5.7: Summary

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The equation of a tangent line to $f(x)$ at $x_{0}$ is given by \[y=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right) . \nonumber \]
If $L(x)$ is the tangent line to a function $f(x)$ at $x_{0}$, then $L(x)$ forms a linear approximation to $f(x)$ near the point $x_{0}$.
In some circumstances, the zero of a tangent line to a function $f(x)$ at a point $x_{0}$ can form an initial approximation to the zero of $f(x)$.
Newton’s method is based on the property of tangent lines. Newton’s method can solve a problem of the form $f(x)=0$. Given an initial guess $x_{0}$, the method generates successive decimal approximations to the zeros of the function to any desired accuracy. The iteration scheme is: \[x_{k+1}=x_{k}-\frac{f\left(x_{k}\right)}{f^{\prime}\left(x_{k}\right)} \text {. } \nonumber \]

Is it possible for two different tangent lines of the same function to be parallel?
When would a tangent line not intersect the $x$-axis?
Consider the graph of the following function, and its tangent line at $x=1$.

$clipboard_ecc3096bfc816273def8289505cf37f79.png$

When would the linear (tangent line) approximation result in an overestimate? Under-estimate?
What is a reasonable interval on which to use this tangent line for approximation?

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