Barrons AP Calculus - David Bock

FIGURE N4–19a

FIGURE N4–19b

L. TANGENT-LINE APPROXIMATIONS

Local linear approximation

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