The Relationship of the Tangent to a Curve and the Area under a Curve

Date: 10/25/95 at 13:26:49
From: "betty.thompson"
Subject:       connection of tangent and finding area

Lindsey Jahn is trying to find the answer to this problem.

The problem of drawing a tangent to a given curve at a given point is 
closely connected with the problem of finding areas ("areas under a 
curve").  

        a. What is this connection?

        b. When was it discovered and by whom?

Betty Thompson

Date: 10/26/95 at 21:55:10
From: Doctor Jonathan
Subject: Re: connection of tangent and finding area

Lindsey:

The tangent of a given curve at a point is the same thing as the slope 
of the curve at that point.  The slope of a curve is also called the 
derivative of the curve (at any given point).  On the other hand, the 
area under a curve is found via the operation of integration.  The 
connection you may be looking for is the fact that the integral is 
exactly the opposite of the derivative.  In fact, the integral is often 
referred to as the anti-derivative. So, the tangent is given by the 
derivative, and the area is given by the anti-derivative.

All of this is basically just calculus, so the answer to your second
question can probably best be answered as "When was calculus 
invented and by who?" Calculus was invented independently by 
Newton and Leibniz around 1666.

-Doctor Jonathan,  The Geometry Forum