrm{d}x}$ are all the same. For the first three think about step by step patterns, for the last one think about actual ratio of change

Algebra	Calculus
$speed = distance / time$	$\frac{d}{dt} distance$
$distance = speed * time$	$\int speed\ \mathrm{d}t$
$area = height * width$	$\int height\ \mathrm{d}w$
$weight = density * length * width * height$	$\iiint density\ \mathrm{d}l\ \mathrm{d}w\ \mathrm{d}h$

Algebra only	logic	Algebra + Calculus	logic
$Area\ of\ square = ?$	Area is unknown	$Area\ of\ circle = ?$	Area is unknown
$\sqrt{area} = 13.3$	but I now the sqrt	$\frac{d}{dr} Area = 2\pi r$	but I know it splits into rings
$(\sqrt{area})^2 = (13.3)^2$	square both sides	$\int \frac{d}{dr} Area = \int 2\pi r$	Integrate both sides
$Area\ of\ square = 176.89$	and receive the area	$Area = \pi\ r^2$	and receive the area

$\int \frac{d}{dr} Area = \int 2\pi r$ is short for $\int_0^r (\frac{d}{dr} Area)\ \mathrm{d}r = \int_0^r 2\pi r\ \mathrm{d}r$

derivative simple example 1
$f(x) = 4x$, derivative of a pattern: $\frac{d}{dx}f(x)$ => find the derivative!

current output : $f(1) = 4$
step forward by $\mathrm{d}x$ (here 1 step) and find the new amount => $f(x + \mathrm{d}x) = f(1+1) = f(2) = 8$
change in output $\mathrm{d}f = f(x + \mathrm{d}x) - f(x)$ : $4(x+ \mathrm{d}x) - 4(x) = 4\mathrm{d}x$
=> increasing the length by $\mathrm{d}x$ increases the cost by $4\mathrm{d}x$ (e.g. total change)
find $\frac{\mathrm{d}f}{\mathrm{d}x}$ ratio of change in output ot change in input : $\frac{\mathrm{d}f}{\mathrm{d}x} = \frac{4dx}{dx} = 4$ (e.g. cost per step, actual ratio of change)

derivative simple example 2
derivative of a pattern: $\frac{d}{dx}x^2$ => find the derivative!

current output : $f(x) = x^2$
step forward by $\mathrm{d}x$ => $f(x + \mathrm{d}x) = (x + \mathrm{d}x)^2 = x^2 + 2x\ \mathrm{d}x + \mathrm{d}x^2$
find change in output $\mathrm{d}f = f(x + \mathrm{d}x) - f(x) = 2x\ \mathrm{d}x + \mathrm{d}x^2$
find the ratio $\frac{\mathrm{d}f}{\mathrm{d}x} = \frac{2x\ \mathrm{d}x + \mathrm{d}x^2}{\mathrm{d}x} = 2x + \mathrm{d}x$
throw away the measurement artifact ($\mathrm{d}x = 0$) => result is $2x$

integral simple example
$2+2+2=6$, which has 3 steps 0-1, 1-2, 2-3 => $\int_0^3 2\ \mathrm{d}x= 6$

References

Azad, K. (2015). Calculus, better explained. https://betterexplained.com/calculus/