4.6 Approximating Values of a Function Using Local Linearity and Linearization

Local linearity

Near a point where a function is differentiable, its graph looks almost like a straight line when you zoom in enough. This means the tangent line at that point gives a good approximation of the function's values for x close to that point — the closer x is, the better the approximation.

Linearization formula

The linearization of f at x = a is L(x) = f(a) + f'(a)(x − a). This is just the tangent line equation. To approximate f(x) near a, plug in the nearby x-value: f(x) ≈ f(a) + f'(a)(x − a).

Over/underestimates from concavity

Since the tangent line only matches the curve at the point of tangency, linearization gives an approximation, not an exact value. If the function is concave up near a, the tangent line lies below the curve, so linearization underestimates. If concave down, it lies above, so linearization overestimates.

Key terms

Local linearity

Near a point of differentiability, a function's graph closely resembles its tangent line when zoomed in.

Linearization formula

L(x) = f(a) + f'(a)(x − a); used to approximate f(x) for x near a.

When linearization underestimates

When the function is concave up near a, the tangent line lies below the curve, underestimating f(x).

When linearization overestimates

When the function is concave down near a, the tangent line lies above the curve, overestimating f(x).

Why does accuracy decrease farther from a?

The tangent line only matches the function exactly at x = a; the curve diverges from the line as x moves away, increasing approximation error.

Try an AP-style question

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

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