Tangent line approximation

Definition

This is a good approximation when ${\displaystyle x}$ is close enough to ${\displaystyle a}$; since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of ${\displaystyle f}$ at ${\displaystyle (a,f(a))}$. For this reason, this process is also called the tangent line approximation. Linear approximations in this case are further improved when the second derivative of a, ${\displaystyle f''(a)}$, is sufficiently small (close to zero) (i.e., at or near an inflection point).