Problem 90 - Thomas Calculus

Question

In Exercises \(89-92,\) 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

Plot, evaluate at midpoints, average, and solve equation using a CAS.

Step 1: Plot the Function

Use a computer algebra system (CAS) to plot the function \( f(x) = \sin^2(x) \) over the interval \([0, \pi]\). This will provide a visual understanding of the function within the specified bounds.

Step 2: Calculate Midpoints for Subintervals

Divide the interval \([0, \pi]\) into \( n = 100, 200, \) and \( 1000 \) subintervals. For each subinterval, calculate the midpoint using the formula \( x_i = a + (i - 0.5)\Delta x \) where \( \Delta x = \frac{b-a}{n} \) and \( a = 0, b = \pi \). Evaluate \( f(x) \) at these midpoints.

Step 3: Average Value Calculation

For each partition \( n = 100, 200, 1000 \), compute the average value of \( f(x) \) at the midpoints. Use the formula: \[\text{Average Value} = \frac{1}{n}\sum_{i=1}^{n}f(x_i)\]where \( x_i \) are the midpoints calculated in the previous step.

Step 4: Solve for f(x) = Average Value

For \( n = 1000 \), use the average value computed in the previous step and solve \( f(x) = \sin^2(x) = \text{Average Value} \). Use a CAS to find the values of \( x \) that satisfy this equation.

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical integration technique used to approximate the area under a curve. This method involves dividing the interval over which the function is integrated into equal subintervals and then calculating the value of the function at the midpoints of these subintervals. The Midpoint Rule gives a more accurate approximation compared to other methods like the Left and Right Riemann Sums because it reduces the error by considering the central point of each subinterval rather than the endpoints.

Here's how it works:

• First, choose the number of subintervals, denoted as \(n\).
• Then, compute the width of each subinterval using the formula \(\Delta x = \frac{b-a}{n}\), where \(a\) and \(b\) are the bounds of the integration interval.
• Next, find the midpoint of each subinterval: \(x_i = a + (i - 0.5)\Delta x\).
• Evaluate the function at these midpoints and sum them up: \(\sum_{i=1}^{n}f(x_i)\).
• The approximate integral is then given by the product of this sum with \(\Delta x\).

By concentrating on these midpoints, the Midpoint Rule reduces discrepancies and achieves a better estimate of the integral, especially when dealing with smooth curves.

Average Value Theorem

The Average Value Theorem deals with finding the average, or mean, value of a function over a certain interval. When you have a continuous function, the theorem states that there exists at least one point within the interval where the function's value equals its average value. This can also be thought of as spreading the function's total output evenly across the interval.

The average value of a function \( f(x) \) over the interval \([a, b]\) is computed using:

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

However, when using numerical methods like in the exercise, where the exact integral may be unknown or hard to compute by hand:

• We instead calculate an approximate average using the function values at chosen points, such as midpoints of divided subintervals: \(\frac{1}{n}\sum_{i=1}^{n}f(x_i)\).

This approach helps us find a practical and useful estimate of the average value, which can be crucial when detailed calculations about the function's behavior are needed.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a sophisticated tool used to perform symbolic and numeric computations. In mathematics, a CAS allows you to solve complex equations, perform integrals, differentiate functions, and even graph functions with high precision. This makes it an invaluable resource for tackling problems that would be tedious and error-prone if done manually.

Some key functionalities include:

• **Equation Solving:** Solve algebraic and differential equations.
• **Integration and Differentiation:** Perform symbolic integration and differentiation.
• **Graph Plotting:** Visualize functions over specified intervals, aiding comprehension of their behaviors.
• **Numerical Computations:** Evaluate functions at specified points or intervals.

In practical terms, using a CAS in an exercise involving multiple calculations and data points, like evaluating a function at many subinterval midpoints, greatly simplifies the work. It can automatically calculate and compile results efficiently, ensuring accuracy and saving a considerable amount of time for the user.

Trigonometric Functions

Trigonometric functions like \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\) play a pivotal role in various mathematical applications. They are essential tools for analyzing periodic phenomena and are characterized by their wave-like properties.

For the function \( f(x) = \sin^2(x) \) specifically:

• This represents the square of the sine function, which retains a sinusoidal pattern but with a wavelength that is half the size of the original sine function.
• Over the interval \([0, \pi]\), \(\sin^2(x)\) has a characteristic shape; it rises from zero to one and back to zero.
• Since \(\sin(x)\) is periodic with a period of \(2\pi\), the squared function repeats the doubled frequency oscillations.

A deep understanding of these functions aids in predicting patterns and behaviors in natural processes and plays a significant role in calculus, as seen in solving integrals and finding averages over specific intervals. When employed in conjunction with tools like a CAS, exploring real-world applications becomes even more accessible and intuitive.