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**Openstax**** **and has been adapted for the GPRC Learning Commons in September 2021. The graphs are generated using Desmos. This work is licensed under a Creative Commons BY 4.0 International License.

The secant line of a curve is:

"an intersecting line, especially one intersecting a curve at two or more points."

From <https://www.dictionary.com/browse/secant>

The tangent line of a curve is:

"a line or a plane that touches a curve or a surface at a point so that it is closer to the curve in the vicinity of the point than any other line or plane drawn through the point."

From <https://www.dictionary.com/browse/tangent>

The above graph shows the tangent line for point * (0.5,0.125)* for

The first step is to define a point near * (a,f(a))* and calculate the slope of the secant line through these two points. We use the line equation to find this slope by taking the ratio of the difference in

After finding the slope of our secant , we can use the line equation to substitute in for known values, ** m_{ac}**,

As we view points closer and closer to ** a**, the secant line trends closer and closer to the expected tangent line:

If we proceed beyond our point, the secant line trends away from the expected tangent line once again:

If we let the point (*x,f(x)*) approach the fixed point (*a,f(a)*), and the slope *m _{ac}* approaches a value

For some purposes, it is convenient to rewrite the formula as such:

- Last Updated: Dec 21, 2023 3:39 PM
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