Back to Infinite Sequences and Series

Lesson PreviewLesson 11 of 15 in this unit

10.11 Finding Taylor Polynomial Approximations of Functions

Part of Infinite Sequences and Series in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

Start free
A Taylor polynomial approximates a function near a specific center x=a using a polynomial built from the function’s derivatives at that point. The degree n Taylor polynomial for f(x) centered at a is
Tn(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)nT_n(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^n
This polynomial matches the function’s value and several derivatives at x=a, which makes it a very good local approximation near that point.

You should think of Taylor polynomials as higher-order linearizations. A linearization uses the tangent line, which only matches the first derivative behavior. A Taylor polynomial goes further by matching curvature and higher-order behavior too. The closer x is to the center a, the better the approximation usually is. When a=0, the Taylor polynomial is called a Maclaurin polynomial. In BC calculus, you often build these by computing derivatives and substituting into the formula.

In this unit

15 lessons

  1. 10.1Defining Convergent and Divergent Infinite SeriesPreview lesson
  2. 10.2Working with Geometric SeriesPreview lesson
  3. 10.3The nth Term Test for DivergencePreview lesson
  4. 10.4Integral Test for ConvergencePreview lesson
  5. 10.5Harmonic Series and p-SeriesPreview lesson
  6. 10.6Comparison Tests for ConvergencePreview lesson
  7. 10.7Alternating Series Test for ConvergencePreview lesson
  8. 10.8Ratio Test for ConvergencePreview lesson
  9. 10.9Determining Absolute or Conditional ConvergencePreview lesson
  10. 10.10Alternating Series Error BoundPreview lesson
  11. 10.11Finding Taylor Polynomial Approximations of FunctionsThis lesson
  12. 10.12Lagrange Error BoundPreview lesson
  13. 10.13Radius and Interval of Convergence of Power SeriesPreview lesson
  14. 10.14Finding Taylor or Maclaurin Series for a FunctionPreview lesson
  15. 10.15Representing Functions as Power SeriesPreview lesson

Continue the full lesson in Stellar

Create a free account to unlock the remaining sections, checkpoints, course practice, and progress tracking for AP® Calculus BC.

Start free