AP Calculus BC Unit 10.2

Taylor Polynomials

Maclaurin Polynomials

• A Maclaurin polynomial is an approximation of a function f(x), centered at x = 0.

• The general form of a Maclaurin polynomial of degree n is:
  $P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots + \frac{f^{(n)}(0)}{n!}x^n$

Taylor Polynomials

• A Taylor polynomial is an approximation of a function f(x), centered at a point c.

• The general form of a Taylor polynomial of degree n is:
  $P_n(x) = f(c) + f'(c)(x - c) + \frac{f''(c)}{2!}(x - c)^2 + \frac{f'''(c)}{3!}(x - c)^3 + \ldots + \frac{f^{(n)}(c)}{n!}(x - c)^n$

Example Application

• To approximate cos(x) using a 6th degree Maclaurin polynomial:

  • The terms are derived from the derivatives of f(x) = cos(x).

Error Analysis

Actual Error and Lagrange Error Bound

• The actual error in an approximation is defined as:
  $E = |f(x) - P_n(x)|$

• The Lagrange error bound provides a maximum error estimate:

  • For a Taylor polynomial of degree n, the Lagrange error bound is given by:
    $|E| = |f^{(n+1)}(z)| \cdot \frac{|x - c|^{n+1}}{(n+1)!}$

  • where z is some value between x and c.

Problems and Examples

Problem Set

1. Approximate cos(0.3) using the first three non-zero terms of the Maclaurin polynomial and find the maximum error bound.

2. Estimate the maximum error bound if the function 1 + x + \frac{x^2}{2!} + \ldots is used to approximate e^x.

3. Let f(x) = √x.

   • a. Find the second-degree Taylor polynomial about x = 4 and estimate f(5.1).

   • b. Use the Lagrange error bound to find a bound on the error for the approximation in part (b).

   • c. Calculate |f(5.1) - P_2(5.1)|.

4. If a function is approximated with a fourth-degree Taylor polynomial about x = 1, and the maximum value of the fifth derivative between x = 1 and x = 3 is 0.01, find the maximum error incurred when computing f(3).

5. Given a function with derivatives of all orders for all real numbers x, assume the following values:

   • f(5) = 6

   • f'(5) = 8

   • f''(5) = 30

   • f'''(5) = 48

   • |f^{(4)}(x)| ≤ 75 for all x such that x ∈ [5, 5.2].

   • a. Find the third-degree Taylor polynomial about x = 5 and estimate f(5.2).

   • b. Determine the maximum possible error in making this estimate.

   • c. Find an interval [a, b] such that a ≤ f(5.2) ≤ b.

Convergence of Series

• A series converges if the sequence of its partial sums converges to a finite limit.

• For a comparison of convergence, consider known series to establish bounds.

  • Example: the series diverges if compared to the harmonic series, Σ (1/n).

Geometric Power Series

1. The geometric series is defined as:
   $ext{Geometric Series} = \sum_{n=0}^{\infty} a r^n$

   • Converges for |r| < 1.

2. Write a geometric power series for:

   • a. f(x) = \frac{1}{3x + 2}` centered at x = 2.

   • b. f(x) = \frac{2}{x^2 - 1}` centered at x = 0.

More Taylor and Maclaurin Series

• To find limits using series, you can apply L'Hôpital's rule or approximations.

  • Example: To find $\lim_{x \to 0} \frac{\sin x}{x}$ using Taylor series expansion for sin(x).

• Approximate the integral: $f(x) = \int_{1}^{12} sin(x^2) dx$