Problem 7 - 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 0 and 10 .

Step-by-Step Solution

Answer

Answer: The average rate of change is 170.

Step 1: Evaluate the function at the endpoints

Evaluate the function \(g(x)=2 x^{3} - 3 x^{2}\) at x = 0 and x = 10: 1. \(g(0) = 2(0)^{3} - 3(0)^{2} = 0\) 2. \(g(10) = 2(10)^{3} - 3(10)^{2} = 2(1000) - 3(100) = 2000 - 300 = 1700\)

Step 2: Calculate the average rate of change

Use the formula for the average rate of change, which is \(\frac{g(b) - g(a)}{b - a}\), where a and b are the endpoints of the interval: Average rate of change = \(\frac{g(10) - g(0)}{10 - 0} = \frac{1700 - 0}{10} = \frac{1700}{10} = 170\) The average rate of change of the function \(g(x) = 2 x^{3} - 3 x^{2}\) on the interval between 0 and 10 is 170.

Key Concepts

Function Evaluation

Function evaluation is a key concept in understanding mathematical functions. It involves finding the value of a function at specific points. In the exercise, we are given the function \(g(x) = 2x^3 - 3x^2\). Evaluating this function means substituting different values of \(x\) into the equation to determine the output or result.

• At \(x = 0\): Substitute \(0\) into the function: \(g(0) = 2(0)^3 - 3(0)^2 = 0\).
• At \(x = 10\): Substitute \(10\) into the function: \(g(10) = 2(10)^3 - 3(10)^2 = 2000 - 300 = 1700\).

By performing these calculations, you can determine the behavior of the function at particular points, which is crucial for further analysis like finding the average rate of change.
It helps to simplify and better understand the given polynomial function's behavior across different intervals.

Polynomial Functions

Polynomial functions, like \(g(x) = 2x^3 - 3x^2\), are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They are widely used due to their simple structure and ability to model a variety of real-world situations.

• Structure: A polynomial is expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\).
• Cubic Polynomial: In this case, \(g(x)\) is a cubic polynomial because the highest power of \(x\) is a cube (\(x^3\)).
• Behavior: The shape of a polynomial curve depends on the degree and the coefficients of the polynomial. Cubic functions can have one or two turning points and tend to resemble a stretched "S" shape.

Knowing how to interpret these polynomials is essential for tackling many algebraic problems. They provide a simple way to describe and interpret more complex relationships, like rate of change over an interval.

Interval Analysis

Interval analysis is crucial when examining functions for behavior over specified ranges. When tasked with finding the average rate of change, understanding the function's behavior in these intervals becomes essential.

• Interval Definition: The exercise specifies an interval as the range between two points, here 0 and 10, denoted as \([0, 10]\).
• Average Rate of Change: This is calculated over the interval and provides insights into the function's behavior. The formula is \(\frac{g(b) - g(a)}{b - a}\), which in the example becomes \(\frac{1700 - 0}{10} = 170\).

The average rate of change gives the function's slope across the interval, illustrating how much the function's output (\(g(x)\)) changes, on average, for each unit increase in \(x\). Understanding interval analysis helps in making informed decisions based on the function's performance within specified ranges.
It also assists in identifying trends and making predictions in real-world applications where polynomial functions are used.