The Lagrange Error Bound for Taylor Polynomials

What the Lagrange Error Bound Is For

A Taylor polynomial gives you an approximation to a function near the center of the polynomial. But an approximation is only useful if you know how accurate it is.

The Lagrange error bound gives you a way to put a guaranteed upper limit on the error. Instead of simply saying that a Taylor polynomial "looks close" to the original function, you can say something much stronger: the actual error is no greater than a specific number.

If is approximated by its degree- Taylor polynomial centered at , the Lagrange error bound tells us how far the approximation can be from the actual value of the function.

The Lagrange Error Bound Formula

Suppose is the degree- Taylor polynomial for centered at . Then the error between and satisfies:

Here, represents the maximum value of the absolute value of the derivative on the interval between and .

The formula is often written more compactly as:

What Each Part Means

• : the maximum possible error.
• : the maximum value of on the relevant interval.
• : one order higher than the degree of the Taylor polynomial.
• : the distance between the point where you are approximating and the center of the Taylor polynomial.

The most important detail is the derivative order. If you use a degree- Taylor polynomial, you need the derivative to calculate the Lagrange error bound.

The Standard AP Calculus Method

Most Lagrange error-bound questions can be solved with the same four-step process.

1. Identify the degree of the Taylor polynomial. If the polynomial has degree , the error formula uses the derivative.
2. Find or bound the maximum value of the next derivative. Determine a value such that throughout the required interval.
3. Substitute into the Lagrange formula.
4. Interpret the result. State that the error is less than or equal to the calculated bound.

Worked Examples

How to Find the Value of

The hardest part of many Lagrange error-bound problems is not the formula itself. It is determining the value of .

You need a number such that:

In other words, must be an upper bound for the absolute value of the required derivative throughout the entire interval.

When the Derivative Has an Obvious Maximum

Sometimes the derivative is something simple like . If the interval is , the largest possible value is clearly .

When the Derivative Is Bounded by 1

This is especially common with sine and cosine. Since:

you can often use whenever the relevant derivative is a sine or cosine function.

Using a Convenient Upper Bound

You do not always need the smallest possible value of . Any valid upper bound works.

For example, if you determine that the derivative is always less than or equal to 7 on the interval, then is valid. If it is also always less than or equal to 10, then is technically valid too, although it produces a weaker error bound.

Why the Interval Matters

The maximum derivative value must be taken over the interval connecting the Taylor center and the point where you are evaluating the approximation.

This is important because the derivative might be small near the center but much larger farther away.

For example, if a Taylor polynomial is centered at and used to approximate a value at , you need a bound valid on the entire interval . You cannot simply evaluate the derivative at the endpoint and assume that is automatically the maximum.

Understanding the Error Bound Intuitively

The Lagrange formula contains two major ingredients:

The first part, , measures how large the relevant derivative can become. The second part, , measures how far you are moving away from the Taylor center.

That second factor is particularly important. Because the distance is raised to the power , increasing the degree of the Taylor polynomial can make the error shrink very quickly when the target point is close to the center.

Common Mistakes

Lagrange Error Bound vs. Actual Error

Suppose a Taylor polynomial gives an approximation of , while the actual value is . The actual error is:

The Lagrange error bound instead gives a number such that:

So the actual error might be much smaller than the bound. The power of the Lagrange method is that the bound is guaranteed.

From Error Bound to Decimal Accuracy

AP Calculus questions sometimes ask you to determine how accurately a Taylor polynomial approximates a value.

If you establish that:

then you know the approximation differs from the actual value by less than .

For example, if the error is guaranteed to be less than , then the approximation is accurate to at least the corresponding decimal place required by the problem.

A Reliable AP Calculus Checklist

1. Find the degree: identify .
2. Increase the derivative order by one: use .
3. Identify the interval: from to .
4. Find a valid maximum: choose so that .
5. Calculate the distance: .
6. Substitute: .
7. Interpret: state that the error is less than or equal to the resulting value.

Practice Problems

The Formula to Remember

If you remember only one workflow for the AP exam, remember this:

The Lagrange error bound is ultimately about turning an approximation into a guarantee. Taylor polynomials tell you what to approximate with; the Lagrange error bound tells you how accurate that approximation is guaranteed to be.