Taylor Polynomials

The definition of limit, $f^{'} (a) = lim_{x \to a} \frac{f (x) - f (a)}{x - a}$ can be thought of, informally, as saying that for $x$ near $a$ . $\frac{f (x) - f (a)}{x - a}$ is approximately equal to $f^{'} (x)$ . That is, $f (x) \approx f (a) + f^{'} (a) (x - a)$ . Of course, $y = f (a) + f^{'} (a) (x - a)$ is the equation of a line through the point $(a, f (a))$ with slope $f^{'} (a)$ , so this is the tangent line approximation. The tangent line at $a$ is the best linear approximation to the function $y = f (x)$ near $a$ . We can try to get a better approximation for $f$ near $a$ by using higher degree polynomials.

Suppose that $f (x)$ is $n$ times differentiable at $a$ . That is, $f (a)$ , $f^{'} (a), f^{″} (a), \dots, f^{(n)} (a)$ all exist. We can look for a polynomial $p (x)$ of degree $n$ whose values for the first $n$ derivatives at $a$ are the sames as the values for the derivatives of $f (x)$ at $a$ . There is only one polynomial that works. To see this, we need to write $p (x)$ as a polynomial in powers of $(x - a)$ . If we write $p (x) = c_{0} + c_{1} (x - a) + c_{2} (x - a)^{2} + \dots + c_{n} (x - a)^{n}$ , then we have $\begin{aligned} p (x) & = c_{0} + c_{1} (x - a) + c_{2} (x - a)^{2} + \dots + c_{n} (x - a)^{n} & p (a) & = c_{0} \\ p^{'} (x) & = c_{1} + 2 c_{2} (x - a) + 3 c_{3} (x - a)^{2} + \dots + n c_{n} (x - a)^{n - 1} & p^{'} (a) & = c_{1} \\ p^{″} (x) & = 2 c_{2} + 3 \cdot 2 c_{3} (x - a) + 4 \cdot 3 (x - a)^{2} \dots + n (n - 1) c_{n} (x - a)^{n - 2} & p^{″} (a) & = 2 c_{2} \\ p^{‴} (x) & = 3 \cdot 2 c_{3} + 4 \cdot 3 \cdot 2 (x - a) + \dots + n (n - 1) (n - 2) c_{n} (x - a)^{n - 3} & p^{‴} (a) & = 3 \cdot 2 c_{3} \\ p^{(4)} (x) & = 4 \cdot 3 \cdot 2 (x - a) + \dots + n (n - 1) (n - 2) (n - 3) c_{n} (x - a)^{n - 4} & p^{(4)} (a) & = 4 \cdot 3 \cdot 2 c_{4} \\ ⋮ & ⋮ \\ p^{(n)} (x) & = n (n - 1) (n - 2) (n - 3) \dots 2 \cdot 1 \cdot c_{n} & p^{(n)} (a) & = n! \cdot c_{n} \end{aligned}$ We can write the general case as $p^{(k)} (a) = k! \cdot c_{k}$ . So if we want $p^{(k)} (a) = f^{(k)} (a)$ for $k = 0, 1, \dots, n$ , then we must have $k! \cdot c_{k} = f^{(k)} (a)$ , that is, $c_{k} = \frac{f^{(k)} (a)}{k!}$ .

Definition:

Let

n \in N

and let

f (x)

be a function that has derivatives up to order

n

x = a

. Then the Taylor Polynomial for

f

of degree

n

a

is defined to be the polynomial

\begin{aligned} p_{n, a} (x) & = \sum_{k = 0}^{n} \frac{f^{(k)} (a)}{k!} (x - a)^{k} \\ = f (a) + f^{'} (a) (x - a) + \frac{f^{″} (a)}{2} (x - a)^{2} + \dots + \frac{f^{(n)} (a)}{n!} (x - a)^{n} \end{aligned}

We expect that a Taylor polynomial $p_{n, a} (x)$ for $f$ can be used as an approximation for $f (x)$ when $x$ is near $a$ . But the question has to be how good the approximation is. That is, we want to put some bound on the error in the approximation. We can define a remainder term, $r_{n, a} (x)$ that represents the error. The remainder is the difference between $p_{n, a} (x)$ and $f (x)$ . So we have $\begin{aligned} f (x) & = p_{n, a} (x) + r_{n, a} (x) \\ = (\sum_{k = 0}^{n} \frac{f^{(k)} (a)}{k!} (x - a)^{k}) + r_{n, a} (x) \\ = f (a) + f^{'} (a) (x - a) + \frac{f^{″} (a)}{2} (x - a)^{2} + \dots + \frac{f^{(n)} (a)}{n!} (x - a)^{n} + r_{n, a} (x) \end{aligned}$

Taylor's Theorem gives a formula for the remainder, with the assumption that

f

has a continuous derivative of order

(n + 1)

on an interval containing

a

. The formula for

r_{n, a} (x)

in this theorem is called the integral form of the remainder.

Theorem (Taylor's Theorem with Remainder): Suppose that $f$ has a continuous derivative of order $(n + 1)$ on an open interval that contains $a$ . Then for any $x$ in that interval, the remainder term $r_{n, a} (x) = f (x) - p_{n, a} (x)$ is given by $r_{n, a} (x) = \int_{a}^{x} \frac{f^{(n + 1)} (t)}{n!} (x - t)^{n} d t$

An interesting case is when it can be shown that the remainder term approaches zero as $n$ approaches infinity. In that case, we will be able to write $f (x)$ near $a$ as an infinite series in powers of $x - a$ . That series will be called the Taylor series for $f$ at $a$ . Taylor series will be covered in Chapter 4.

There are two other ways to write the remainder in Taylor's Theorem, still assuming that $f$ has a continuous derivative of order $(n + 1)$ . The Cauchy form of the remainder says that $r_{n, a} (x) = \frac{f^{(n + 1)} (ξ)}{n!} (x - t)^{n} (x - a), for some ξ between x and a$ while the Lagrange form of the remainder says $r_{n, a} (x) = \frac{f^{(n + 1)} (ξ)}{(n + 1)!} (x - a)^{n + 1}, for some ξ between x and a$

(Note, by the way, that if $f$ has a continuous $(n + 1)^{s t}$ derivative on some interval, then all of the derivatives of order less than $n$ must exist and be continuous on that interval. We say that $f$ is `` $n$ times continuously differentiable'' on the interval.)

(back to contents)