Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for each function. $$f(x)=x^{2}-1$$

Step-by-Step Solution

Answer

The evaluated sum is 26.

Step 1: Identify the expression to evaluate

We need to evaluate the sum \( \sum_{i=1}^{4} f(x_i) \Delta x \) for the function \( f(x) = x^2 - 1 \) with the given \( x_i \) values and \( \Delta x = 0.5 \).

Step 2: Calculate \( f(x_1) \)

Substitute \( x_1 = 0 \) into the function: \[ f(x_1) = (0)^2 - 1 = -1 \]

Step 3: Calculate \( f(x_2) \)

Substitute \( x_2 = 2 \) into the function: \[ f(x_2) = (2)^2 - 1 = 4 - 1 = 3 \]

Step 4: Calculate \( f(x_3) \)

Substitute \( x_3 = 4 \) into the function: \[ f(x_3) = (4)^2 - 1 = 16 - 1 = 15 \]

Step 5: Calculate \( f(x_4) \)

Substitute \( x_4 = 6 \) into the function: \[ f(x_4) = (6)^2 - 1 = 36 - 1 = 35 \]

Step 6: Evaluate the sum

Multiply each function result by \( \Delta x = 0.5 \) and sum them up:\[ 0.5(-1) + 0.5(3) + 0.5(15) + 0.5(35) \]Calculating each term, we get:- \(0.5(-1) = -0.5\)- \(0.5(3) = 1.5\)- \(0.5(15) = 7.5\)- \(0.5(35) = 17.5\)Summing these results, we find:\[ -0.5 + 1.5 + 7.5 + 17.5 = 26 \]

Key Concepts

Piecewise Functions

Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval within the domain. These are crucial for more complex real-world scenarios where a function's behavior changes over its range.
For example, consider a shipping cost function that charges different rates based on weight brackets, or a tax rate that changes with income levels. These functions are defined piece-by-piece, over specific intervals.

• Each 'piece' is a distinct function, valid only over a particular interval.
• The boundaries between intervals are often critical points where the rule of the function changes.
• Understanding piecewise functions helps in modeling and interpreting real-life data that doesn't follow a single straightforward pattern.

The Riemann Sum in the given exercise isn't a piecewise function directly but helps us understand how breaking a function into parts (like integrating segment by segment) can yield important results.

Definite Integrals

Definite integrals evaluate the net area under a curve from one point to another. They provide insight into total values within a finite interval, such as distances, areas, and accumulated quantities.
In mathematics, the definite integral of a function from point \( a \) to \( b \) is represented as \( \int_{a}^{b} f(x) \, dx \).
This concept underpins many areas of calculus and aids in understanding functional changes over intervals.

• Definite integrals involve limits and the function being continuous over the interval.
• They can be visualized as the signed area between the function and the x-axis.
• The Fundamental Theorem of Calculus links antiderivatives with definite integrals, simplifying their calculation.

The exercise's Riemann Sum is a numerical method to approximate definite integrals, breaking the interval into smaller parts for ease of calculation.

Numerical Integration

Numerical integration is a broad term encompassing techniques used to approximate the value of definite integrals, especially when an exact solution is difficult to obtain. These methods come in handy when dealing with complex functions that are arduous to integrate analytically.
Riemann Sums, in particular, estimate the integral by summing up areas of rectangles:

• Each rectangle's width can be any small \( \Delta x \), capturing the interval between evaluation points.
• For the given exercise, these intervals (or subintervals) are represented by the sequence of \( x_i \) values: \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6\).
• The Riemann Sum effectively calculates small areas, approximating the total area under the curve.

While Riemann Sums offer a straightforward way to do this, more sophisticated methods like the Trapezoidal Rule or Simpson's Rule may provide even better approximations for the same integral, often with less work. This makes numerical integration a versatile and powerful tool in science and engineering.