# Linear approximation

Near x = a, a curve looks like its tangent line. So f(x) is about f(a) + f'(a)(x - a) for x close to a.

Maths (university) · Part 04: Applications of derivatives · Card 8 of 9 · Last updated 2026-10-19

## Analogy

A small piece of a round Earth looks flat, so a straight map works for your town.

## Example

**Estimate the square root of 4.1**

f(x) = sqrt(x), f(4) = 2.
f'(x) = 1/(2 sqrt(x)), so f'(4) = 1/4.
Estimate: 2 + (1/4)(0.1) = 2.025.
True value: 2.0248.

## In practice

Gives quick estimates of roots, powers and small errors without a calculator.

## Practice

### Flashcards

- **Write the linear approximation formula.** f(x) is about f(a) + f'(a)(x - a).
- **Estimate the square root of 4.1 by linear approximation.** 2.025.

## Questions people ask

### How accurate is linear approximation?

It is very good very close to a and worsens as you move away. The error grows roughly with the square of the distance. For the square root of 4.1, the estimate 2.025 differs from the true value by less than 0.0002.

### Does the approximation overestimate or underestimate?

It depends on the bend. For a concave down curve like square root, the tangent line lies above the curve, so the estimate is slightly too high. For a concave up curve, the tangent lies below, so the estimate is slightly too low.

## Related concepts

- [Critical point](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-1)
- [First derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-2)
- [Second derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-3)
- [Mean value theorem](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-5)

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