Problem 9 - Calculus

Question

(a) Using a calculator, make a table of values to four decimal places of \(\sin x\) for $$x=-0.5,-0.4, \ldots,-0.1,0,0.1, \ldots, 0.4,0.5$$ (b) Add to your table the values of the error \(E_{1}=\) \(\sin x-x\) for these \(x\) -values. (c) Using a calculator or computer, draw a graph of the quantity \(E_{1}=\sin x-x\) showing that $$\left|E_{1}\right|<0.03 \text { for }-0.5 \leq x \leq 0.5$$

Step-by-Step Solution

Answer

The error \(E_1 = \sin x - x\) is less than 0.03 for \(-0.5 \leq x \leq 0.5\).

Step 1: Creating the Table of Sine Values

First, use a calculator to find the sine of each given value of \(x\). Start with \(x = -0.5\) and proceed to \(x = 0.5\), using increments of 0.1. Ensure that each sine value is rounded to four decimal places. For example, for \(x = -0.5\), \(\sin(-0.5) \approx -0.4794\). Record these values in a table.

Step 2: Calculating Error E1

For each value of \(x\), calculate the error \(E_1 = \sin x - x\). Subtract the value of \(x\) from its corresponding sine value. Record this error in a new column next to the sine values in the table.

Step 3: Graphing the Error

Using graphing software or a calculator, plot the values of \(E_1\) against \(x\). This involves creating a set of points \((x, E_1)\) for each value of \(x\) in your table. Observe how the graph behaves within the interval \(-0.5 \leq x \leq 0.5\). Verify if the magnitude of \(E_1\) stays below 0.03 across this interval.

Key Concepts

Error Analysis

Error analysis in mathematics is used to understand how closely our computations approximate real values. For this exercise, we are examining the error between the actual sine function and the identity line. The error term, denoted as \(E_1 = \sin x - x\), allows us to evaluate the difference between the sine of an angle \(x\) and \(x\) itself.

To perform error analysis:

• First, calculate \(\sin x\) for each specified value of \(x\), like \(-0.5, -0.4, \ldots, 0.5\).
• Subtract \(x\) from \(\sin x\) to get \(E_1\). This will help in assessing how much \(\sin x\) deviates from the straight line \(y = x\).
• The goal is to determine the range of \(x\) over which this error remains small, specifically within the bounds \(-0.5 \leq x \leq 0.5\) and is less than \(0.03\).

Error analysis is crucial for verifying mathematical models and ensuring they predict or match real-world data accurately, particularly in calculus where functions often approximate other functions.

Graphing Functions

Graphing functions is a visual technique for displaying relationships between variables. In this exercise, plotting the error \(E_1 = \sin x - x\) helps us understand the behavior of \(E_1\) over the interval \(-0.5 \leq x \leq 0.5\).

Steps for creating the graph:

• Plot each calculated point from your table on a graph, with \(x\) as the horizontal axis and \(E_1\) as the vertical axis.
• As you chart these points, you'll observe whether the graph maintains a small consistent deviation, specifically verifying \(|E_1| < 0.03\).
• This visual can quickly show where the sine function closely mimics its linear approximation, which is \(x\), across the interval.

Graphing is not only useful for understanding mathematical concepts like this but also serves as a powerful tool to spot trends, verify calculations, and illustrate proofs in calculus.

Calculus Problem Solving

Solving calculus problems involves understanding both the theoretical and practical applications of functions, derivatives, and integrals. This exercise specifically aligns with finding how close a function (sine function) approximates a linear function in a small neighborhood around zero.

Problem-solving tips:

• When tasked with a problem like this, start by calculating or approximating the functions involved, here being \(\sin x\) and \(x\).
• Next, identify the error by determining how much the actual function and its approximation diverge.
• Follow through with a visual or numerical verification to confirm patterns or trends—you can use a plot as in the error analysis phase.
• Such exercises often relate to Taylor series expansion or local linear approximation in calculus, stressing how calculus concepts predict or simplify complex behaviors of functions.

Concrete practice in these problem-solving steps develops intuition and efficiency when faced with larger calculus challenges, often requiring layers of approximation and estimation.