Question

47\. Challenge Problem A Geometric Series In calculus, you will learn that certain functions can be approximated by polynomial functions. We will explore one such function now. (a) Using a graphing utility, create a table of values with $$ \begin{array}{l} Y_{1}=f(x)=\frac{1}{1-x} \text { and } \\\ Y_{2}=g_{2}(x)=1+x+x^{2}+x^{3} \end{array} $$ for \(-1

Short answer

As more polynomial terms are added, the approximation of \(\frac{1}{1-x}\) becomes more accurate, especially near \(x = 0\).

Step-by-step solution

- Define the functions

Define the functions to be used: Function 1: \(Y_1 = f(x) = \frac{1}{1-x}\)Function 2 (for part a): \(Y_2 = g_2(x) = 1 + x + x^2 + x^3\)Function 2 (for part b): \(Y_2 = g_3(x) = 1 + x + x^2 + x^3 + x^4\)Function 2 (for part c): \(Y_2 = g_4(x) = 1 + x + x^2 + x^3 + x^4 + x^5\)

- Graph the functions for part (a)

Using a graphing utility, create a table of values for \(Y_1 = \frac{1}{1-x}\) and \(Y_2 = 1 + x + x^2 + x^3\) for \(-1 < x < 1\) with \(\Delta \text{Tbl} = 0.1\). Record the values for comparison.

- Graph the functions for part (b)

Create a table of values for \(Y_1 = \frac{1}{1-x}\) and \(Y_2 = 1 + x + x^2 + x^3 + x^4\) for \(-1 < x < 1\) with \(\Delta \text{Tbl} = 0.1\). Record the values for comparison.

- Graph the functions for part (c)

Create a table of values for \(Y_1 = \frac{1}{1-x}\) and \(Y_2 = 1 + x + x^2 + x^3 + x^4 + x^5\) for \(-1 < x < 1\) with \(\Delta \text{Tbl} = 0.1\). Record the values for comparison.

- Analyze the data

Compare the tables of values from parts (a), (b), and (c). Observe how the approximations (polynomials) get closer to the actual function \(\frac{1}{1-x}\) as more terms are added. Note any specific values of \(x\) where the polynomials provide better approximations.

Key concepts

polynomial approximation

Polynomial approximation is a powerful tool in calculus that helps to represent complex functions with simpler polynomial functions. It involves expressing a function as a sum of polynomial terms. This method can help in simplifying calculations and understanding the behavior of functions.
In the given exercise, we approximate the function \( f(x) = \frac{1}{1-x} \) using different polynomials with increasing degrees:

• For part (a), we use \( g_2(x) = 1 + x + x^2 + x^3 \).
• For part (b), we use \( g_3(x) = 1 + x + x^2 + x^3 + x^4 \).
• For part (c), we use \( g_4(x) = 1 + x + x^2 + x^3 + x^4 + x^5 \).

Observing these polynomials, we see that as we add more terms (higher degree polynomials), they become better approximations of the original function \( f(x) = \frac{1}{1-x} \). We notice that the approximation becomes more accurate for values of \(x\) near 0. This concept plays a critical role in numerical analysis and various applications in engineering and science.

graphing utility

A graphing utility is a valuable tool for visualizing and comparing functions. It helps to create tables of values, plot graphs, and analyze the behavior of functions.
In this exercise, we used a graphing utility to create tables of values for the functions \( f(x) = \frac{1}{1-x} \) and the polynomial approximations. Here is what we did:

• For each polynomial approximation in parts (a), (b), and (c), we set the range for \( -1 < x < 1 \) with a step size of 0.1.
• We then recorded the values of \( f(x) \) and the corresponding polynomial approximation \( g_2(x) \), \( g_3(x) \), and \( g_4(x) \).

By comparing the tables of values, we observed that as more terms are added, the polynomial approximation becomes more precise, especially for values of x close to 0. This hands-on approach using a graphing utility makes it easier to understand how polynomial approximations work and how they improve.

calculus

Calculus is the branch of mathematics that deals with continuous change and includes the study of limits, derivatives, integrals, and infinite series. In this exercise, we touched upon the concept of infinite series and polynomial approximations.
The function \( f(x) = \frac{1}{1-x} \) can be expressed as an infinite geometric series:
\( f(x) = 1 + x + x^2 + x^3 + x^4 + ... \)
By truncating this series to a finite number of terms, we get polynomial approximations. This is crucial in calculus because it helps in approximating functions that are difficult to work with.
In practical applications, we often use the first few terms of a series to get an approximate value that is good enough for the task at hand. As seen in this exercise, the polynomial approximation improves with the addition of more terms.
Understanding these basic calculus concepts is fundamental for further studies in mathematics, physics, and engineering.