What linear approximation is
Close to a point, a smooth curve looks almost straight. Linear approximation replaces f with its tangent line at x = a, called the linearization L(x), and uses that line to estimate f at nearby points. It's how you estimate √9.1 or sin(0.1) without a calculator, and it's the idea behind differentials.
The linearization formula
f(a) is the height at the center and f′(a) is the slope there. The estimate is f(x) ≈ L(x) for x near a.
Worked example
Estimate √9.1 using a = 9, where √9 = 3 is known.
- f(x) = √x, so f(9) = 3.
- f′(x) = 1/(2√x), so f′(9) = 1/6.
- L(x) = 3 + (1/6)(x − 9).
- L(9.1) = 3 + (1/6)(0.1) ≈ 3.016667.
The true value is √9.1 ≈ 3.016621, so the estimate is too high by about 0.000046. It overshoots because √x is concave down: the tangent line sits above the curve.
How accurate is it?
The error grows with the distance from a and with how sharply f bends. A rough guide is the next term of the Taylor series, f″(a)(x − a)²/2, which the calculator shows. Doubling the distance from a roughly quadruples the error.
Common linear approximations near 0
| f(x) | a | L(x) |
|---|---|---|
| sin x | 0 | x |
| tan x | 0 | x |
| eˣ | 0 | 1 + x |
| ln(1 + x) | 0 | x |
| √(1 + x) | 0 | 1 + x/2 |
| (1 + x)ⁿ | 0 | 1 + nx |
Limitations
- It only works well near a. Far away, the tangent line and the curve part ways.
- f must be differentiable at a. At a corner or a vertical tangent, such as ∛x at 0, there is no useful linearization.
- Today it's easy to compute f(x) directly. The value of linearization is in understanding how a small change in x moves f, which is what the calculator's error rows show.