Linear Approximation

Formula

\[L(x) = f(a) + f'(a)(x - a)\]

Proof

\(m\) is the gradient of a line and recall the gradient-of-a-line is \({y_2-y_1} \over {x_2 - x_1}\)

\[{f'(x)} = {{y_2 - y_1} \over {x_2 - x_1}}\]

Writing \(f(x)\) in terms of a value at a sepific point \(f(a)\) gives us

\[{f'(a)} = {{y_2 - f(a)} \over {x_2 - a}}\]

Simplifying we get

\[{f'(a)(x_2 - a)} + f(a) = y_2\]

Writing \(y_2\) as a function of \(x_2\)

\[L(x) = {f'(a)(x - a)} + f(a)\]