Problem 25 - Contemporary Precalculus

Question

Find the average rate of change of the function. \(f(x)=3\left(4^{x}\right)\) as \(x\) goes from 1 to 3

Step-by-Step Solution

Answer

Answer: The average rate of change of the function is 90.

Step 1: Determine the initial and final x values.

The x values are given as 1 and 3. So, we will label them as \(x_1 = 1\) and \(x_2 = 3\). Our task is to find the average rate of change of the function \(f(x) = 3(4^x)\) as \(x\) goes from 1 to 3.

Step 2: Calculate the function values at \(x_1\) and \(x_2\).

We will substitute the x values into the function to find the function values at \(x_1\) and \(x_2\). - For \(x_1 = 1\), we have $$ f(x_1) = f(1) = 3(4^1)=(3)(4)=12 $$ - For \(x_2 = 3\), we have $$ f(x_2) = f(3) = 3(4^3)=(3)(64)=192 $$

Step 3: Calculate the average rate of change.

Now that we have the function values, we can calculate the average rate of change by finding the difference in function values divided by the difference in x values. $$ Average\ Rate\ of\ Change = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{192 - 12}{3 - 1} = \frac{180}{2} = 90 $$ So, the average rate of change of the function \(f(x) = 3(4^x)\) as x goes from 1 to 3 is 90.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a power that is a variable. In the function given here, \( f(x) = 3(4^x) \), 4 is the base, and \( x \) is the exponent or power. This means that the function will grow rapidly as \( x \) increases. Exponential functions are characterized by their constant multiplicative rate of growth or decay.

Some important properties of exponential functions include:

• Base Greater than One: When the base is greater than one, like 4 in our example, the function grows as \( x \) increases.
• Vertical Stretch/Compression: The coefficient 3 in front of \( 4^x \) acts as a vertical stretch or compression factor, affecting how steeply the function rises.
• Rapid Change in Values: Small increments in \( x \) result in significant changes in the value of the function, illustrating the powerful growth of exponential expressions.

Understanding exponential functions is crucial because they commonly describe real-world phenomena, including population growth, radioactive decay, and financial compound interest.

Function Evaluation

Function evaluation is the process of determining the output of a function for a given input. In our exercise, we evaluate the function \( f(x) = 3(4^x) \) at specific values, \( x_1 = 1 \) and \( x_2 = 3 \). This involves substituting these values into the function to find corresponding results.

Let's take a closer look at the steps involved in function evaluation:

• Substitution: To find \( f(1) \), substitute 1 into the function: \( f(1) = 3(4^1) \). Similarly, for \( f(3) \), substitute 3: \( f(3) = 3(4^3) \).
• Simplification: Calculate the power of the base: \( 4^1 = 4 \) and \( 4^3 = 64 \). Then multiply by the coefficient 3: \( f(1) = 3 \times 4 = 12 \) and \( f(3) = 3 \times 64 = 192 \).
• Result Interpretation: These calculated values, 12 and 192, represent the outputs of the function at the given inputs, allowing us to proceed with further calculations like finding the average rate of change.

Accurate function evaluation is essential in various mathematical contexts. It helps us understand how changes in input lead to changes in output, a fundamental concept in calculus and algebra.

Algebraic Calculations

Algebraic calculations involve using algebra to solve problems, including simplifying expressions and solving equations. In our exercise, the final step involves calculating the average rate of change between two points on the function \( f(x) = 3(4^x) \).

This calculation is done by using the formula:\[Average\ Rate\ of\ Change = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]Here's how we carry out algebraic calculations in this context:

• Identify \(x_1\) and \(x_2\): These are the starting and ending values of \( x \), which are 1 and 3, respectively.
• Evaluate Function Values: As calculated earlier, \( f(x_1) = 12 \) and \( f(x_2) = 192 \).
• Apply the Formula: Substitute these values into the formula: \( \frac{192 - 12}{3 - 1} = \frac{180}{2} = 90 \).
• Interpret the Result: The result, 90, represents the average rate at which the function changes between \( x_1 = 1 \) and \( x_2 = 3 \).

Understanding algebraic calculations is fundamental for comparing how quantities relate to each other. It helps in determining trends, rates of change, and various other mathematical and real-world applications.