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Secant Lines, Tangent Lines, and Limit Definition of a Derivative

 (Note: this page is just a brief review of the ideas covered in Group. It is meant to serve as a summary only.)

 

A secant line is a straight line joining two points on a function. (See below.) It is also equivalent to the average rate of change, or simply the slope between two points.

The average rate of change of a function between two points and the slope between two points are the same thing.

Secant line = Average Rate of Change = Slope

 

A tangent line is a straight line that touches a function at only one point. (See above.) The tangent line represents the instantaneous rate of change of the function at that one point. The slope of the tangent line at a point on the function is equal to the derivative of the function at the same point (See below.)

 

Tangent Line = Instantaneous Rate of Change = Derivative

 

Let's see what happens as the two points used for the secant line get closer to one another. Let Dx represent the distant between the two points along the x-axis and determine the limit as Dx approaches zero.

 

As the two points used for the secant line get closer to one another, the average rate of change becomes the instantaneous rate of change and the secant line becomes the tangent line. (See animation above.)