Tangent Line Approximation

The Tangent Line Approximation

For any function $f(x)$, if the observed section of the graph is sufficiently localized, it will look like a graph of a linear function. Therefore, the equation of the "linear" function around the point $\left(a,\ f\left(a\right)\right)$ will be

${\begin{matrix}y\ -\ f\left(a\right)&=&f^{\prime }\left(a\right)\left(x\ -\ a\right)\\y&=&f\left(a\right)\ +\ f^{\prime }\left(a\right)\left(x-a\right)\end{matrix}}$

The expression $f\left(a\right)\ +\ f^{\prime }\left(a\right)\left(x-a\right)$ is called the local linearization of f near x = a. Thus, the error E(x) in the approximation is

$E\left(x\right)\ =\ f\left(x\right)\ -\ \left\{f\left(a\right)\ +\ f^{\prime }\left(a\right)\left(x-a\right)\right\}$

Error of Approximation

When a graph is viewed in a increasingly localized manner, the graph will eventually become indistinguishable from a linear function. However, since this phenomenon occurs locally, and not at a point, the ratio of E(x) to (x-a) must be extremely small. To prove this, we will take the limit as x approaches a of the ratio.

${\begin{matrix}\lim _{x\rightarrow a}{\frac {E\left(x\right)}{x-a}}&=&\lim _{x\rightarrow a}{\frac {f\left(x\right)\ -\ \left\{f\left(a\right)\ +\ f^{\prime }\left(a\right)\left(x-a\right)\right\}}{x-a}}\\\\&=&\lim _{x\rightarrow a}{\frac {f\left(x\right)\ -\ f\left(a\right)}{x-a}}\ -\ \lim _{x\rightarrow a}{\frac {f^{\prime }\left(a\right)\left(x\ -\ a\right)}{x-a}}\\\\&=&\lim _{h\rightarrow 0}{\frac {f\left(a\ +\ h\right)\ -\ f\left(a\right)}{a\ +\ h\ -\ a}}\ -\ \lim _{x\rightarrow a}f^{\prime }\left(a\right)\\\\&=&\lim _{h\rightarrow 0}{\frac {f\left(a\ +\ h\right)\ -\ f\left(a\right)}{h}}\ -\ f^{\prime }\left(a\right)\\\\&=&f^{\prime }\left(a\right)\ -\ f^{\prime }\left(a\right)\\\\&=&0\end{matrix}}$

See also

• Tangent Plane Approximation
• Taylor's series