Problem 57 - 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. $$ f(x)=\sqrt{2 x-1}, x_{1}=1, \text { and } x_{2}=3 $$

Step-by-Step Solution

Answer

The average rate of change from 1 to 3 is approximately 0.62, representing the slope of the secant line between these points on the graph.

Step 1: Evaluate the Function at Given Points

First, we need to compute the values of the function at the given points, \(x_1 = 1\) and \(x_2 = 3\). Evaluate \(f(x_1)\) and \(f(x_2)\).\[f(x_1) = f(1) = \sqrt{2(1) - 1} = \sqrt{1} = 1\]\[f(x_2) = f(3) = \sqrt{2(3) - 1} = \sqrt{5}\]

Step 2: Apply the Average Rate of Change Formula

The average rate of change of a function \(f\) from \(x_1\) to \(x_2\) is given by the formula:\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]Substituting the values we calculated:\[\text{Average Rate of Change} = \frac{\sqrt{5} - 1}{3 - 1} = \frac{\sqrt{5} - 1}{2}\]

Step 3: Calculate and Approximate

Next, we need to approximate the value of \(\sqrt{5}\) and calculate the average rate of change.\(\sqrt{5} \approx 2.24\)Substitute into the formula:\[\text{Average Rate of Change} \approx \frac{2.24 - 1}{2} \approx \frac{1.24}{2} \approx 0.62\]So, the average rate of change is approximately \(0.62\).

Step 4: Interpret Graphically

Graphically, the average rate of change represents the slope of the secant line that passes through the points \((1, f(1))\) and \((3, f(3))\) on the curve \(y = \sqrt{2x - 1}\). This slope indicates how fast the function is changing, on average, between these two points.

Key Concepts

Square Root Function

The square root function is an important part of understanding various mathematical concepts, including the rate of change. In the function given, \(f(x) = \sqrt{2x - 1}\), the square root affects how the function behaves. As \(x\) increases, the value inside the square root increases, causing \(f(x)\) to change, but not in a linear fashion. This means that changes in \(x\) will result in non-equally spaced changes in \(f(x)\).

This function takes a number, multiplies it by 2, subtracts 1, and then takes the square root.

• When \(x = 1\), \(f(x) = \sqrt{1} = 1\).
• When \(x = 3\), \(f(x) = \sqrt{5} \approx 2.24\).

This specific form of the function is continuous and increases gradually. It shows a a growing pattern represented graphically by a curve that starts at \((1, 1)\) and rises to \((3, \sqrt{5})\). The function doesn't have a constant slope, unlike linear functions, which is why we compute average rate of change instead.

Secant Line

The secant line is a simple yet powerful tool to approximate changes in a function over a specific interval. In this exercise, you compute the average rate of change of the function \(f(x) = \sqrt{2x - 1}\) using the values at \(x_1 = 1\) and \(x_2 = 3\).

The formula for the average rate of change basically gives us the slope of the secant line. The secant line here passes through two points on the curve: \((1, 1)\) and \((3, \sqrt{5})\).
The slope of this secant line is: \[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{\sqrt{5} - 1}{3 - 1} = \approx 0.62 \]
This slope tells us how much \(f(x)\) changes, on average, with each unit increase in \(x\) from \(x_1\) to \(x_2\). The secant line thus serves as a linear approximation of the function over a segment of the domain.

Graphical Interpretation

Graphical interpretation is key to visualizing mathematical concepts. In this case, understanding the average rate of change involves visualizing the secant line on the graph of the square root function.

Imagine the graph of \(y = \sqrt{2x - 1}\). It starts gently then gets steeper. At specific points \((1, 1)\) and \((3, \sqrt{5})\), plot these on the graph. Draw a straight line connecting these two points—this is your secant line.

The slope of this line illustrates the average rate of change, giving you insight into how quickly the function's value is increasing over that interval from \(x_1\) to \(x_2\). As a concept, this visualization helps us estimate and understand the rate at which the function changes without exploring every point in between, offering a broader perspective through graphical interpretation. The steeper the line, the greater the increase in function value for each unit of \(x\).