Welcome to Taylor Polynomials and Error Bounds!

Have you ever wondered how your calculator knows the value of \(\sin(1)\) or \(e^{0.5}\) so quickly? It doesn't have an infinite list of numbers stored in its memory. Instead, it uses simple addition and multiplication—the building blocks of polynomials—to approximate these complex functions. In this chapter, we will learn how to build these "mathematical clones" (Taylor Polynomials) and how to calculate exactly how much of a "safety margin" we need to ensure our answer is accurate (Lagrange Error Bound).

10.11 Finding Taylor Polynomial Approximations

A Taylor Polynomial is a polynomial designed to mimic the behavior of a non-polynomial function (like \(\ln(x)\), \(\cos(x)\), or \(e^x\)) near a specific point, called the center.

The Idea: Matching Derivatives

If we want a polynomial \(P(x)\) to act like a function \(f(x)\) at a center \(c\), we make sure that at that specific point, they share the same height, the same slope, the same "curviness," and so on. We do this by matching their derivatives.

The General Formula

The Taylor polynomial of degree \(n\) for a function \(f\) centered at \(x = c\) is defined as:

\(P_n(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \dots + \frac{f^{(n)}(c)}{n!}(x-c)^n\)

Note: A Maclaurin Polynomial is simply a Taylor polynomial where the center is \(c = 0\). It looks like this:

\(P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots + \frac{f^{(n)}(0)}{n!}x^n\)

Step-by-Step: How to Construct a Taylor Polynomial

Don't worry if this seems like a lot of symbols! Just follow these four steps:

  1. Find the Derivatives: Calculate \(f'(x)\), \(f''(x)\), \(f'''(x)\), etc., up to the degree \(n\) requested.
  2. Evaluate at the Center: Plug the center \(c\) into each of those derivatives to get numerical values.
  3. Apply the Factorials: Divide each numerical value by its corresponding factorial (\(0!\), \(1!\), \(2!\), \(3!\), etc.). Remember that \(0! = 1\) and \(1! = 1\).
  4. Assemble: Attach the \((x-c)^k\) terms to your coefficients.

Example: Find the 3rd-degree Maclaurin polynomial for \(f(x) = e^x\).
1. Derivatives: \(f'(x) = e^x, f''(x) = e^x, f'''(x) = e^x\).
2. Evaluate at \(c=0\): All derivatives are \(e^0 = 1\).
3. Build: \(P_3(x) = 1 + 1x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 = 1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3\).

Key Takeaway: Taylor polynomials are just a sum of terms following the pattern: \(\frac{\text{Derivative at Center}}{\text{Factorial}} \cdot (\text{Distance from Center})^{\text{Power}}\).

10.12 The Lagrange Error Bound

Since a Taylor polynomial is only an approximation, there is usually a small difference between the actual function value \(f(x)\) and the polynomial value \(P_n(x)\). This difference is called the remainder or the error: \(R_n(x) = f(x) - P_n(x)\).

While the Alternating Series Error Bound (which you may have learned in Topic 10.10) only works for alternating series, the Lagrange Error Bound works for any Taylor polynomial!

The Formula

The maximum possible error when using \(P_n(x)\) to estimate \(f(x)\) is given by:

\(|R_n(x)| \leq \frac{M}{(n+1)!} |x-c|^{n+1}\)

Where:

  • \(n\) is the degree of the polynomial you used.
  • \(c\) is the center.
  • \(x\) is the point where you are evaluating the function.
  • \(M\) is the "worst-case scenario" value. It is the maximum possible value of the absolute value of the next derivative, \(|f^{(n+1)}(t)|\), for all \(t\) between the center \(c\) and the point \(x\).

How to Find \(M\) (The Trickiest Part)

On the AP Exam, \(M\) is usually provided in one of three ways:

  1. The "Given" Method: The problem explicitly states "The \((n+1)\)-th derivative satisfies \(|f^{(n+1)}(x)| \leq 5\)." In this case, \(M = 5\).
  2. The Graph Method: You are given a graph of the \((n+1)\)-th derivative and must find its highest point (peak) on the interval between \(c\) and \(x\).
  3. The Trigonometric Method: If the next derivative involves \(\sin(x)\) or \(\cos(x)\), you can usually use \(M = 1\), because those functions never exceed 1.

Analogy: Think of Lagrange Error Bound like a weather forecast. If the forecast says it will be \(70^\circ\) with an error bound of \(3^\circ\), you know the actual temperature is somewhere between \(67^\circ\) and \(73^\circ\). The Lagrange bound gives you that \(\pm\) range.

Key Takeaway: To find the Lagrange error bound, always look for the next derivative (\(n+1\)) and find its maximum value \(M\) over the interval.

Common Mistakes to Avoid

  • Forgetting Factorials: Students often forget the \(n!\) in the denominator of the Taylor terms or the \((n+1)!\) in the Lagrange formula. Don't let those factorials disappear!
  • Center Confusion: Make sure you are using \((x-c)\) and not just \(x\) if the polynomial is centered somewhere other than zero.
  • Using the Wrong Derivative for \(M\): For a degree \(n\) polynomial, the error bound always uses the \((n+1)\)-th derivative.
  • Mixing up "Terms" and "Degree": If a Taylor polynomial is \(P(x) = 1 - \frac{x^2}{2}\), it is a 2nd-degree polynomial (even though it only has two terms). The error bound would use the 3rd derivative.

Quick Review Box

Taylor Term: \(\frac{f^{(n)}(c)}{n!}(x-c)^n\)
Maclaurin: Center is \(0\).
Lagrange Error Bound: \(\frac{\max|f^{(n+1)}|}{(n+1)!}|x-c|^{n+1}\)

For more information on how these polynomials turn into infinite series, check out Topic 10.14: Finding Taylor or Maclaurin Series for a Function.