Taylor Polynomial Approximation of a Function (College Board AP® Calculus BC): Revision Note

Taylor & Maclaurin polynomials

What is a Taylor polynomial approximation of a function?

• A Taylor approximation about the point $x=a$ is a way of approximating a function near to that point using a polynomial

  • This polynomial is called a Taylor polynomial

• Let $f$ be a function

  • If the function and its first $n$ derivatives all exist at $x=a$, then the nth degree Taylor polynomial for $f$ about $x=a$ is

$p_n\left(x\right)=f\left(a\right)+f^{\text{'}}\left(a\right)\left(x-a\right)+\frac{f^{\text{'}\text{'}}\left(a\right)}{2!}{\left(x-a\right)}^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}{\left(x-a\right)}^n$

• Note that

  • $p_n\left(a\right)=f\left(a\right)+f^{\text{'}}\left(a\right)\left(a-a\right)+\frac{f^{\text{'}\text{'}}\left(a\right)}{2!}{\left(a-a\right)}^2+...+\frac{f^{\left(n\right)}\left(a\right)}{n!}{\left(a-a\right)}^n=f\left(a\right)$

    • A Taylor polynomial is always exactly equal to its function at $x=a$

  • $p_1\left(x\right)=f\left(a\right)+f^{\text{'}}\left(a\right)\left(x-a\right)$

    • This is the equation of a line with slope $f^{\text{'}}\left(a\right)$, that goes through the point $\left(a,f\left(a\right)\right)$

    • The first-degree Taylor polynomial is the equation of the tangent line to the graph of $f$ at the point $\left(a,f\left(a\right)\right)$

• For example, the first few Taylor polynomials for $f\left(x\right)=\ln x$ about $x=2$ are:

  • $p_1\left(x\right)=\ln 2+\frac{x-2}{2}$

  • $p_2\left(x\right)=\ln 2+\frac{x-2}{2}-\frac{{\left(x-2\right)}^2}{8}$

  • $p_3\left(x\right)=\ln 2+\frac{x-2}{2}-\frac{{\left(x-2\right)}^2}{8}+\frac{{\left(x-2\right)}^3}{24}$

  • $p_4\left(x\right)=\ln 2+\frac{x-2}{2}-\frac{{\left(x-2\right)}^2}{8}+\frac{{\left(x-2\right)}^3}{24}-\frac{{\left(x-2\right)}^4}{64}$

• The graphs of those polynomials about $x=2$ and $f\left(x\right)=\ln x$ can be seen in the following diagrams

• The diagrams illustrate the following general facts about Taylor polynomials:

  • Their accuracy as an approximation decreases as you move away from $x=a$

  • They become a more accurate approximation (and more accurate further away from $x=a$) if you increase the degree of the polynomial

    • Adding additional terms in higher powers of $x$ increases the accuracy

What is a Maclaurin polynomial approximation of a function?

• A Maclaurin polynomial approximation of a function is a special case of a Taylor approximation

• It is the Taylor approximation of a function about the point $x=0$

  • If the function $f$ and its first $n$ derivatives all exist at $x=0$, then the nth degree Maclaurin polynomial for $f$ about $x=0$ is

  $p_n\left(x\right)=f\left(0\right)+f^{\text{'}}\left(0\right)x+\frac{f^{\text{'}\text{'}}\left(0\right)}{2!}{x}^2+...+\frac{f^{\left(n\right)}\left(0\right)}{n!}{x}^n$