Problem 58 - College Algebra with Modeling and Visualization

Question

Compute the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\). Round your answer to two decimal places when appropriate. Interpret your result graphically. $$ f(x)=\sqrt[3]{x+1}, x_{1}=7, \text { and } x_{2}=26 $$

Step-by-Step Solution

Answer

The average rate of change is approximately 0.053. This indicates a gentle increase in the function value from \(x_1 = 7\) to \(x_2 = 26\).

Step 1: Understand the Formula

The average rate of change of a function \( f(x) \) from \( x_1 \) to \( x_2 \) is calculated using the formula: \[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \] This represents the slope of the secant line passing through the points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) on the graph of the function.

Step 2: Evaluate the Function at Given Points

First, we calculate \( f(x_1) \) and \( f(x_2) \). Here, \( f(x) = \sqrt[3]{x+1} \). Thus: - \( f(7) = \sqrt[3]{7+1} = \sqrt[3]{8} = 2 \) - \( f(26) = \sqrt[3]{26+1} = \sqrt[3]{27} = 3 \)

Step 3: Calculate the Average Rate of Change

Substitute \( f(x_1) = 2 \), \( f(x_2) = 3 \), \( x_1 = 7 \), and \( x_2 = 26 \) into the average rate of change formula: \[ \frac{3 - 2}{26 - 7} = \frac{1}{19} \approx 0.053 \] So, the average rate of change is approximately 0.053, rounded to three decimal places.

Step 4: Interpret Graphically

Graphically, the average rate of change is the slope of the secant line that connects the points \((7, 2)\) and \((26, 3)\) on the graph of \( f(x)=\sqrt[3]{x+1} \). The small positive value indicates a gentle upward slope, reflecting a gradual increase in the function's value over this interval.

Key Concepts

Slope of Secant Line

The concept of the slope of a secant line is crucial in understanding the average rate of change for a function between two points. Imagine you have a curved line, representing a function, on a graph. Now, if you draw a straight line connecting two points on this curve, that line is called a 'secant line'.

When calculating the slope of this secant line, you essentially measure how steep the line is as it stretches between those two points. This is done using the formula:

• The change in the function's value, which is \(f(x_2) - f(x_1)\)
• Divided by the change in the x-values, \(x_2 - x_1\)

So essentially, the formula looks like this: \[ \text{Slope of secant line} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]In our specific example, plotting a secant line from \((7, 2)\) to \((26, 3)\) reflects how the function \(\sqrt[3]{x+1}\) changes as \(x\) changes from 7 to 26.

Function Evaluation

Function evaluation is the process of finding the output of a function given a specific input value. For the function \(f(x) = \sqrt[3]{x+1}\), function evaluation means replacing \(x\) with particular values and calculating the result.

Let's break it down:

• For \(x_1 = 7\), replace \(x\) with 7 to find \(f(7)\). You get: \(\sqrt[3]{7+1} = \sqrt[3]{8} = 2\)
• For \(x_2 = 26\), replace \(x\) with 26 to find \(f(26)\). You get: \(\sqrt[3]{26+1} = \sqrt[3]{27} = 3\)

By calculating these values, you're essentially pinpointing exact spots on the graph of the function that correspond to the given inputs \(x_1\) and \(x_2\). This step is absolutely necessary before you can compute the average rate of change.

Graph Interpretation

Graph interpretation involves visualizing the mathematical problem and understanding what the computed values imply about the graph of the function. After finding the slope of the secant line, translating the numerical result into a visual representation on a graph gives you a clearer picture of the function's behavior over an interval.

In the given example, plotting \(f(x) = \sqrt[3]{x+1}\) and identifying the points \((7, 2)\) and \((26, 3)\) should create a line that slightly tilts upwards. This gentle incline signifies a positive slope, meaning the function's values are increasing as \(x\) increases from 7 to 26. The calculated average rate of change, approximately 0.053, suggests this increase is mild.

Thus, despite the curve in the function, over the stretch between x-values, we can visualize and confirm the gradual upward trajectory sketched by the secant line.