Linear Approximation

Unit 4 · Contextual Applications of Differentiation

Linear Approximation (Linearization)

The linearization of f at x = a is the tangent line at that point: L(x) = f(a) + f′(a)(x − a). For x near a, f(x) ≈ L(x). Whether this is an overestimate or underestimate depends on concavity.

Linearization Formula

L(x) = f(a) + f'(a)(x - a)

Differential form

\Delta y \approx dy = f'(a)\,\Delta x

Over/Underestimate Rule

Concave up (f″ > 0): tangent below curve →

\text{underestimate}

Concave down (f″ < 0): tangent above curve →

\text{overestimate}

AP Tip: Choose a = the nearest value where f(a) and f′(a) are easy to compute exactly (e.g., a perfect square or cube).

Type 1

Approximating Function Values

Identify a nearby anchor point a, compute L(x), then evaluate at the target x.

Example 1

Use the linearization of f(x) = √x at a = 9 to approximate √9.1.

Example 2

f(3) = 7 and f′(3) = −2. Approximate f(3.4) using local linearization.

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Type 2

Overestimate vs. Underestimate

After computing the linear approximation, determine whether it overestimates or underestimates by checking the sign of f″ at a.

Example 3

Use linearization at

a = 0

to estimate

e^{0.1}

. Is it an overestimate or underestimate?

Example 4

Use linearization of f(x) = ln x at a = 1 to approximate ln(1.04). Overestimate or underestimate?

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Type 3

Differentials and Error Estimation

Use dy = f′(x) dx to estimate the propagated error in a calculated quantity when the input has a small measurement error.

Example 5

A sphere's radius is measured as r = 5 cm with possible error dr = 0.05 cm. Estimate the maximum error in surface area S = 4πr².

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← Related Rates L'Hôpital's Rule →