Problem 97 - University Calculus: Early Transcendentals

Question

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right]$$

Step-by-Step Solution

Answer

Graph the function, partition the interval, evaluate at midpoints, compute average, solve for x.

Step 1: Plot the Function

First, input the function \( f(x) = x \sin\left(\frac{1}{x}\right) \) in a CAS and set the interval to \( \left[ \frac{\pi}{4}, \pi \right] \). Plot the graph of this function over the specified interval. Observe the behavior and key features of the graph, such as zero crossings, maxima, or any oscillations.

Step 2: Partition the Interval

Divide the interval \( \left[ \frac{\pi}{4}, \pi \right] \) into \( n = 100, 200, \) and \( 1000 \) subintervals. For each partition, calculate the midpoint of each subinterval. The midpoint for the \( i^{\text{th}} \) subinterval is found using the formula \( x_i = \frac{x_{i-1} + x_i}{2} \), where \( x_{i-1} \) and \( x_i \) are the endpoints of the \( i^{\text{th}} \) subinterval.

Step 3: Evaluate the Function at Midpoints

For each partition of \( n = 100, 200, \) and \( 1000 \) subintervals, evaluate the function \( f(x) = x \sin\left(\frac{1}{x}\right) \) at each midpoint calculated in the previous step. Record these function values.

Step 4: Calculate the Average Value

Sum all the function values obtained for each partition with \( n = 100, 200, \) and \( 1000 \). Divide the sum by the number of subintervals to find the average value of the function. This is calculated as:\[ \text{average value} = \frac{1}{n} \sum_{i=1}^{n} f\left( x_i \right) \]

Step 5: Solve the Equation for x

Use the average value obtained from the \( n=1000 \) partition to solve the equation \( f(x) = \text{average value} \). Substitute the average value into the function \( f(x) \) and find the solutions for \( x \) numerically or graphically using a CAS.

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical method used in calculus to estimate the integral of a function over a given interval. It involves partitioning the interval into smaller subintervals, then evaluating the function at the midpoint of each subinterval.

By using these midpoints, we can approximate the total area under the curve. This method provides a balance between accuracy and computational simplicity, making it a preferred choice for many practical calculations.

Let's consider how to apply this with the function given as \( f(x) = x \sin\left( \frac{1}{x} \right) \) over the interval \( \left[ \frac{\pi}{4}, \pi \right] \). First, the interval is divided into \(n\) equal subintervals. We calculate the midpoint \( x_i \) of each subinterval using the formula:

• \( x_i = \frac{x_{i-1} + x_i}{2} \)

At each of these midpoints, we evaluate the function \( f(x) \) which will later help in calculating the total area or average value of the function.

The Midpoint Rule is particularly advantageous because it minimizes error more effectively than other numerical integration methods such as the rectangular method, especially when the function is smooth over the given interval.

Average Value of a Function

To find the average value of a function across an interval, we sum up all the function values at various points in that interval and divide by the number of points. This helps us understand the function's behavior across the interval.

The average value formula for a continuous function \( f(x) \) over the interval \( [a, b] \) is given by:

• \( \text{Average Value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \)

In numerical calculations, like the exercise with the function \( f(x) = x \sin\left( \frac{1}{x} \right) \), we approximate this by computing the average of function values from the midpoints calculated in each subinterval. This process entails:

• Summing up all evaluated function values at midpoints for a chosen partition \( n \).
• Dividing the sum by the total number of subintervals \( n \).

This calculation provides a single value representing how the function "averages" across the specified interval. In our example, for the partition \( n = 1000 \), we find this average value and use it to further analyze the function's behavior.

Partitioning Intervals

Partitioning intervals is a critical step in numerical integration techniques such as the Midpoint Rule. It involves dividing a larger interval into smaller, equal-length subintervals.

This allows for more manageable calculations and improves the precision of numerical estimates.

In the given exercise, the interval \( \left[ \frac{\pi}{4}, \pi \right] \) is divided into \( n = 100, 200, \) and \( 1000 \) smaller sections. This systematic division is key to implementing many numerical methods correctly.

When partitioning, each subinterval's length \( \Delta x \) can be computed as follows:

• \( \Delta x = \frac{b-a}{n} \)

Where \( a = \frac{\pi}{4} \) and \( b = \pi \) define the interval's endpoints. Each subinterval will have its midpoint, and the function will be evaluated precisely at these midpoints, contributing to more accurate approximations of integrals and averages.

This method is robust and widely used in computational mathematics, enabling complex functions to be processed efficiently.

Computer Algebra Systems (CAS)

Computer Algebra Systems (CAS) are powerful tools that assist mathematicians, engineers, and scientists in dealing with various mathematical computations. These systems provide functionalities to perform symbolic and numerical calculations efficiently.

For the exercise in question, a CAS is utilized to:

• Plot the function \( f(x) = x \sin(\frac{1}{x}) \) over the interval \( \left[ \frac{\pi}{4}, \pi \right] \).
• Partition the interval and calculate midpoints.
• Evaluate the function at these points with high precision.
• Compute the average values swiftly and accurately.
• Solve equations involving computed values.

Using CAS, these tasks can be automated, significantly reducing the time and effort required compared to manual computation.

Students and professionals often employ CAS for complex calculations due to its accuracy and speed. Popular systems include Mathematica, Matlab, and Maple, each offering unique features and capabilities that cater to a wide range of mathematical operations.

CAS not only helps in obtaining results but also enhances learning by allowing users to visualize and manipulate mathematical concepts dynamically.