Problem 96 - 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. $$f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi]$$

Step-by-Step Solution

Answer

Computed plots, evaluated midpoints, and solved for \(x\) using the average function value from \(n=1000\).

Step 1: Plot the Function

Using a Computer Algebra System (CAS), plot the function \(f(x) = \sin^2 x\) over the interval \([0, \pi]\). This visual representation will help us understand the behavior of the function over the given range.

Step 2: Partition the Interval

Divide the interval \([0, \pi]\) into \(n = 100\), \(n = 200\), and \(n = 1000\) subintervals of equal length. For each case, compute the midpoints \(x_i\) of these subintervals where \(x_i = \frac{x_{i-1} + x_i}{2}\).

Step 3: Evaluate the Function at Midpoints

Calculate \(f(x_i) = \sin^2(x_i)\) for each midpoint \(x_i\) for the three partition cases (\(n = 100\), \(n = 200\), \(n = 1000\)). Store the resultant values for future calculations.

Step 4: Compute the Average Value

For each partition size, find the average value of the function by computing \(\text{Average} = \frac{1}{n} \sum_{i=1}^{n} f(x_i)\). In this step, ensure to calculate the average for \(n = 1000\), which will be used in the next part.

Step 5: Solve for x using Average Value

Using the average value calculated for \(n=1000\), solve the equation \(f(x) = \text{Average}\) using the CAS. This step involves finding all possible values of \(x\) such that \(\sin^2 x = \text{Average}\) within the interval \([0, \pi]\).

Key Concepts

Midpoint Rule

The **Midpoint Rule** is a numerical integration technique used to approximate the definite integral of a function. This method provides a simple yet effective way to estimate the area under a curve when finding an exact integral is challenging.

• The interval over which the function is integrated is divided into small subintervals.
• The function is evaluated at the midpoint of each subinterval, rather than at the endpoints.
• These values are averaged to approximate the integral.

By choosing the midpoint as the point of evaluation, we gain a more balanced representation of the function's behavior over each subinterval. This technique often gives better approximations than simply using the endpoints for many practical calculations.

Average Value of a Function

The **average value of a function** over a given interval provides a single representative value that summarizes the behavior of the function on that interval. To find this average value, follow these steps:

• Evaluate the function at specific points in the interval (like midpoints).
• Sum all these values together.
• Divide the total sum by the number of points (subintervals).

This gives an average that is useful for simplifying complex functions into more digestible terms. In our task, this technique has been crucial for simplifying the data obtained from evaluating the function at various points within the intervals.

Partitioning Intervals

**Partitioning intervals** is the initial step in applying methods like the Midpoint Rule for numerical integration. It involves breaking down a range into smaller, equal parts, which makes calculations more manageable.

• Select the interval of interest, such as \[0, \pi\].
• Decide the number of subintervals, for example, 100, 200, or 1000.
• Each subinterval will have an equal length, determined by dividing the total length of the interval by the number of subintervals.

Partitioning helps in gaining a finer resolution of the function's behavior, thus allowing for more accurate computations when determining areas or averages.

Computer Algebra System (CAS)

A **Computer Algebra System (CAS)** is instrumental in solving complex algebraic expressions, performing plots, and tackling integration challenges that are cumbersome by hand. These systems assist in:

• Automatically plotting functions over specified intervals.
• Executing large calculations needed for evaluating functions numerous times over subdivided intervals.
• Solving equations where manual methods would be tedious, especially when dealing with large datasets or complicated functions.

CAS tools, such as Mathematica or Maple, are especially beneficial for students and educators in demonstrating and learning mathematical concepts effectively. They save time and reduce the potential for human error in calculations.

Trigonometric Functions

**Trigonometric functions** are fundamental in mathematics, especially when dealing with periodic phenomena such as waves or circles. In our solution, the function we are working with is \(f(x) = \sin^2 x\). Understanding trigonometric functions involves:

• Recognizing that they repeat over specific intervals due to their periodic nature.
• Knowing their key properties, like symmetry and period (for sine, this is \[0, \pi\]).
• Applying these properties to simplify expressions, computations, and solve related equations.

Functions like \(\sin(x)\) appear often, and their transformations, such as squaring in \(\sin^2(x)\), highlight additional characteristics like amplitude and frequency, which are crucial in both pure and applied mathematical contexts.