Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, find the average rate of change of each function on the interval specified. \(g(x)=3 x^{3}-1\) on [-3,3]

Step-by-Step Solution

Answer

The average rate of change is 27.

Step 1: Identify the interval endpoints

The given interval is [-3, 3]. Thus, the endpoints are \(x_1 = -3\) and \(x_2 = 3\).

Step 2: Calculate function values at endpoints

Calculate \(g(x)\) at \(x_1\) and \(x_2\).First, calculate \(g(-3)\):\[g(-3) = 3(-3)^3 - 1 = 3(-27) - 1 = -81 - 1 = -82\]Then, calculate \(g(3)\):\[g(3) = 3(3)^3 - 1 = 3(27) - 1 = 81 - 1 = 80\]

Step 3: Apply the average rate of change formula

The average rate of change formula is given by:\[\text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1}\]Substitute \(g(-3) = -82\), \(g(3) = 80\), \(x_1 = -3\), and \(x_2 = 3\) into the formula:\[\text{Average Rate of Change} = \frac{80 - (-82)}{3 - (-3)} = \frac{80 + 82}{6} = \frac{162}{6} = 27\]

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression involving a sum of powers of variables, each multiplied by a coefficient. In simpler terms, it looks like an equation made up of lots of terms separated by plus or minus signs. Each term is a product of a constant coefficient and a variable raised to a whole number power. For example, the function given in the exercise, \(g(x) = 3x^3 - 1\), is a polynomial. Let's look more closely:

• The highest power of the variable \(x\) determines the degree of the polynomial. Here, \(g(x)\) is a third-degree polynomial because the highest power is 3.
• This function is made up of two terms: \(3x^3\) and \(-1\).
• These functions can model various phenomena such as the trajectory of a projectile or even economic growth.

By understanding polynomial functions, we can predict how the function behaves at different values of \(x\).

Rate of Change

The rate of change is a key concept in mathematics and everyday life. It tells us how one quantity changes in relation to another. Usually expressed in the form of a ratio or fraction, it tells you how much the function value changes as the input value changes. In this exercise, we are finding the average rate of change, which is essentially the slope of the line that connects two points on the graph of a function.

• For the polynomial \(g(x) = 3x^3 - 1\), it helps us understand how \(g(x)\) changes as \(x\) increases from \(-3\) to \(3\).
• The average rate of change is calculated by taking the difference in function values and dividing it by the difference in \(x\) values.
• This concept is quite similar to finding the slope in a linear function.

In this problem, the average rate of change gives a single number that represents the overall change of the function over a specified interval.

Interval Endpoints

Interval endpoints are the specific starting and ending values of \(x\) for which we want to evaluate the function and its rate of change. In practical problems, they define a finite range over which calculations are made.In the specific exercise problem, the interval given is \([-3, 3]\). This means the endpoints are:

• \(x_1 = -3\)
• \(x_2 = 3\)

The endpoints are critical because when calculating the average rate of change, they dictate the \(x\) values to plug into the function. Without these endpoints, it would not be possible to determine where to start and stop evaluating the function.

Function Values

Function values refer to the outputs you get after plugging specific \(x\) values into the function. They are the "\(y\)" in potential \(x, y\) coordinate pairs.In the original problem, we calculate function values at the given endpoints to understand how the function \(g(x)\) changes from \(x_1 = -3\) to \(x_2 = 3\):

• For \(x = -3\), \(g(-3) = -82\).
• For \(x = 3\), \(g(3) = 80\).

These values are crucial for calculating the average rate of change. Each function value represents a point on the graph, and together, they give us a sense of how the function behaves over the interval. Understanding function values helps in graphing the curve and performing further analysis on how the function progresses.