11.11

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11.11 Applications of Taylor

Polynomials

Approximating Functions

by Polynomials

Approximating Functions by Polynomials

Suppose that f(x) is equal to the sum of its Taylor series at a:

The notation T_n(x) is used to represent the nth partial sum of this series and we can call it as it the nth-degree Taylor polynomial of f at a.

Thus

Approximating Functions by Polynomials

Since f is the sum of its Taylor series, we know that T_n(x) f(x) as n and so T_n can be used as an approximation to f:

f(x) ≈ T_n(x).

Notice that the first-degree Taylor polynomial T₁(x) = f(a) + f′(a)(x – a)

Approximating Functions by Polynomials

Notice also that T₁ and its derivative have the same values at a that f and f′ have. In general, it can be shown that the derivatives of T_n at a agree with those of f up to and

including derivatives of order n.

To illustrate these ideas let’s take another look at the graphs of y = e^x and its first few Taylor

polynomials, as shown in Figure 1.

Figure 1

Approximating Functions by Polynomials

The graph of T₁ is the tangent line to y = e^x at (0, 1); this tangent line is the best linear approximation to e^x near (0, 1).

The graph of T₂ is the parabola y = 1 + x + x²/2, and the graph of T₃ is the cubic curve y = 1 + x + x²/2 + x³/6, which is a closer fit to the exponential curve y = e^x than T₂.

The next Taylor polynomial T₄ would be an even better approximation, and so on.

Approximating Functions by Polynomials

The values in the table give a numerical demonstration of the convergence of the Taylor polynomials T_n(x) to the

function y = e^x. We see that when x = 0.2 the convergence is very rapid, but when x = 3 it is somewhat slower.

In fact, the farther x is from 0, the more slowly T_n(x) converges to e^x. When using a Taylor polynomial T_n to approximate a function f, we have to ask the questions:

How good an approximation is it?

Approximating Functions by Polynomials

How large should we take n to be in order to achieve a

desired accuracy? To answer these questions we need to look at the absolute value of the remainder:

| R_n(x)| = | f(x) – T_n(x)|

There are three possible methods for estimating the size of the error:

1. If a graphing device is available, we can use it to graph

|R_n(x)| and thereby estimate the error.

Approximating Functions by Polynomials

2. If the series happens to be an alternating series, we can use the Alternating Series Estimation Theorem.

3. In all cases we can use Taylor’s Inequality which says

that if then

Example 1

(a) Approximate the function by a Taylor polynomial of degree 2 at a = 8.

(b) How accurate is this approximation when 7 ≤ x ≤ 9?

Solution:

(a)

Example 1 – Solution

Thus the second-degree Taylor polynomial is

The desired approximation is

cont’d

Example 1 – Solution

(b) The Taylor series is not alternating when x < 8, so we can’t use the Alternating Series Estimation Theorem in this example.

But we can use Taylor’s Inequality with n = 2 and a = 8:

where | f′″(x)| ≤ M.

Because x ≥ 7, we have x^8/3 ≥ 7^8/3 and so

cont’d

Example 1 – Solution

Therefore we can take M = 0.0021. Also 7 ≤ x ≤ 9, so –1 ≤ x – 8 ≤ 1 and |x – 8| ≤ 1.

Then Taylor’s Inequality gives

Thus, if 7 ≤ x ≤ 9, the approximation in part (a) is accurate to within 0.0004.

cont’d

Approximating Functions by Polynomials

Let’s use a graphing device to check the calculation in

Example 1. Figure 2 shows that the graphs of and y = T₂(x) are very close to each other when x is near 8.

Figure 2

Approximating Functions by Polynomials

Figure 3 shows the graph of |R₂(x)| computed from the expression

We see from the graph that

|R₂(x)| < 0.0003 when 7 ≤ x ≤ 9.

Thus the error estimate from graphical methods is slightly better than the error estimate

from Taylor’s Inequality in this case.

Figure 3

Applications to Physics

Applications to Physics

Taylor polynomials are also used frequently in physics. In order to gain insight into an equation, a physicist often

simplifies a function by considering only the first two or three terms in its Taylor series.

In other words, the physicist uses a Taylor polynomial as an approximation to the function. Taylor’s Inequality can then be used to gauge the accuracy of the approximation.

The following example shows one way in which this idea is used in special relativity.

Example 3

In Einstein’s theory of special relativity the mass of an object moving with velocity v is

where m₀ is the mass of the object when at rest and c is the speed of light. The kinetic energy of the object is the

difference between its total energy and its energy at rest:

K = mc² – m₀c²

Example 3

(a) Show that when v is very small compared with c, this expression for K agrees with classical Newtonian physics: K = m₀v².

(b) Use Taylor’s Inequality to estimate the difference in these expressions for K when | v | ≤ 100 m/s.

Solution:

(a) Using the expressions given for K and m, we get

cont’d

Example 3 – Solution

With x = –v²/c², the Maclaurin series for (1 + x)^–1/2 is

most easily computed as a binomial series with k = (Notice that | x | < 1 because v < c.) Therefore we have

and

cont’d

Example 3 – Solution

If v is much smaller than c, then all terms after the first are very small when compared with the first term. If we omit them, we get

(b) If x = –v²/c², f(x) = m₀c² [(1 + x)^–1/2 – 1], and M is a

number such that | f″(x)| ≤ M, then we can use Taylor’s Inequality to write

cont’d

Example 3 – Solution

We have f″(x) = m₀c²(1 + x)^–5/2 and we are given that

| v | ≤ 100 m/s, so

Thus, with c = 3 × 10⁸ m/s,

So when |v | ≤ 100 m/s, the magnitude of the error in using the Newtonian expression for kinetic energy is at

cont’d