Constructing Taylor Polynomials Centered at a Value

What a Taylor Polynomial Does

A Taylor polynomial is a polynomial designed to approximate a function near a particular value. Instead of trying to reproduce the function everywhere, we construct a polynomial that matches the function's behavior at one specific point.

The key idea is simple: if two functions have the same value, the same first derivative, the same second derivative, and so on at a point, then they have increasingly similar behavior near that point.

For example, if we want to approximate a function near , we can build a polynomial whose value and derivatives agree with the original function at .

The word centered is important. A Taylor polynomial centered at is built using powers of , not powers of .

The Taylor Polynomial Formula

The Taylor polynomial of degree centered at is:

This formula may look complicated, but every term follows exactly the same pattern.

Here, is the function, is the center, and means the derivative of the function.

The factorials are also important. The coefficient of the degree- term is the corresponding derivative evaluated at the center, divided by .

The Most Important Pattern

When constructing a Taylor polynomial, it helps to write out the first few terms rather than trying to memorize the entire formula at once.

Notice the pattern:

• The constant term uses .
• The coefficient of uses .
• The coefficient of uses .
• The coefficient of uses .
• Every derivative is evaluated at the center .

This gives you a reliable construction method: differentiate, evaluate at the center, divide by the appropriate factorial, and multiply by the appropriate power of .

A Step-by-Step Construction Method

1. Identify the center. Find the value around which the polynomial is being constructed.
2. Identify the degree. A degree- Taylor polynomial requires derivatives from the zeroth through the derivative.
3. Find the derivatives. Compute , , , and so on through the required order.
4. Evaluate every derivative at the center. Calculate , , etc.
5. Divide each derivative value by the appropriate factorial.
6. Multiply by the appropriate power of .
7. Write the terms in descending or ascending degree order. Either is mathematically correct.

Worked Examples

Why the Center Matters

A common source of errors is treating the center as though it were always zero.

When , the Taylor polynomial becomes a Maclaurin polynomial. In that special case, the formula simplifies to powers of .

But when the center is some other value, such as , the polynomial must be written in powers of .

So do not automatically write . First ask: What value is the polynomial centered at?

Understanding the Coefficients

There is a useful reason behind the Taylor formula. The coefficients are chosen so that the polynomial matches the function's derivatives at the center.

For a Taylor polynomial centered at , we want:

That means the polynomial has the same value as the function at the center, the same slope at the center, the same concavity at the center, and increasingly higher-order behavior as the degree increases.

For example, a degree-1 Taylor polynomial matches the function's value and first derivative. A degree-2 polynomial additionally matches the second derivative. A degree-3 polynomial additionally matches the third derivative, and so forth.

Degree Determines How Many Derivatives You Need

This is an excellent AP Calculus shortcut. If the problem asks for a degree-5 Taylor polynomial, you need derivatives through the fifth derivative—but you do not need the sixth derivative.

The zeroth derivative is simply the original function itself:

Using a Taylor Polynomial to Approximate a Function

Once you have constructed a Taylor polynomial, you can use it to approximate the original function near the center.

Suppose is a Taylor polynomial for centered at . To approximate , simply evaluate the polynomial at .

The approximation is generally most accurate when is close to the center . As you move farther away from the center, the approximation may become less accurate.

Worked Approximation Example

Common Mistakes

AP Calculus BC Strategy

On the AP exam, Taylor polynomial questions often combine several skills. A problem may give you a function, a center, derivative information, and an approximation target all in the same question.

A reliable strategy is to separate the problem into two tasks:

1. Construct the polynomial. Determine the coefficients from the function and its derivatives.
2. Use the polynomial. Substitute the requested value into the polynomial to obtain the approximation.

If the problem gives a table of derivative values, you often do not need to differentiate anything yourself. Simply read the values from the table and place them into the Taylor formula.

Also pay attention to the phrase "centered at". If the center is not zero, immediately write somewhere on your scratch work. This prevents one of the most common Taylor polynomial mistakes.

Practice Problems