Problem 4 - Calculus: Early Transcendental Functions

Question

List the evaluation points corresponding to the midpoint of each sub interval, sketch the function and approximating rectangles and evaluate the Riemann sum. $$f(x)=4-x^{2}, \quad \text { (a) }[-1,1], n=4 ; \quad \text { (b) }[-3,-1], n=4$$

Step-by-Step Solution

Answer

Riemann sums for the intervals [-1,1] and [-3,-1] with n=4 are respectively calculated as \(0.5 * (f(-0.75) + f(-0.25) + f(0.25) + f(0.75))\) and \(0.5 * (f(-2.75) + f(-2.25) + f(-1.75) + f(-1.25))\), where \(f(x) = 4 - x^{2}\). Approximating rectangles are formed with width of 0.5 (as n=4) and heights are the function value at the midpoints.

Step 1: Identifying subintervals and Midpoints

Step 1: For sub interval [-1,1], the total interval is of length 2 and it is divided into 4 equal parts. Thus, each subinterval would be of length 2/4 = 0.5. The midpoints of each subinterval will be -0.75, -0.25, 0.25, 0.75. Similarly for sub interval [-3,-1], each subinterval would be of length (-1 - (-3))/4 = 0.5. The midpoints will be -2.75, -2.25, -1.75, -1.25.

Step 2: Sketch Function and Rectangles

Step 2: Now one can sketch the function \( y = 4 - x^{2} \) along the two intervals and draw rectangles. The rectangles would have widths equal to length of the subintervals (0.5) and heights equal to the value of the function at the midpoints.

Step 3: Calculate the Riemann sums

Step 3: Now proceed to calculate the Riemann sums for the function \( f(x) = 4 - x^{2} \) over the given intervals. For each interval, multiply the height of each rectangle (which is the function evaluated at its midpoint) by the width of each rectangle (0.5). Add these to calculate the Riemann sum. For instance, for [-1,1] we calculate \(0.5 * (f(-0.75) + f(-0.25) + f(0.25) + f(0.75))\), where \(f(x) = 4 - x^{2}\). This process should be repeated for the second interval [-3,-1].

Key Concepts

Midpoint Rule

The Midpoint Rule is a method for approximating the integral of a function. Instead of using the endpoints of each subinterval, we take the midpoint to evaluate the function. This often provides a more accurate estimate of the area under a curve.

To apply the Midpoint Rule, follow these steps:

• Divide the entire interval into equal subintervals.
• Find the midpoint of each subinterval.
• Calculate the function's value at these midpoints.
• Multiply by the width of the subintervals.

This method is particularly beneficial when the function is relatively smooth and well-behaved over the interval.

Subintervals

Subintervals are crucial in breaking down larger intervals into manageable pieces. They help in calculating approximations, like Riemann sums, accurately.

For example, to evaluate the function over the interval [-1,1] with four subintervals, you divide the total interval length (2) by the number of subintervals (4), resulting in a width of 0.5. Each of these subintervals can be considered separately, and when summed together, they help approximate the area under a curve effectively.

Function Sketching

Function sketching is visualizing the function on a graph. This provides insights into its behavior. For instance, sketch the function \( y = 4 - x^2 \) over the given intervals.

By plotting this, you quickly observe that it is a downward-opening parabola. Drawing rectangles based on the midpoints help visualize how the Midpoint Rule works. Each rectangle's width equals the subinterval's length (0.5), and the height equals the function value at the midpoint.

This sketch aids in understanding how the Riemann sum estimates the integral, as the sum of the areas of these rectangles.

Calculus Problem Solving

In calculus, problem solving often involves breaking down complex tasks into simpler parts. When dealing with integrals and sums, the process requires thorough understanding. Here's how:

• First, define your function and interval.
• Divide the interval into subintervals and identify midpoints.
• Sketch the function to visualize the area you are approximating.
• Calculate the Riemann sum using the Midpoint Rule, multiplying the function values at midpoints by the width of subintervals and summing them up.

This approach not only simplifies the calculation but also enhances comprehension of how integral approximations work.