Problem 39 - Thomas Calculus in SI Units

Question

Use a CAS to perform the following steps for the given graph of the function over the closed interval. a. Plot the curve together with the polygonal path approximations for \(n=2,4,8\) partition points over the interval. (See Figure 6.22.) b. Find the corresponding approximation to the length of the curve by summing the lengths of the line segments. c. Evaluate the length of the curve using an integral. Compare your approximations for \(n=2,4,8\) with the actual length given by the integral. How does the actual length compare with the approximations as \(n\) increases? Explain your answer. $$f(x)=\frac{x-1}{4 x^{2}+1}, \quad-\frac{1}{2} \leq x \leq 1$$

Step-by-Step Solution

Answer

Increasing partition points \( n \) improves approximation of curve length.

Step 1: Define the Function

The function given is \( f(x) = \frac{x-1}{4x^2+1} \). We'll be considering this function over the interval \(-\frac{1}{2} \leq x \leq 1\). This will help us set up the graph and perform further calculations.

Step 2: Plot the Curve and Polygonal Approximations

Using a Computer Algebra System (CAS), plot the graph of the function \( f(x) \) over the given interval. Then, create polygonal path approximations by choosing partition points for \( n = 2, 4, 8 \), resulting in piecewise linear segments that approximate the curve.

Step 3: Calculate Approximate Lengths

For each value of \( n \), calculate the sum of lengths of each linear segment. This involves finding the Euclidean distance between consecutive partition points on the graph of the function. Sum these distances to obtain an approximation for the length of the curve for each value of \( n \).

Step 4: Calculate Exact Length of Curve via Integral

The exact length of the curve can be determined by evaluating the integral: \[ L = \int_{-\frac{1}{2}}^{1} \sqrt{1+ \left(\frac{d}{dx}\left(\frac{x-1}{4x^2+1}\right)\right)^2} \, dx \]. First, find the derivative \( \frac{d}{dx}(f(x)) \), then substitute into the integral and evaluate to obtain the exact length of the curve.

Step 5: Compare Approximations to Exact Length

Compare the approximate lengths obtained from the polygonal approximations for \( n = 2, 4, 8 \) with the exact length calculated via the integral. Note how the approximation improves (approaches the true length) as \( n \) increases because finer partitions create a closer approximation to the actual curve.

Step 6: Explanation of Findings

Explain that as \( n \) increases, the polygonal approximations tend to become more accurate, converging to the actual length of the curve. This occurs because increasing \( n \) results in more, smaller line segments that fit the curve more closely.

Key Concepts

Polygonal Path Approximation

Polygonal path approximation is a method to approximate a curve by connecting a series of straight lines between a set of points on the graph. Imagine trying to walk along a wavy path; you could ask a friend to place evenly spaced flags along the path and simply walk in straight lines between them. These lines form a polygonal path approximation of the wavy curve.

In mathematical terms, we choose partition points on the interval of interest, creating segments that form a zigzag along the function. For example, if a function is defined on the interval \([-\frac{1}{2}, 1]\), picking partition points based on the number \ n = 2, 4, \ or \ 8 \ creates different levels of approximation. The more partitions, the more lines you have, and the closer the approximation will be to the actual curve.

This method is simple yet effective. It provides a visual understanding and a practical way to estimate curve lengths, which leads us to the need for calculating those segment lengths using Euclidean distance.

Integral Calculus

Integral calculus allows us to find the exact length of a curve by integrating over the interval where the function is defined. This process considers the minute changes along the curve to generate a precise measure of its length.

To calculate the length of a curve defined by a function like \ f(x) = \frac{x-1}{4x^2+1} \, we first find the derivative of the function. This represents the rate at which the function's output changes with respect to changes in input. The derivative is squared and added to 1 inside a square root to account for all the infinitesimal "straight segments" along the curve.

Here's the integral formula used:

• \ L = \int_{-\frac{1}{2}}^{1} \sqrt{1+ \( \frac{d}{dx}\left(\frac{x-1}{4x^2+1}\right) \)^2} \, dx \ represents the calculation of the exact length.

This method provides the real length without estimations, unlike polygonal approximation, and serves as a perfect benchmark to compare our approximations.

Euclidean Distance Calculation

Euclidean distance calculation helps us find the straight-line distance between two points in the Cartesian coordinate system, which is crucial when you're constructing the polygonal path approximation of a curve.

The formula for calculating Euclidean distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

• \ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \ .

This distance corresponds to the length of each segment between partition points chosen along the function's path.

In our scenario, for each partition number like \( n = 2, 4, 8 \), we calculate the Euclidean distance for each segment of the polygonal path approximation. Summing these segment distances gives an estimate for the total curve length. As you increase \ n \, the approximation improves due to smaller distances between points, which more accurately follow the curve's shape.