Chapter 4, Problem 14PS

a.

To determine

Graph the arctangent function and its polynomial approximation by using graphing utility.

Answer to Problem 14PS

  

Explanation of Solution

Given information:

Using calculus, it can be shown that the arctangent function can be approximated by the polynomial

  $\arctan x\approx x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}$

Where $x$ is in radians.

Use a graphing utility to graph the arctangent function and its polynomial approximation in the same viewing window. How do the graphs compare?

Calculation:

Here, we will consider the approximation of the arctangent function:

  $\arctan \left(x\right)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}$

Now, we will use graphing utility to compare the arctangent function and its approximation as stated above.

We will press $\boxed{Y=}$ button on graphing calculator and input the two functions:

  

Now, we will use $\boxed{GRAPH}$ button to get the graphs of both the functions:

  

Hence, from above we can see that the approximation and the actual function of arctangent follow the same trend but the values are differing by minute values.

b.

To determine

Find the changes in the accuracy when an additional term is added.

Answer to Problem 14PS

  $\boxed{Itismoreaccuratethantheoneinpart\left(a\right)}$

Explanation of Solution

Given information:

Using calculus, it can be shown that the arctangent function can be approximated by the polynomial

  $\arctan x\approx x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}$

Where $x$ is in radians.

Study the pattern in the polynomial approximation of the arctangent function and predict the next term. Then repeat part $(a)$ . How does the accuracy of the approximation change when an additional term is added?

Calculation:

Here, we will consider the following the approximation:

  $\arctan \left(x\right)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}$

Now, there is a common trend following in the above series that alternatively there is one positive and one negative and the powers are increasing by the factor of $2$ and the denominator are same. Thus, the next term will be $\left(+\frac{x^9}{9}\right)$ so that the approximation becomes:

  $\arctan \left(x\right)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}$

Now, we will use graphing utility to compare the arctangent function and its approximation stated above.

Now, we will press $\boxed{Y=}$ button on graphing calculator and input the two functions:

  

Now, we will press $\boxed{GRAPH}$ button to obtain the graph:

  

Hence, we can see that the above graph is more accurate than the one drawn in part $\left(a\right)$ as a result of which both the graph almost coincide with each other.