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Mathematics LibreTexts

5: Tangent lines, Linear Approximation, and Newton’s Method

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In Chapter 3 , we defined the tangent line as the line we see when we zoom into the graph of a (continuous) function, y=f(x), at some point. In much the same sense, the tangent line approximates the local behavior of a function near the point of tangency., x0. Given x0,y0=f(x0), and the slope m= f(x0) (the derivative), we can find the equation of the tangent line

 rise  run =yy0xx0=m=f(x0)

y=f(x0)+f(x0)(xx0).

(See Appendix A for a review of straight lines.)

We use Equation ??? in several applications, including linear approximation, a method for estimating the value of a function near the point of tangency. A further application of the tangent line is Newton’s method which locates zeros of a function (values of x for which f(x)=0 ).


This page titled 5: Tangent lines, Linear Approximation, and Newton’s Method is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Leah Edelstein-Keshet via source content that was edited to the style and standards of the LibreTexts platform.

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