Linearization of a Function at a Point

Problem 167: Find the linearization of $f(x)$ at $x=a$ :

(1) $\begin{equation*} f(x) = x^3 - 2x + 3, \qquad a = 2\end{equation*}$

Solution: The linearization of a differentiable function at a point is given by

(2) $\begin{equation*}L(x) = f(a) + f'(a)(x-a). \end{equation*}$

Hence,

(3) $\begin{align*}f(a) & = f(2)= (2)^3 - 2(2) + 3 = 8 - 4 + 3 = 7, \ \text{and} \\ f'(a) & = f'(2) = 3(2)^2 - 2 =12 - 2 = 10. \end{align*}$

Therefore,

(4) $\begin{align*} L(x) & = f(a) + f'(a) (x - a) \\ & = 7 + 10(x-2) \\ & = 10x - 20 + 7 \\ & = 10x - 13.\end{align*}$

That is,

(5) $\begin{equation*} L(x) = 10x - 13. \end{equation*}$

One Problem at a Time