Approximating Values Using Local Linearity

What Local Linearity Means

Near any point where a function is differentiable, the function behaves approximately like a straight line. If you zoom in closely enough on a smooth curve, the curve starts to look almost indistinguishable from its tangent line.

This idea is called local linearity. It gives us a powerful way to approximate a function value that would otherwise be difficult to calculate directly.

Suppose you know the value of a function and its derivative at . If you want the value of the function at a nearby point , you can use the tangent line at as an approximation.

The key idea is simple: start with a known function value, then use the derivative to estimate how much the function changes.

The Linearization Formula

The tangent line to at is called the linearization of at .

Once you have the linearization, you can approximate a nearby function value by evaluating the line instead of the original function:

Notice what makes this useful. The original function might be difficult to evaluate, but the linearization is just a line. You only need multiplication and addition.

Why the Approximation Works

The tangent line has exactly the same function value and slope as the original function at the point of tangency.

At :

• The function and tangent line have the same height: .
• The function and tangent line have the same slope: .

Because the two graphs agree at the point and initially move in the same direction, the tangent line is often a very good approximation for points that are close to the point of tangency.

A Step-by-Step Method

1. Identify the function. Write down .
2. Choose the nearby known point. Find a value where is easy to determine.
3. Find the derivative. Calculate .
4. Evaluate the function and derivative at the known point. Find and .
5. Build the linearization. Use .
6. Evaluate the linearization at the desired point. Use to approximate .

Worked Examples

Recognizing the Best Point to Use

On AP Calculus questions, the most important step is often choosing the point . Look for a nearby value where the function can be evaluated exactly or very easily.

For example, if you need to approximate , a natural choice is because is easy to evaluate. You do not want to choose an arbitrary point simply because it is close.

The closer is to , the more likely the linear approximation is to be accurate.

Estimating Error

Local linearity gives an approximation, not necessarily the exact answer. The difference between the true value and the linear approximation is called the approximation error.

For a sufficiently smooth function, the tangent-line approximation usually becomes more accurate as the target point gets closer to the point of tangency.

On an AP exam, you should generally report the approximation to the requested number of decimal places rather than pretending the linearization is exact.

Local Linearity and Differentials

Local linearity is closely connected to differentials. Starting from the linear approximation, a small change in the input, , produces an estimated change in the output:

Therefore, for a small change from to , we can estimate:

This is another way of expressing the same fundamental idea: small changes in the input produce approximately linear changes in the output.

Common Mistakes

Practice Problems