Fundamental Theorem of Calculus and Initial Value Problems
Fundamental Theorem of Calculus and Initial Value Problems
This Demonstration helps to visualize the fundamental theorem of calculus. It includes the animation of a particle's motion on the axis and a plot of its height as a function of time, which is the solution to the initial value problem with differential equation =v(t) and initial condition . You can change the particle's initial position and its continuous velocity function . As the animation runs, Mathematica graphs +v(τ)dτ on the right and the accumulated area that is colored on the left; when the animation stops, the new graph shows a function that solves the corresponding initial value problem.
y
ds
dt
s(0)=
s
0
s
0
v(t)
s
0
t
∫
0