Problem 15 - 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. $$ \frac{f(x+h)-f(x)}{h} \text { given } f(x)=2 x^{2}-3 x \text { on }[x, x+h] $$

Step-by-Step Solution

Answer

The average rate of change is \(4x + 2h - 3\).

Step 1: Identify the Function and Interval

The function given is \( f(x) = 2x^2 - 3x \). We need to find the average rate of change on the interval \([x, x+h]\).

Step 2: Apply the Difference Quotient Formula

The difference quotient for the average rate of change is \( \frac{f(x+h) - f(x)}{h} \). Our goal is to compute this expression.

Step 3: Calculate \(f(x+h)\)

Substitute \(x+h\) into the function: \[f(x+h) = 2(x+h)^2 - 3(x+h)\]Expanding this gives:\[= 2(x^2 + 2xh + h^2) - 3x - 3h\]\[= 2x^2 + 4xh + 2h^2 - 3x - 3h\]

Step 4: Subtract \(f(x)\) from \(f(x+h)\)

Compute \(f(x+h) - f(x)\): \[f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 - 3x - 3h) - (2x^2 - 3x)\]Simplifying:\[= 4xh + 2h^2 - 3h\]

Step 5: Divide by \(h\) to Find the Rate of Change

Now, divide the expression obtained by \(h\):\[\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 - 3h}{h}\]Simplifying further:\[= 4x + 2h - 3\]

Step 6: Interpret the Result

The expression \(4x + 2h - 3\) represents the average rate of change of the function \(f(x) = 2x^2 - 3x\) over the interval \([x, x+h]\). This result is in its simplest form.

Key Concepts

Difference Quotient

The average rate of change of a function on a given interval is calculated using the difference quotient. This concept helps us understand how a function's output value changes when the input value changes. It is particularly useful for measuring how steep a graph is over an interval.To find the difference quotient, follow this formula:

• Identify the function, say \( f(x) \).
• Choose two points on the function: \( x \) and \( x + h \), where \( h \) is the change in \( x \).
• Compute the change in the function's output: \( f(x + h) - f(x) \).
• Divide the change in the output by the change in the input, \( h \), forming \( \frac{f(x + h) - f(x)}{h} \).

This quotient essentially tells us the average slope of the function between \( x \) and \( x + h \).

Function Evaluation

When working with functions, evaluating them at specific points is a critical step. Function evaluation means substituting a value into the function's variable.For example, if you have a function \( f(x) = 2x^2 - 3x \), evaluating it at a point means:

• If \( x = a \), then \( f(a) = 2(a)^2 - 3(a) \).
• This calculation returns a specific number that is the function's value at \( a \).

Evaluating functions when solving problems like the difference quotient helps you find values at different points on the interval, such as \( f(x + h) \). This is essential for calculating the changes in values over intervals.

Expanding Expressions

Expanding expressions is a key skill in algebra that involves rewriting expressions in an extended form. In our problem, we dealt with expanding expressions by working with squares and distributing products.For the expression \( f(x+h) = 2(x+h)^2 - 3(x+h) \):

• The term \( (x + h)^2 \) is expanded as \( x^2 + 2xh + h^2 \).
• Next, you distribute 2 over all the terms in \( x^2 + 2xh + h^2 \) to get \( 2x^2 + 4xh + 2h^2 \).
• Similarly, the \( -3(x + h) \) yields \( -3x - 3h \).

Expansion allows us to deconstruct complex expressions into simpler, more manageable parts, facilitating further simplification and calculation.

Simplifying Expressions

After expanding expressions, the next logical step is simplifying them. Simplification involves combining like terms and eliminating unnecessary parts to produce a more concise expression.For instance, starting from our example, after expanding \( f(x+h) \) and then computing \( f(x+h) - f(x) \), we find:

• The expanded form: \( 2x^2 + 4xh + 2h^2 - 3x - 3h - 2x^2 + 3x \).
• Remove the common terms like \( 2x^2 \) and \( -3x \) to simplify.
• It simplifies to: \( 4xh + 2h^2 - 3h \).

Finally, dividing this expression by \( h \), we simplify \( \frac{4xh + 2h^2 - 3h}{h} \) to \( 4x + 2h - 3 \). Each step of simplification makes the expressions easier to work with and understand, allowing us to interpret results with clarity.