Barrons AP Calculus

Figure  N4–19b

†Local linear approximation is also referred to as “local linearization” or even “best linear approximation”
(the latter because it is better than any other linear approximation).

L. TANGENT-LINE APPROXIMATIONS

If f ′(a) exists, then the local linear approximation of f (x) at a is

f   (a) + f ′(a)(x  −   a).

Since the equation of the tangent line to y = f (x) at x = a is

y   −   f   (a) =   f ′(a)(x    −   a),

we see that the y value on the tangent line is an approximation for the actual or
true value of f.

Tangent-line    approximation

Local linear approximation is therefore also called tangent-line approximation.†
For values of x close to a, we have

f   (x)  f  (a) + f ′(a)(x  −   a),

(1)