Problem 208 - Precalculus

Question

For the following exercises, find the exact value without the aid of a calculator. Graph the function \(f(x)=\frac{x}{1}-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}\) on the interval \([-1,1]\) and compare the graph to the graph of \(f(x)=\sin x\) on the same interval. Describe any observations.

Step-by-Step Solution

Answer

The polynomial approximates \(\sin x\) well near the origin, but differences grow at \([-1, 1]\).

Step 1: Understand the Function

The function given is a polynomial function: \[f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}\]This polynomial is a partial Taylor series expansion of \(\sin x\) centered at 0. The general form of the Taylor series for sine is:\[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \cdots\] The given function uses terms through \(x^7\).

Step 2: Graph the Polynomial Function

Graph the function \(f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}\) on the interval \([-1, 1]\). This involves plotting several points, computing the values of \(f(x)\) at each point, and then drawing a smooth curve through these points.

Step 3: Graph the Sine Function

Graph the function \(f(x) = \sin x\) on the same interval \([-1, 1]\). This function is the well-known sine wave which repeats every \(2\pi\) and has the characteristic 'S' shape within \([-1, 1]\).

Step 4: Compare the Graphs

Observe both graphs on the same interval and notice their similarities. The polynomial \(f(x)\) closely approximates \(\sin x\) within \([-1, 1]\), especially near the origin. As we move towards the boundaries \(-1\) and \(1\), the minor differences between the series expansion and the actual sine function become more noticeable.

Step 5: Describe Observations

The polynomial \(f(x)\) approximates \(\sin x\) quite accurately within the small interval of \([-1, 1]\). Near the origin, both functions almost coincide perfectly because the approximation is a truncated Taylor series of \(\sin x\). The further we move from the origin, the approximation slightly deviates as higher-order terms not included in the polynomial affect \(\sin x\).

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients organized in terms of powers, usually written in descending order of exponents. These functions can be as simple as linear functions, like \(f(x) = x\), or more complex like the one in our exercise:

• \(f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}\)

This specific polynomial is part of a Taylor series expansion for \(\sin x\), where each term uses factorial notation to adjust the power of \(x\). Factorials grow quickly, contributing progressively smaller terms, helping accurately shape the polynomial to imitate more complex functions like trigonometric ones. Understanding the behavior of polynomial functions is crucial since they serve as the building blocks for approximating other functions across various disciplines. By using higher-degree polynomials, or adding more terms, one can achieve an even closer approximation.

Sine Function

The sine function, denoted as \(\sin x\), is a periodic function that arises frequently in trigonometry. It maps an angle to a coordinate on the unit circle, making it fundamental to not only mathematics but also physics and engineering. The sine function has particular characteristics:

• It has a period of \(2\pi\), meaning the pattern repeats every \(2\pi\) radians.
• Its range is bounded between -1 and 1.
• Its graph produces a smooth, oscillating wave pattern.

When thinking about functions such as \(\sin x\), it's interesting to note how the Taylor series for \(\sin x\) is developed. It involves finding derivatives of \(\sin x\), evaluating them at 0, and forming an infinite polynomial. By truncating this series at a finite number, we create polynomials like the one in our exercise that closely resemble \(\sin x\) over small intervals such as \([-1, 1]\). Understanding the sine function's geometry and properties is vital for recognizing how polynomial approximations can be used effectively.

Graph Comparison

Comparing graphs is an essential technique in understanding the accuracy and limitations of approximations like the Taylor series. For our exercise, we have two graphs to compare:

• \(f(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}\)
• \(f(x) = \sin x\)

Both are plotted over the interval \([-1, 1]\), and this limited interval allows us to see how well the polynomial approximates the sine curve. Near the origin, both graphs nearly overlap, indicating a strong approximation due to the lower-order terms predominance. However, as the input value moves further from zero, the polynomial gradually starts diverging from the actual sine wave. This happens because higher-order terms, omitted from the polynomial, become significant and impact the sine curve. This comparison offers valuable insight into how polynomial functions can be used to approximate other functions, providing a balance between simplicity and accuracy depending on the interval of consideration.