Problem 17 - Precalculus Essentials

Question

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$f(x)=\sqrt{x} \text { from } x_{1}=4 \text { to } x_{2}=9$$

Step-by-Step Solution

Answer

The average rate of change of the function from \(x_{1}=4\) to \(x_{2}=9\) is \(\frac{1}{5}\).

Step 1: Determine the value of f at \(x_{1}\) and \(x_{2}\)

First, replace the \(x\) in \(f(x)\) by \(x_{1}=4\) and \(x_{2}=9\) respectively to find the corresponding function values. This yields \(f(x_{1}) = f(4) = \sqrt{4} = 2\) and \(f(x_{2}) = f(9) = \sqrt{9} = 3\).

Step 2: Apply the formula

Next, substitute \(f(x_{1})=2\), \(f(x_{2})=3\), \(x_{1}=4\), and \(x_{2}=9\) into the formula \(\frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\). The calculation will then be \(\frac{3-2}{9-4}= \frac{1}{5}\).

Step 3: Interpret the result

The result \(\frac{1}{5}\) represents the average rate of change of the function \(f(x) = \sqrt{x}\) from \(x_{1}=4\) to \(x_{2}=9\). This means that, on average, the function increases by \(\frac{1}{5}\) for each unit the x-value changes on the interval from 4 to 9.

Key Concepts

Function Evaluation

To find the average rate of change, the first step is to evaluate the function at specific points. This process is known as "Function Evaluation." Let's break it down with an example. When you have a function, such as \( f(x) = \sqrt{x} \), you can determine its output values by substituting specific input values. For our exercise, we need to find \( f(x) \) at two different points: \( x_1 = 4 \) and \( x_2 = 9 \). This is done as follows:

• Substitute \( x_1 \) into the function: \( f(4) = \sqrt{4} = 2 \).
• Substitute \( x_2 \) into the function: \( f(9) = \sqrt{9} = 3 \).

By performing function evaluation at these specific points, we have established that the outputs are 2 and 3 when the inputs are 4 and 9, respectively.

Formula Application

Once the function values have been obtained from function evaluation, the next step is to apply the formula for the average rate of change. Knowing this formula is key:\[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]Here, the formula calculates the difference in function values divided by the difference in x-values. Let's break it down:

• "subtract \( f(x_1) \) from \( f(x_2) \)" means the difference between the function values at these two points. In our case, \( 3 - 2 \).
• "subtract \( x_1 \) from \( x_2 \)" refers to the change in x-values or the interval length. This equates to \( 9 - 4 \).

Now, by applying the formula, we substitute these values:- Numerator becomes: \( 3 - 2 = 1 \).- Denominator becomes: \( 9 - 4 = 5 \).Thus, the average rate of change is \( \frac{1}{5} \).

Interval Calculation

Interval calculation involves determining the range over which we are measuring the change in the function. It's the denominator in our earlier formula, and it represents the distance or gap between two x-values. For our problem:

• The interval starts at \( x_1 = 4 \).
• The interval ends at \( x_2 = 9 \).

Calculating the interval involves simply finding the difference between these endpoints:\[ x_2 - x_1 = 9 - 4 = 5 \]This tells us that the interval is 5 units long. Knowing the interval is crucial because it directly affects the calculation of the average rate of change. If the interval was different, the average rate of change could also differ, emphasizing the importance of precise interval calculation.