4.6 Linearization

Local linearity

Near a point, a differentiable function looks like its tangent line (local linearity). The linearization is L(x) = f(a) + f'(a)(x − a), which approximates f(x) for x near a.

Using it

Linearization gives quick estimates of function values close to a. Example: for f(x) = √x at a = 4 (f(4) = 2, f'(4) = 1/4), L(x) = 2 + (1/4)(x − 4). The approximation is best when x is near a.

Key terms

Local linearity

Near a point, a curve resembles its tangent line.

Linearization

L(x) = f(a) + f'(a)(x − a).

Purpose

Approximate function values near a.

Accuracy

Best when x is close to a.

Example

√x near 4: L(x) = 2 + (1/4)(x − 4).

Try an AP-style question

Use the tangent line to f(x) = √x at x = 9 to approximate √9.2.

1. 3.01
2. 3.0667
3. 3.2
4. 3.0333