AP Calculus AB — calculusStandard1 min read

Linearization and Differentials

Approximate function values using linearization and differentials for AP Calculus AB.

Tutor AI Team2026-02-25T02:12:59.900Z

Linearization uses the tangent line to approximate function values near a known point.

Linear Approximation

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

This is the tangent line at $x = a$ , used to approximate $f(x)$ for $x$ near $a$ .

$dy = f'(x)\,dx$

$\Delta y \approx dy$ for small $\Delta x = dx$ .

Problem

Approximate $\sqrt{4.1}$ using linearization of $f(x) = \sqrt{x}$ at $a = 4$ .

$f(4) = 2$ , $f'(x) = \frac{1}{2\sqrt{x}}$ , $f'(4) = \frac{1}{4}$ .

$L(4.1) = 2 + \frac{1}{4}(0.1) = 2.025$ . (Actual: 2.0248...)

Solution

Problem

$y = x^3$ . $dy = 3x^2\,dx$ .

At $x = 2$ , $dx = 0.01$ : $dy = 12(0.01) = 0.12$ .

Solution

1. Approximate $\sin(0.1)$ using $L(x)$ at $a = 0$ .
1. Use differentials to estimate the change in volume of a sphere when $r$ changes from 5 to 5.02.

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#ap#calculus#applications#linearization#differentials

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