Physical Intuition
Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or over some other quantity) adds up to the net change in the quantity.
In the case of a particle traveling in a straight line, its position, x, is given by x(t) where t is time and x(t) means that x is a function of t. The derivative of this function is equal to the infinitesimal change in quantity, dx, per infinitesimal change in time, dt (of course, the derivative itself is dependent on time). This change in displacement per change in time is the velocity v of the particle. In Leibniz's notation:
Rearranging this equation, it follows that:
By the logic above, a change in x (or Δx) is the sum of the infinitesimal changes dx. It is also equal to the sum of the infinitesimal products of the derivative and time. This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative. It can be concluded that this operation works in reverse; the result of the integral can be differentiated to recover the original function.
Read more about this topic: Fundamental Theorem Of Calculus
Famous quotes containing the words physical and/or intuition:
“Good schools are schools for the development of the whole child. They seek to help children develop to their maximum their social powers and their intellectual powers, their emotional capacities, their physical powers.”
—James L. Hymes, Jr. (20th century)
“All appeared new, and strange at first, inexpressibly rare and delightful and beautiful. I was a little stranger, which at my entrance into the world was saluted and surrounded with innumerable joys. My knowledge was divine. I knew by intuition those things which since my Apostasy, I collected again by the highest reason.”
—Thomas Traherne (1636–1674)