Error Propagation Using Differentials

Why Differentials Are Useful for Error

Measurements are never perfectly exact. If you measure the radius of a circle with a ruler, for example, your measurement might be off by a small amount. That small error in the input can produce an error in the quantity you calculate from it.

Differentials give us a fast way to estimate how much that measurement error will affect the final answer. The key idea is simple: a small change in the input produces an approximately predictable change in the output.

If , then a small change in the input produces an approximate change in the output.

For error propagation, represents the uncertainty or error in the measured input, while estimates the resulting error in the calculated output.

The Basic Error-Propagation Formula

Suppose and the measured value of has a small error of approximately . Since

the approximate error in is:

In practice, you will usually take the absolute value when reporting the size of the error:

This is an estimate, not an exact error. The approximation works well when the measurement error is small relative to the measured value.

Absolute Error vs. Relative Error

There are two different ways you might describe an error.

• Absolute error: the approximate size of the error in the calculated quantity.
• Relative error: the error compared with the size of the quantity itself.
• Percent error: relative error expressed as a percentage.

If is the approximate error in , then the relative and percent errors are approximately:

Notice that the denominator is the calculated quantity itself, .

A Step-by-Step Method

1. Write the formula connecting the quantity with the measured variable.
2. Differentiate the formula.
3. Identify the input error, which is the uncertainty in the measured quantity.
4. Use the differential relationship to estimate the resulting output error.
5. Evaluate at the measured value, not at the unknown exact value.
6. Take the absolute value if the question asks for the size or maximum approximate error.
7. If requested, convert the error into relative or percent error.

Worked Examples

Why the Differential Gives an Error Estimate

The differential comes from the tangent-line approximation. If , then near a point , the function can be approximated by its tangent line.

Therefore, a small change in produces an approximate change in given by:

That is why differential-based error propagation is especially useful when the measurement error is small. We are essentially replacing the actual curve with its tangent-line approximation near the measured value.

A Useful Shortcut for Percent Error

For many AP Calculus problems, you can make percent-error calculations especially efficient by starting with the relative differential.

For example, if , then:

Dividing both sides by gives:

This tells us something important: the relative error in the area is approximately twice the relative error in the radius.

Likewise, for the volume of a sphere, , so:

Therefore, the relative error in the volume is approximately three times the relative error in the radius.

Common Mistakes

Practice Problems