Linear Approximation: Estimating Values with a Tangent Line

Some values are hard to compute by hand, like √4.1. Linear approximation lets you estimate them quickly using the tangent line of a function near a point where the values are easy.

The Idea

Near a point x = a, a smooth curve looks almost like its tangent line. So instead of evaluating the function at a hard value, we evaluate the tangent line, which is much easier.

The Formula

The tangent line to f(x) at x = a is:

For x close to a, the function is approximately equal to the line:

What You Need to Start

You need three things, all taken at the same point a:

1. The base point a
2. The function value f(a)
3. The derivative value f'(a)

Both f(a) and f'(a) must come from the same a. The line has to pass through the point (a, f(a)) with slope f'(a).

Worked Example: Approximating √4.1

Step 1: Choose the function. Let f(x) be the square root of x:

Step 2: Choose a base point. Pick a = 4, since it is close to 4.1 and its square root is easy.

Step 3: Find the derivative.

Step 4: Evaluate at a = 4.

Step 5: Build the tangent line.

Step 6: Plug in the target value. Since 4.1 is 0.1 away from 4:

The true value is about 2.02485, so the estimate is off by only about 0.00015.

Tips for Choosing a Good Base Point

• Pick a as close as possible to the value you want.
• Make sure f(a) and f'(a) are easy to compute by hand.
• Remember the approximation gets worse as x moves farther from a.

Quick Summary

Linear approximation replaces a curve with its tangent line near a chosen point. Choose a nearby a with easy values, compute f(a) and f'(a), build L(x), and plug in your target x.

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