5: Tangent lines, Linear Approximation, and Newton’s Method
- Page ID
- 121105
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., \(x_{0}\). Given \(x_{0}, y_{0}=f\left(x_{0}\right)\), and the slope \(m=\) \(f^{\prime}\left(x_{0}\right)\) (the derivative), we can find the equation of the tangent line
\[\frac{\text { rise }}{\text { run }}=\frac{y-y_{0}}{x-x_{0}}=m=f^{\prime}\left(x_{0}\right) \nonumber \]
\[\Rightarrow \quad y=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right) . \label{eq1}\]
(See Appendix A for a review of straight lines.)
We use Equation \ref{eq1} 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 \(f(x)=0\) ).
- 5.3: Approximating a Function by its Tangent Line
- We have seen that the tangent line approximates the local behavior of a function, at least close enough to the point of tangency. Here we use this idea in a formal procedure called linear approximation. The idea is to chose a point (often called the base point) where the value of the function and its derivative are known, or are easy to calculate, and use the tangent line at that point to estimate values of the function in the vicinity.
- 5.4: Tangent Lines for Finding Zeros of a Function - Newton’s Method
- In many cases, it is impossible to compute a value of a zero, x∗ analytically. Based on tangent line approximations, we now explore Newton’s method, an approximation that does the job.