FIGURE N4–20
where f (a) + f ′(a)(x − a) is the linear or tangent-line approximation for f (x), and f ′(a)(x − a) is the
approximate change in f as we move along the curve from a to x. See Figure N4–20.
In general, the closer x is to a, the better the approximation is to f (x).
EXAMPLE 30
Find tangent-line approximations for each of the following functions at the values indicated:
(a) sin x at a = 0 (b) cos x at a =
(c) 2x^3 − 3x at a = 1 (d) at a = 8
SOLUTIONS:
(a) At a = 0, sin x sin (0) + cos (0)(x − 0) 0 + 1 · x x
(b)
(c) At a = 1, 2x^3 − 3x − 1 + 3(x − 1) 3 x − 4
(d)
† 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).
EXAMPLE 31
Using the tangent lines obtained in Example 30 and a calculator, we evaluate each function, then
its linear approximation, at the indicated x-values:Example 31 shows how small the errors can be when tangent lines are used for approximations
and x is near a.