Problem 5 - Algebra Form and Function

Question

Find the average rate of change of \(g(x)=2 x^{3}-3 x^{2}\) on the intervals indicated. Between 1 and 3 .

Step-by-Step Solution

Answer

Answer: The average rate of change is 14.

Step 1: Identify the endpoints of the interval

The interval mentioned in the exercise is between x = 1 and x = 3. We will represent these points as \((x_1, y_1)\) and \((x_2, y_2)\), where \(x_1 = 1\) and \(x_2 = 3\).

Step 2: Find the function values at the endpoints

Next, we need to find the function values at the endpoints of the interval. To do this, we will plug \(x_1 = 1\) and \(x_2 = 3\) into the function \(g(x) = 2x^3 - 3x^2\). For \(x_1 = 1\), g(x) = \(2(1)^3 - 3(1)^2 = 2 - 3 = -1\). So, \(y_1 = -1\). For \(x_2 = 3\), g(x) = \(2(3)^3 - 3(3)^2 = 54 - 27 = 27\). So, \(y_2 = 27\). The coordinates of the two endpoints are \((1, -1)\) and \((3, 27)\).

Step 3: Calculate the average rate of change

To find the average rate of change, we will use the formula: $$\frac{y_2 - y_1}{x_2 - x_1}$$ Substitute the coordinates of the endpoints we calculated in the previous step: $$\frac{27 - (-1)}{3 - 1} = \frac{28}{2} = 14$$ The average rate of change of the function \(g(x) = 2x^3 - 3x^2\) between x = 1 and x = 3 is 14.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to powers and coefficients, combined using addition, subtraction, and multiplication. A single polynomial can be expressed as: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \] where \(a_n, a_{n-1}, ..., a_1, a_0\) are constants, and \(n\) is a non-negative integer. In the given exercise, we are working with the polynomial function: \[ g(x) = 2x^3 - 3x^2 \] This is a simple polynomial of degree 3, which means the highest power of the variable \(x\) is 3.

• Cubic Term: The term \(2x^3\) is called the cubic term because it involves \(x\) raised to the third power.
• Quadratic Term: The term \(-3x^2\) is quadratic, meaning it involves \(x\) squared.

Polynomial functions like \(g(x)\) are continuous and smooth, which makes them suitable for discussing concepts such as the average rate of change across an interval. Understanding polynomial functions is crucial because they serve as the foundation for much of algebra and calculus.

Intervals in Mathematics

In mathematics, an interval is a way of representing a continuous set of numbers between two endpoints. The endpoints define the bounds or limits of the interval. In this exercise, we're focusing on the interval between \(x = 1\) and \(x = 3\). In the context of functions, an interval helps us examine how a function behaves between specific values of \(x\). The chosen interval \([1, 3]\) includes both endpoints and is known as a closed interval.

• Closed Interval: Includes both endpoints, denoted by square brackets \([a, b]\).
• Open Interval: Does not include endpoints, denoted by parentheses \((a, b)\).

By evaluating the function on an interval, like from \(x = 1\) to \(x = 3\), we can determine the average rate at which function values change over that span. This concept is particularly useful when analyzing functions that model real-world phenomena, as it provides insight into how quickly changes occur within a specific range.

Function Values

Function values represent the output of a function for a given input value of \(x\). They are crucial when finding the average rate of change over an interval. In this exercise, we evaluated the function \(g(x) = 2x^3 - 3x^2\) at two points: \(x_1 = 1\) and \(x_2 = 3\). By substituting these values into the function, we calculated:

• At \(x = 1\): \(g(1) = 2(1)^3 - 3(1)^2 = -1\).
• At \(x = 3\): \(g(3) = 2(3)^3 - 3(3)^2 = 27\).

These function values \(y_1 = -1\) and \(y_2 = 27\) are key to calculating the average rate of change. The average rate of change measures how much the output \(y\) values differ, given a change in \(x\) over an interval: \[ \text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \] This concept aids in assessing how much a function's value increases or decreases between two specific inputs, providing insight into the behavior of the function over a defined span.