Problem 23 - Calculus: Early Transcendental Functions

Question

Compute sums of the form \(\sum_{i=1}^{11} f\left(x_{i}\right) \Delta x\) for the given values. $$f(x)=x^{2}+4 x ; x=0.2,0.4,0.6,0.8,1.0 ; \Delta x=0.2 ; n=5$$

Step-by-Step Solution

Answer

The final sum of the form \(\sum_{i=1}^{5} f(x_{i})\Delta x\) is obtained by adding together \(\Delta x \cdot f(0.2)\), \(\Delta x \cdot f(0.4)\), \(\Delta x \cdot f(0.6)\), \(\Delta x \cdot f(0.8)\), and \(\Delta x \cdot f(1.0)\).

Step 1: Substitute the Values of x in the Function

First, substitute each value of \(x\) into the function \(f(x)=x^{2}+4 x\) to obtain \(f(x_{i})\) for each \(x_{i}\). That is, calculate \(f(0.2)\), \(f(0.4)\), \(f(0.6)\), \(f(0.8)\), and \(f(1.0)\).

Step 2: Multiply Each Result with \(\Delta x\)

Next, multiply each \(f(x_{i})\) by \(\Delta x\) which is 0.2: \(\Delta x \cdot f(0.2)\), \(\Delta x \cdot f(0.4)\), \(\Delta x \cdot f(0.6)\), \(\Delta x \cdot f(0.8)\), and \(\Delta x \cdot f(1.0)\). This will give the term to be summed up in \(\sum_{i=1}^{5} f(x_{i})\Delta x\).

Step 3: Sum the Results

Lastly, sum all the resulting terms: \(\sum_{i=1}^{5} f(x_{i})\Delta x = \Delta x \cdot f(0.2) + \Delta x \cdot f(0.4) + \Delta x \cdot f(0.6) + \Delta x \cdot f(0.8) + \Delta x \cdot f(1.0)\).

Key Concepts

Definite Integrals

Definite integrals are used in calculus to find the area under a curve over a given interval. They essentially provide us with a way to sum infinitely many infinitely small quantities, which is practically impossible to do on paper.

The definite integral of a function from point \(a\) to point \(b\) is symbolized by \[\int_{a}^{b} f(x) \, dx\]where \(f(x)\) represents the function, and \(dx\) is a very small width. This means that you add up the areas of infinitesimally narrow rectangles that reach from the \(x\)-axis up to the curve.

• The function \(f(x)\) specifies the height of each rectangle.
• The integral sign represents the summation of those rectangles over the interval \([a, b]\).
• If the function is above the \(x\)-axis, the integral gives us a positive area; if it's below, the result is negative.

Understanding definite integrals is crucial for applications in physics, engineering, and probability studies, as they allow for apprehending and computing changes over time or space.

Numerical Integration

Numerical integration is a mathematical technique that provides approximate solutions for definite integrals, especially when finding an exact solution is difficult or impossible.

It utilizes methods like Riemann sums, Trapezoidal Rule, and Simpson's Rule to estimate the area under a curve. In computational scenarios, these approximations become essential when dealing with real-world data or complex functions.

• One common approach is to break up the area under the curve into slices, approximate the area of each slice, and then sum these areas to get a total.
• Each method differs in how these slices' areas are calculated, ranging from assuming them as rectangles (Riemann sums) to using trapezoids or parabolic approximations (Simpson's Rule).
• The main goal is to get a close statistically significant estimate with reduced computation error.

Numerical integration is fundamental in fields like economics and engineering, where performing exact integrations is often infeasible.

Piecewise Constant Functions

A piecewise constant function is a type of function that is composed of constant values over specific intervals. These functions take on one constant value for each interval specified in the domain.

These functions can be visualized as a series of flat steps or jumps on a graph, with each separate part representing a constant value.

• They are useful in simplifying the analysis and integration of complex functions by approximating them in segments.
• When using Riemann sums for integration, piecewise constant functions simplify the process by making it easy to calculate the area of each rectangular slice.
• In reality, models such as step functions in economics (tax brackets) or applied math (signal processing) often use piecewise constant functions.

This concept not only aids in breaking down the complexity of a function but also turns problems into manageable calculations, making it a valuable tool in both theoretical and practical applications.