Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(f(x)=\frac{3}{x^{3}+x}\). Let \(a\) and \(b\) be positive constants, \(0

Step-by-Step Solution

Answer

Step sizes \(\Delta w\) and \(\Delta t\) are given by \(\Delta w=(b-a)/8\) and \(\Delta t=(b-a)/12\) respectively.

Step 1: Understanding the problem

The task is to calculate the step sizes (\(\Delta w\) and \(\Delta t\)) for two different partitions of the interval [a, b]. One partition divides the interval into 8 equal subintervals (using \( w \)'s) and the other divides the interval into 12 equal subintervals (using \( t \)'s). Both partitions start at \( a \) and end at \( b \) and \( 0 < a < b \) are positive constants.

Step 2: Calculate \(\Delta w\)

The step size \(\Delta w\) for the partition using \( w \)'s is calculated by dividing the total length of the interval (b - a) by the number of subintervals (8). This can be written as: \(\Delta w=(b-a)/8\).

Step 3: Calculate \(\Delta t\)

Similarly, the step size \(\Delta t\) for the partition using \( t \)'s is calculated by dividing the total length of the interval (b - a) by the number of subintervals (12). This can be written as: \(\Delta t=(b-a)/12\).

Key Concepts

Partitioning of Intervals

Partitioning an interval involves dividing it into smaller, equal subintervals, which can be useful for approximating areas under curves in calculus. In this context, an interval \([a, b]\) is divided into multiple subintervals. Partitioning helps us study the behavior of functions over a range. This understanding is particularly beneficial in calculus, where we approximate integral values using sums.

Each subinterval is like a tiny fragment of the whole interval, and partitioning helps make things manageable and computable. For example, our problem divides \([a, b]\) into two different sets of intervals with either 8 or 12 pieces.

• Using 8 subintervals gives us one set of partitions (\( w_i \)), each with a certain length.
• Similarly, using 12 subintervals results in another set of partitions (\( t_i \)).

Understanding how intervals are partitioned is the first step toward using techniques like Riemann sums, which approximate total areas under curves.

Step Size Calculation

Calculating the step size is crucial in understanding how finely or coarsely we are dividing an interval. The step size, often represented by \( \Delta \) (Delta), determines the width of each partitioned piece. To compute it, we take the full length of the interval \([b - a]\) and divide it by the desired number of subintervals. This provides a uniform step size, which is the same for each subinterval.

• For \( \Delta w \), which involves dividing the interval into 8 equal parts, the step size is: \( \Delta w = \frac{b-a}{8} \).
• For \( \Delta t \), dividing into 12 equal parts, the calculation changes slightly: \( \Delta t = \frac{b-a}{12} \).

By understanding the step size calculation, students can better apprehend how changes in the number of subintervals affect the granularity of the partition, influencing the precision of approximations in calculations like the Riemann Sum.

Riemann Sum

The Riemann Sum is a way to approximate the area under a curve, meaning it helps us estimate an integral. It's like adding up the areas of rectangles over partitioned intervals. When partitioning \([a, b]\), each subinterval gets a rectangle whose height is determined by the function value at certain points within the subinterval.

There are different types of Riemann sums based on where you sample the height:

• Left Riemann Sum: Takes the height from the start of each subinterval.
• Right Riemann Sum: Takes the height from the end of each subinterval.
• Midpoint Riemann Sum: Uses the height from the middle of each subinterval.

The precision of a Riemann Sum is heavily influenced by the number of subintervals and their respective widths. Larger numbers of subintervals with smaller step sizes generally provide a better approximation. This concept forms the foundational idea behind definite integrals in calculus, and understanding it is key to grasping integral calculus in its entirety.