Section 4.11

Linear Approximations

Complex curves are hard to compute. Straight lines are easy. Near a specific point, they are almost the same thing. This is the art of "close enough."

The Linearization Formula

The Linearization of a function at is just the equation of the tangent line:

Use when is near .

Worked Example

Approximating Roots

Estimate using linearization.

The Divergence Gap

Actual: 2.1213Approx: 2.1250Gap: 0.0037

Move

away from

. Notice how the Red Gap (Error) grows quadratically.

At , both are 2. At 4.1, they diverge slightly.

1. Choose Function & Anchor

Let .
Choose (easy to square root).

2. Find Point & Slope

3. Linearize & Evaluate

(Calculator says . Error < 0.0002!)

Level Up Examples

Example A: Small Angle Approximation

Estimate .

1. Setup:

2. Values:

3. Result:

Physics students call this the small-angle approx!

Example B: Percent Change