3.9 | Linear Approximation and the Derivative


Suppose \(f\) is differentiable at \(a\). Then, for values of \(x\) near \(a\), the tangent line approximation of \(f(x)\) is
$$
f(x) \approx f(a) + f'(a)(x-a).
$$
The expression \(f(a) + f'(a)(x-a)\) is called the local linearization of \(f\) near \(x = a\). We are thinking of \(a\) as fixed, so that \(f(a)\) and \(f'(a)\) are constant.
The error, \(E(x)\), in the approximation is defined by
$$
E(x) = f(x) – f(a) – f'(a)(x-a).
$$
Suppose \(f\) is differentiable at \(x = a\) and \(E(x)\) is the error in the tangent line approximation, that is:
$$
E(x) = f(x) – f(a) -f'(a)(x-a).
$$
Then
$$
\lim_{x \rightarrow a} \frac{E(x)}{x-a} = 0.
$$

E 3.9 Exercises