Taylor polynomials

Section 7.4 Taylor polynomials

In the previous section we studied the linear approximation of a function at a point. But why stop at linear polynomials? Wouldn't we be able to achieve better approximations if we looked for higher degree polynomials?

The answer is yes, and the result is what is called the “Taylor polynomials” of a function at a point. The degree \(d\) Taylor polynomial of a function at a point approximates that function near this point by a degree \(d\) polynomial. It turns out that the Taylor polynomials of a function at a point can be fully calculated using differentiation. I told you that calculus was a theory of approximations!

Objectives

You should be able to:

Calculate the Taylor polynomials of a function, and compute its corresponding higher degree approximation at a given point.
Illustrate the relation between the function and its higher degree approximations.
Use the higher degree approximations of a function at a point to approximate the value of the function near this point.

Subsection 7.4.1 Instructional video

Subsection 7.4.2 Key concepts

Concept 7.4.1. Taylor polynomials.

The idea of Taylor polynomials is to approximate a function \(f(x)\) at a point \(x=a\) by higher degree polynomials. The polynomial

\begin{equation*} T_d(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 +\dots +\frac{f^{(d)}(a)}{d!}(x-a)^d, \end{equation*}

where \(k!= 1 \cdot 2 \cdot 3 \dots k\text{,}\) is called the degree \(d\) Taylor polynomial of \(f(x)\) at \(x = a\text{.}\)

It satisfies the properties:

\begin{equation*} T_d(a)=f(a) \qquad \text{and} \qquad T_d^{(k)}(a)=f^{(k)}(a) \end{equation*}

for all \(k=1,2,\dots, d\text{.}\)

Concept 7.4.2. The degree \(d\) polynomial approximation of a function at a point.

The degree \(d\) approximation of \(f(x)\) at \(x = a\) is

\begin{equation*} f(x)\approx T_d(x), \end{equation*}

where \(T_d(x)\) is the degree \(d\) Taylor polynomial of \(f(x)\) at \(x=a\text{.}\) When \(d=1\) we get the linear approximation of \(f\) at \(x=a\text{,}\)

\begin{equation*} f(x)\approx f(a) + f'(a)(x-a). \end{equation*}

When \(d=2\) we get the quadratic approximation of \(f\) at \(x=a\text{,}\)

\begin{equation*} f(x)\approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2. \end{equation*}

Concept 7.4.3. The Taylor series.

In fact, we could try to send \(d \to \infty\text{;}\) that would give an “approximation” of \(f(x)\) at \(x=a\) by an “infinite degree polynomial”. This can be made rigorous using sequence and series; namely, taking \(d\to \infty\) means taking the limit of the sequence of Taylor polynomials \(T_d(x)\text{,}\) and the result is the Taylor series of \(f(x)\) at \(x=a\). Reversing the process, one can think of the Taylor polynomials above as truncations (so-called “partial sums”) of the Taylor series of \(f(x)\) at \(x=a\text{.}\)

MATH 144: Calculus for the Physical Sciences I