Using Taylor Series to Approximate Function Values

Why Use a Taylor Polynomial to Approximate a Function?

Some function values are difficult or inconvenient to calculate exactly. For example, evaluating by hand is not especially useful if your goal is a decimal approximation. A Taylor polynomial gives us another approach: replace the complicated function with a polynomial that behaves very similarly near a chosen center.

Polynomials are easy to evaluate. If we can find a Taylor polynomial for a function, we can use that polynomial to estimate the original function at nearby values.

The key idea is simple:

The approximation becomes better as we use more terms—provided the point we're evaluating is within a region where the Taylor series converges to the function.

The Taylor Polynomial

The Taylor polynomial of degree centered at is:

When , this becomes a Maclaurin polynomial:

For approximation problems, you usually do not need to derive the entire polynomial from scratch. Very often, the problem gives you a Taylor polynomial, gives you a known series, or asks you to construct the first few terms.

The Basic Approximation Process

1. Identify the Taylor polynomial. Determine the center and degree you are supposed to use.
2. Substitute the target value into the polynomial.
3. Calculate the resulting number. This is your approximation to the original function.
4. Check the error requirement. If the problem asks for a certain accuracy, use an error bound to justify the approximation.

Think of the Taylor polynomial as a calculator replacement. Instead of evaluating the original function directly, you evaluate a polynomial that closely imitates it near the center.

Worked Examples

How the Degree Affects Accuracy

Suppose you have several Taylor polynomials for the same function:

Higher-degree polynomials contain more information about the function. Near the center, the additional terms generally make the approximation more accurate.

For example, consider the Maclaurin series for :

If we approximate using only , the approximation is:

Adding the next term gives:

The second approximation is much closer because we included more of the series.

Choosing a Good Center

A Taylor polynomial is usually most accurate near its center. If the polynomial is centered at , then it is naturally designed to approximate the function well when is close to .

This is why the center matters so much.

As the distance between and increases, more terms may be needed to obtain the same level of accuracy.

Using a Known Series to Build an Approximation

AP Calculus BC problems often give you a known Taylor series and ask you to manipulate it. You may need to substitute, multiply, differentiate, or integrate a known series before using it for approximation.

For example, start with the Maclaurin series:

To approximate , substitute for :

Therefore, a polynomial approximation can be obtained without calculating a completely new set of derivatives.

Approximation Error

An approximation is useful only if we understand how close it is to the actual value. The error is the difference between the exact function value and the approximation:

For a Taylor polynomial, the error is represented by the remainder:

In many AP Calculus problems, you are not asked to calculate the exact remainder. Instead, you are asked to find an upper bound for the error.

The Lagrange Error Bound

If is the Taylor polynomial of degree centered at , then the Lagrange error bound is:

Here:

• is the maximum value of on the interval between and .
• is the degree of the Taylor polynomial.
• is the center of the polynomial.
• is the point where you are approximating the function.

The bound tells you that the actual error cannot be larger than the number you calculate.

Worked Error-Bound Example

How to Decide How Many Terms You Need

Sometimes the question does not tell you which degree to use. Instead, it asks you to approximate a value within a specified error.

The strategy is:

1. Start with the Taylor polynomial of a reasonable degree.
2. Use the Lagrange error bound to estimate the maximum possible error.
3. If the error bound is too large, increase the degree.
4. Repeat until the error bound is below the required tolerance.

For example, if the problem says the approximation must be accurate to within , you need to find a degree for which:

Be careful with language. "Within " means the absolute error must be no greater than , not that the answer simply needs to be rounded to a certain number of decimal places.

Common AP Calculus Mistakes

A Reliable AP Exam Strategy

When you see a Taylor-series approximation problem on the AP exam, use this checklist:

1. Find the center. Look for in the polynomial.
2. Find the target. Identify the value where the function is being approximated.
3. Substitute the target into the polynomial.
4. Calculate the approximation.
5. Check whether an error bound is required.
6. If an error bound is needed, use the next derivative. A degree- polynomial requires the derivative in the Lagrange bound.
7. State what the bound means. For example: "The approximation differs from the actual value by less than..."

Practice Problems