Problem 10 - College Algebra

Question

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. $$ g(x)=3 x^{2}-2 \text { on }[x, x+h] $$

Step-by-Step Solution

Answer

The average rate of change is \( 6x + 3h \).

Step 1: Identify Function Values

First, calculate the value of the function \( g(x) \) at the endpoints of the interval. Given \( g(x) = 3x^2 - 2 \), identify \( g(x) \) and \( g(x+h) \).

Step 2: Compute \( g(x+h) \)

Substitute \( x+h \) into the function \( g(x) = 3x^2 - 2 \) to find \( g(x+h) \):\[ g(x+h) = 3(x+h)^2 - 2 \]Expand \( (x+h)^2 \) to get \( x^2 + 2xh + h^2 \), leading to:\[ g(x+h) = 3(x^2 + 2xh + h^2) - 2 = 3x^2 + 6xh + 3h^2 - 2 \]

Step 3: Find the Change in Function Values

Calculate the change in the function values, \( g(x+h) - g(x) \):\[ g(x+h) - g(x) = (3x^2 + 6xh + 3h^2 - 2) - (3x^2 - 2) \]Simplify this to:\[ g(x+h) - g(x) = 6xh + 3h^2 \]

Step 4: Calculate the Average Rate of Change

The average rate of change of the function \( g(x) \) on the interval \([x, x+h]\) is given by:\[ \frac{g(x+h) - g(x)}{h} = \frac{6xh + 3h^2}{h} \]Simplify the expression by dividing each term by \( h \):\[ \frac{6xh}{h} + \frac{3h^2}{h} = 6x + 3h \]

Step 5: Simplified Average Rate of Change

The average rate of change of \( g(x) = 3x^2 - 2 \) on the interval \([x, x+h]\) in simplest form is:\[ 6x + 3h \]

Key Concepts

Understanding Functions

Functions are mathematical expressions that relate an input to an output. In simple terms, a function tells you how to get from one number (the input) to another number (the output). A key feature of functions is that each input is related to exactly one output.

• The function in our problem is defined as \( g(x) = 3x^2 - 2 \).
• An input \( x \) into the function yields an output by applying the function rules: square \( x \), multiply by 3, and finally subtract 2.
• Functions can be thought of as machines that take in numbers, process them according to a formula (like \( 3x^2 - 2 \)), and produce another number as output.

In this exercise, we look at how the function behaves between two values, \( x \) and \( x + h \). Knowing how a function changes across an interval is crucial for analyzing its behavior.

The Process of Expanding Binomials

Expanding binomials is an essential algebraic skill, especially when dealing with quadratic expressions.In our example, we need to expand \((x+h)^2\), which requires us to apply what is called the "FOIL" method. "FOIL" stands for:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outer terms in the product.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

So, when expanding \((x+h)^2\), this becomes:

• \(x^2\) from the first terms.
• \(xh + xh = 2xh\) from the outer and inner terms.
• \(h^2\) from the last terms.

Thus, \((x+h)^2\) expands to \(x^2 + 2xh + h^2\). This expansion becomes a part of finding \(g(x+h)\) and contributes to understanding how the function changes over an interval.

Simplifying Algebraic Expressions

Simplifying algebraic expressions streamlines complex calculations and makes working with equations more manageable. This process often involves combining like terms and reducing expressions to their simplest form.In the given problem, once we expand \(g(x+h)\), we calculate the change in function values by subtracting \(g(x)\) from \(g(x+h)\). After organizing the terms, we ended up with:\[6xh + 3h^2\]To find the average rate of change, we divide all parts of this expression by \(h\):

• \(\frac{6xh}{h} = 6x\)
• \(\frac{3h^2}{h} = 3h\)

After simplifying, the expression yields the average rate of change: \(6x + 3h\).This process of simplifying helps us find an expression that describes the rate at which \(g(x)\) changes over a specific interval. Understanding these simplifications clarifies how functions behave and change, making complex algebra more accessible.