Chapter 9.3, Problem 21E

$\text{(a)}$

To determine

To find: The bound based on alternating series error bound.

Answer to Problem 21E

The bound for the maximum error is $\frac{1}{2^{4}4!}$ and the approximation by using $P_2(x)$ is too small.

Explanation of Solution

Given information:

  $P_2(x)=1-\left(x^2/2\right)$ is used to approximate $\cos x$ for $\left|x\right|<0.5$ .

Calculation:

Because the terms of the series meet all three of the characteristics indicated in Chapter 10—they alternate in sign, decrease in absolute value, and approach $0$ as a limit— establish a bound on the error based on the Alternating Series Error Bound method.

The truncation error is therefore smaller than the absolute value of the term at $x$ :

  $\text{error }\le \left|\frac{x^4}{4!}\right|$

Exercise $20(a)$ shows that the error of the $\cos (0.5)$ approximation using $P_2(x)$ is less than $\frac{1}{2^{4}4!}$ The error for such $x$ values is if $\left|x\right|<0.5$ .

So,

  $\begin{array}{c}\text{error }\le \left|\frac{x^4}{4!}\right|=\frac{|x{|}^4}{4!}<\frac{{0.5}^4}{4!}\\ =\frac{1}{2^{4}4!}\end{array}$

The approximation by using $P_2(x)$

  $\text{(}$ and likewise by using $\left.P_3(x)\right)$ underestimates the real value of $\text{cosx}$

  $\text{(}$ i.e. it is too small $\text{)}$ , and the approximation oscillates because the next term $\left(x^4/4!\right)$ is a positive integer.

Therefore, the required bound for the maximum error is $\frac{1}{2^{4}4!}$ and the approximation by using $P_2(x)$ is too small.

$\text{(b)}$

To determine

To find: The Lagrange error formula.

Answer to Problem 21E

The bound for the maximum error is $\frac{1}{2^{4}4!}$ and the approximation by using $P_2(x)$ is too small.

Explanation of Solution

Given information:

  $P_2(x)=1-\left(x^2/2\right)$ is used to approximate $\cos x$ for $\left|x\right|<0.5$ .

Calculation:

From exercise $20(b)$ , that $P_2(x)=P_3(x),$

But

  $P_4(x)$ differ,

thus

  $n=3$

And by using the formula the error at $x=0.5$ is:

  $\left|R_4(0.5)\right|\le \frac{1}{2^4\cdot 4!}$

If $\left|x\right|<0.5$ , then the error

for such $x$ values is:

  $\begin{array}{c}\left|R_4(x)\right|=\left|\frac{f^{(4)}(c)}{4!}{x}^4\right|\\ =\frac{\left|\cos c\right|}{4!}{x}^4<\frac{1\cdot {0.5}^4}{4!}\\ =\frac{1}{2^4\cdot 4!}\end{array}$

The approximation by using $P_2(x)$

  $\text{(}$ and likewise by using $\left.P_3(x)\right)$ underestimates the real value of $\text{cosx}$

  $\text{(}$ i.e. it is too small $\text{)}$ , and the approximation oscillates because the next term $\left(x^4/4!\right)$ is a positive integer.

Therefore, the required bound for the maximum error is $\frac{1}{2^{4}4!}$ and the approximation by using $P_2(x)$ is too small.