sensing the link between the tangent problem and the area problem

While we know that the fundamental theorem of calculus (FTC) establishes the link between differentiation and integration, what could have inspired Newton or  Leibniz to formulate the theorem?

It is said that even before the FTC was derived, mathematicians have already sensed a link between the tangent problem and the area problem, but what was this intuition that hovered behind the FTC?

Consider that the differentiation of displacement would give velocity, then summing the area of the velocity graph would ‘intuitively’ give the amount of distance traveled, ie. displacement. One can also consider that the accumulation of wealth over time would simply be the sum of the rate of accumulation at each point of time. (or the total amount of haze over time = the sum of PSI concentration at each point in time).

For some idea of the FTC: Let A(x) be the area obtained under the function f(t) starting from x=a.

A(x+h)-A(x) \approx f(t)h

\frac{A(x+h)-A(x)}{h} \approx f(t)

A'(x) = f(t) \blacksquare

Advertisements
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s