Problem 72 - Calculus Single Variable

Question

In later chapters, we will investigate the approximation of complicated functions by simpler ones. For instance, the function \(x \mapsto x\) is a good approximation to the function \(x \mapsto \sin (x) \cos \left(x^{2}\right)\) for \(|x|\) sufficiently small. Use a graphing utility to find an interval on which \(x\) approximates \(\sin (x) \cos \left(x^{2}\right)\) with an absolute error of at most 0.01

Step-by-Step Solution

Answer

The interval where \(|x - \sin(x)\cos(x^2)| \leq 0.01\) is approximately \([-0.085, 0.085]\).

Step 1: Define the Functions

Begin by defining the two functions involved in the problem. Let \( f(x) = x \) and \( g(x) = \sin(x)\cos(x^2) \). Our task is to find the interval where \(|f(x) - g(x)| \leq 0.01 \).

Step 2: Set Up the Inequality

The absolute error condition can be expressed as \(|x - \sin(x)\cos(x^2)| \leq 0.01 \). We need to determine for which values of \(x\) this inequality holds true.

Step 3: Graph the Functions

Use a graphing calculator or software to plot \( y = x \) and \( y = \sin(x)\cos(x^2) \) on the same set of axes. Visual inspection will help identify where they are close to each other.

Step 4: Analyze the Graph

Observe the graph to locate an interval where the two functions seem to overlap closely. For precise analysis, look for segments where the graphs are indistinguishable within the y-axis tolerance of 0.01 units.

Step 5: Use a Graphing Utility for Precise Calculation

Utilize a graphing utility to zoom into the identified regions and refine the values where \(|x - \sin(x)\cos(x^2)| \leq 0.01\). Tweak parameters until the specific interval is found.

Step 6: Confirm with Calculations

Perform calculations at the endpoints of identified interval to ensure \(|f(x) - g(x)| \leq 0.01\) holds true. This ensures that the interval boundaries are set accurately.

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that assist in visualizing mathematical functions. When dealing with function approximations, like the one in our exercise, a graphing utility can effectively help to compare two functions visually.

A graphing utility may be a software application or a handheld device, like a calculator that can plot graphs. These utilities allow you to input mathematical functions and automatically generate their graphs over a specified range of values. By doing so, they help you see the shape and behavior of functions.

In our exercise, you use a graphing utility to plot the line \(y = x\) and the curve \(y = \sin(x)\cos(x^2)\). By graphing them together, you can easily see the intervals over which these two functions approximate each other closely.

Even though visual inspection does provide great insight, the true power of a graphing utility is revealed in its ability to zoom in on specific parts of the graph. This zooming ability allows us to pinpoint where the functions coincide with precision, making it easier to find the intervals where the approximation is within the specified absolute error.

Absolute Error

Absolute error is a concept that measures how much one value differs from another. It is an essential idea when assessing how closely one function can approximate another.

In mathematical terms, absolute error is calculated as the absolute value of the difference between two values. For functions, the absolute error when approximating one with another is expressed as \(|f(x) - g(x)|\).

In the context of our exercise, we're interested in the absolute error between the functions \( f(x) = x \) and \( g(x) = \sin(x)\cos(x^2) \). The goal is to find an interval where this error remains below 0.01. This means that the values of \(f(x)\) should be no more than 0.01 units away from the values of \(g(x)\) within the entire interval.

Maintaining a small absolute error is crucial in function approximation as it ensures the function we use as a substitute does not significantly deviate from the target function. This ensures a high level of accuracy where needed.

Inequality Solving

Inequality solving is a crucial mathematical process that involves finding all the values of a variable that satisfy a given inequality condition.

In this exercise, we encounter the inequality \(|x - \sin(x)\cos(x^2)| \leq 0.01\). Solving inequalities involves finding the set of \(x\)-values that make the expression true, which in this case ensures that the absolute error between the approximating function and the target function does not exceed 0.01.

Working with inequalities often requires both analytical and graphical methods. Graphically, a graphing utility can provide insights into where functions overlap or closely match the needed criteria.

After identifying promising intervals graphically, analytical techniques can be used to refine solutions. This might involve solving algebraic equations that arise from setting the absolute error to the upper allowed limit, here being 0.01. By verifying the solutions with actual calculations, we confirm that the inequality condition is satisfied across the identified interval.