# Calculus as dx = f(t)dt as x = integral f(t) dt

Watch and play with this code. containing the essence of Calculus in few lines.

The Calculus of Newton/Leibniz opened to the Industrial Revolution of the late 17th century forever changing human conditions in the form of mathematics of change as the initial value problem of finding a function $x(t)$ of the (time) variable $t$ which solves the initial value problem

• $\frac{dx}{dt} = f(t)$ for $t>0$
• $x(0)=x_0$

where $f(t)$ is a given function, $x_0$ a given initial value and $dt$ a given time step.

Computing $x(t)$ by time stepping in the form

• $x(t+dt) = x(t) + f(t)\, dt$,

successively computes $x(t+dt)$ at a next time $t+dt$ from $x(t)$ at former time t plus the change of $f(t)dt$,  which can be captured by summation over all time steps to a final time $T=Ndt$ after N steps:

• $x(T) =\sum_{n=1}^{N-1}f(ndt)\, dt + x_0$.

To capture this summation formula Leibniz introduced the notation

•  $x(T)=\int_0^Tf(t)\, dt + x_0$

where the integral sign $\int$ represents the summation with t as integration variable and dt as time step.

We can now express the Fundamental Theorem of Calculus as:

1. Time stepping of $\frac{dx}{dt} = f(t)$ constructs the integral $x=\int f(t)\, dt$ of the integrand $f(t)$.
2. Differentiation of the integral $x=\int f(t)\, dt$ produces the integrand $f(t)$.

Here the constructive step is the time stepping 1 with $f(t)$ given, captured as summation by repetition,  while 2 expresses the starting point as the initial value problem with $dx =f(t)dt$ or $\frac{dx}{dt}=f(t)$.

Extension to $f(x,t)$ depending also on x direct.

A basic case is $f(x,t)=x$ for which the integral is $x =\exp(t)$ with the remarkable property of being equal to its derivative as shown here.

To do:

• Study experimentally by computing the dependence of $\int_0^T f(t)\, dt$ on the time step $dt$.