Question

Errors in approximations Suppose you approximate \(\sin x\) at the points \(x=-0.2,-0.1,0.0,0.1,\) and 0.2 using the Taylor polynomials \(p_{3}=x-x^{3} / 6\) and \(p_{5}=x-x^{3} / 6+x^{5} / 120 .\) Assume that the exact value of \(\sin x\) is given by a calculator. a. Complete the table showing the absolute errors in the approximations at each point. Show two significant digits. $$\begin{array}{|c|l|l|} \hline x & \text { Error }=\left|\sin x-p_{3}(x)\right| & \text { Error }=\left|\sin x-p_{5}(x)\right| \\ \hline-0.2 & & \\ \hline-0.1 & & \\ \hline 0.0 & & \\ \hline 0.1 & & \\ \hline 0.2 & & \\ \hline \end{array}$$ b. In each error column, how do the errors vary with \(x\) ? For what values of \(x\) are the errors the largest and smallest in magnitude?

Short answer

Question: Calculate the absolute errors for each approximation of 𝑠𝑖𝑛𝑥 at the given points 𝑥=−0.2, −0.1, 0.0, 0.1, and 0.2 using the Taylor polynomials 𝑝3(𝑥)=𝑥−𝑥𝑐3/6 and 𝑝5(𝑥)=𝑥−𝑥𝑐3/6+𝑥𝑐5/120, and analyze how the errors vary with 𝑥. Identify where the errors are largest and smallest in magnitude.

Step-by-step solution

Find the sine values of the given points using a calculator

Use a calculator to find the sine values of the given points \(x=-0.2,-0.1,0.0,0.1,\) and \(0.2\).

Compute the Taylor polynomials approximations at the given points

Use the Taylor polynomials \(p_{3}(x)\) and \(p_{5}(x)\) to approximate the sine values of the given points. For each point \(x\), compute \(p_{3}(x) = x - \frac{x^3}{6}\) and \(p_{5}(x) = x - \frac{x^3}{6} + \frac{x^5}{120}\).

Calculate the absolute errors

Now, we'll find the absolute errors for each point. Absolute error is given by the formula \(| \sin x - p_n(x) |\), where \(n\) is the degree of the Taylor polynomial (\(3\) or \(5\)). For each point and each Taylor polynomial, find the absolute error and round it to two significant digits. Now complete the table: $$\begin{array}{|c|l|l|} \hline x & \text { Error }=\left|\sin x-p_{3}(x)\right| & \text { Error }=\left|\sin x-p_{5}(x)\right| \\ \hline-0.2 & & \\ \hline-0.1 & & \\ \hline 0.0 & & \\ \hline 0.1 & & \\ \hline 0.2 & & \\ \hline \end{array}$$

Analyze the errors

Observe the error values in both columns and determine how the errors vary with \(x\). Identify, for each Taylor polynomial approximation, the values of \(x\) where the errors are largest and smallest in magnitude.

Key concepts

Sin Function Approximation

The sine function, often denoted as \(\sin x\), is a fundamental mathematical function with a periodic wave-like pattern. Approximating \(\sin x\) using Taylor polynomials can provide a useful estimate of this function’s value near a given point. This technique is especially powerful when computing the sine function with limited computational resources. For example, in this exercise, we're using Taylor polynomials \(p_3(x)\) and \(p_5(x)\) to approximate \(\sin x\) at the points \(x = -0.2, -0.1, 0.0, 0.1,\) and \(0.2\).

Each Taylor polynomial uses a series expansion derived from the function's derivatives at a specific point, usually zero. This means that for small values of \(x\), the approximations \(p_3(x) = x - \frac{x^3}{6}\) and \(p_5(x) = x - \frac{x^3}{6} + \frac{x^5}{120}\) become particularly effective. Knowing how these expansions work can greatly help to understand their applications.

Absolute Error Calculation

When we approximate a function, it’s important to measure how close our approximation is to the actual value. This is where absolute error comes into play. Absolute error is the difference between the exact value and the approximated value, expressed as an absolute value to ensure it’s always positive.

In this exercise, the absolute errors are calculated using the formula \(|\sin x - p_n(x)|\), where \(p_n(x)\) is our Taylor polynomial approximation. Performing this calculation for each value of \(x\) gives us a sense of the accuracy of the approximation.

For instance, using a calculator to find \(\sin(-0.1)\) and subtracting it from its approximation \(p_3(-0.1)\) results in a specific error value, which must then be rounded to two significant digits for precision consistency.

Taylor Polynomials

Taylor polynomials, named after mathematician Brook Taylor, provide a method to approximate functions using polynomials. They employ derivatives of a function at a single point to create polynomials that closely estimate the function around that point.

• Third-Degree Polynomial \(p_3(x)\): This uses the first two non-zero terms of the series, \(x\) and \(-\frac{x^3}{6}\).
• Fifth-Degree Polynomial \(p_5(x)\): This extends the approximation with an additional term \(+\frac{x^5}{120}\).

Since these polynomials include higher degrees of \(x\), they often provide a closer match to \(\sin x\), especially as more terms are added. The degree of the polynomial directly influences the range and accuracy of the approximation. With the degree increasing, like from 3 to 5, we generally get smaller errors, meaning closer approximations for larger \(|x|\) values.

Function Approximation Errors

Approximating functions often comes with errors, and understanding these errors helps us improve the approximations. Two main factors affect the errors: the degree of the Taylor polynomial and the point of approximation.

Generally, the error becomes smaller when a higher-degree polynomial is used. This is evident when comparing \(p_3(x)\) to \(p_5(x)\) across our selected \(x\) values.

• Errors are naturally minimized around \(x = 0\) due to the nature of Taylor series expansions centered at this point.
• Largest errors tend to occur as \(x\) moves further away from the center (0 in our case), as the series represents the function less accurately.

This exercise shows how the choice of polynomial impacts the size of approximation error, prompting considerations on balance between polynomial complexity and desired accuracy for different scenarios.