Problem 8 - 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 -0.1 and 0.1 .

Step-by-Step Solution

Answer

Answer: The average rate of change of the function on the given interval is 0.02.

Step 1: Define the function and interval

We are given the function \(g(x) = 2x^3 - 3x^2\) and want to find the average rate of change of the function on the interval between -0.1 and 0.1.

Step 2: Compute the value of g(x) at the two points

Evaluate the function g(x) at the endpoints of the interval. That is, calculate \(g(-0.1)\) and \(g(0.1)\). \(g(-0.1) = 2(-0.1)^3 - 3(-0.1)^2 = -0.002 - 0.03 = -0.032\) \(g(0.1) = 2(0.1)^3 - 3(0.1)^2= 0.002 - 0.03 = -0.028\)

Step 3: Calculate the average rate of change

The average rate of change of a function is the change in the y values divided by the change in the x values. In this case, we have: Average rate of change \(= \frac{g(0.1) - g(-0.1)}{0.1 - (-0.1)} = \frac{-0.028 - (-0.032)}{0.1 + 0.1} = \frac{0.004}{0.2}= 0.02\) The average rate of change of the function \(g(x) = 2x^3 - 3x^2\) on the interval between -0.1 and 0.1 is 0.02.

Key Concepts

Understanding Cubic Functions

Cubic functions are a type of polynomial function that have the highest power of three, which means they include terms like \(x^3\). These functions can take many forms, but all will express either an increase or decrease in value that isn't steady – instead, it often increases and decreases at different rates due to the curve's shape.
For instance, the given function \(g(x) = 2x^3 - 3x^2\) is a cubic function.
This means it can display interesting behaviors such as multiple turning points or symmetric features.

• In a graph, a cubic function typically forms an "S" shape, showing the potential for varied rates of change.
• The coefficient of the \(x^3\) term (which is 2 in this instance) influences the steepness and direction of the curve.
• The other terms will shift this curve around the plot vertically and horizontally.

Understanding the dynamics of cubic functions helps in analyzing their rate of change and approximating their patterns over specific intervals.

Function Evaluation: Calculating Values

Function evaluation refers to the process of determining what value a function has at specific points in its domain.
This operation is crucial for finding items such as the function's rate of change. In our exercise, you were asked to evaluate the function \(g(x) = 2x^3 - 3x^2\) at two specific points: \(-0.1\) and \(0.1\).
Here's how you generally evaluate a function:

• Replace the variable \(x\) in the expression with the given numerical value.
• Simplify the expression to solve for \(g(x)\).

For example:

• \(g(-0.1)\) means substituting \(-0.1\) into the expression. Calculations show it's simplified to \(-0.032\).
• \(g(0.1)\) substitutes \(0.1\), resulting in a value of \(-0.028\).

This evaluation is central in subsequent calculations like finding average rate of change, and it highlights how the function behaves on a specific interval.

Algebraic Expressions: Simplifying and Solving

An algebraic expression involves a combination of numbers, variables, and arithmetic operators (like +, -, *, /)).
In many math problems, you deal with expressions that require simplifying and solving. For the current problem, simplifying the function evaluation is an essential step.

• The expression \(2x^3 - 3x^2\) needed simplification by inserting numerical values.
• For example, substituting \(-0.1\) into the expression \(2(-0.1)^3 - 3(-0.1)^2\), simplifies to \(-0.032\).
• Similarly, \(2(0.1)^3 - 3(0.1)^2\) becomes \(-0.028\).

Simplifying helps to clearly understand and work with the function's expression, allowing for accurate evaluations. Each term in an algebraic expression plays a role. Variables for representing unknowns and constants provide the values to calculate with.
This skill is crucial for higher-level math, as it allows the transformation of complex structures into manageable problems that are easier to interpret and solve.