Question

activity 2 maths connection average rate of change the average rate of change of a function is the ratio of the change in the dependent variable to the change in the independent variable over a specific interval. the formula for average rate of change is \\( \frac{f(b) - f(a)}{b - a} \\) for the interval a, b. the expression \\( f(b) - f(a) \\) represents the change in the input values of the function f. the expression \\( b - a \\) represents the change in the output values of the function f as the input values change from a to b. worked example you can determine the average rate of change of zoranas profit for the time interval 3.25, 4.25. zoranas profit over years 0–5 profit (thousands of dollars) time (years) substitute the input and output values into the average rate of change formula. evaluate the expression. the average rate of change for the time interval 3.25, 4.25 is approximately $634,000 per year. ask yourself for which other function types have you previously determined the average rate of change? is the formula the same? sam i can quickly estimate the average rate of change for intervals that are above and below the x - axis because they add to zero. for example, at year 1, the profit is about $300,000 and at year 2.25 the profit is about - $300,000. therefore, the average rate of change for the time interval (1, 2.25) is approximately $0. 1 analyze the worked example ① explain why the average rate of change is $600,000 per year and not $600. ② explain why the average rate of change is positive over this interval. ③ what does the average rate of change represent in this problem situation? 2 determine the average rate of change of zoranas profit for the time interval 1, 3. 3 sam has a theory about the average rate of change. describe the error in sams reasoning.

Explanation:

To address the problem of finding the average rate of change for Zorana's profit over the time interval \((1, 3)\), we follow these steps:

Step 1: Recall the Average Rate of Change Formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Here, \( a = 1 \) and \( b = 3 \). We need the profit values at \( t = 1 \) and \( t = 3 \) from the graph.

Step 2: Identify Profit Values from the Graph

From the graph (Worked Example context):

• At \( t = 1 \) (year 1), the profit \( f(1) \approx 300 \) (thousands of dollars, so \( 300,000 \)).
• At \( t = 3 \) (year 3), the profit \( f(3) \approx 800 \) (thousands of dollars, so \( 800,000 \)).

Step 3: Substitute into the Formula

Substitute \( a = 1 \), \( b = 3 \), \( f(1) = 300 \), and \( f(3) = 800 \) into the formula:
\[
\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{800 - 300}{2} = \frac{500}{2} = 250
\]
(Note: The units are thousands of dollars per year, so \( 250,000 \) dollars per year.)

Final Answer

The average rate of change of Zorana’s profit over the time interval \((1, 3)\) is \(\boldsymbol{250}\) (thousand dollars per year, or \(\boldsymbol{\$250,000}\) per year).

Answer:

To address the problem of finding the average rate of change for Zorana's profit over the time interval \((1, 3)\), we follow these steps:

Step 1: Recall the Average Rate of Change Formula

The average rate of change of a function \( f(x) \) over the interval \([a, b]\) is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Here, \( a = 1 \) and \( b = 3 \). We need the profit values at \( t = 1 \) and \( t = 3 \) from the graph.

Step 2: Identify Profit Values from the Graph

From the graph (Worked Example context):

• At \( t = 1 \) (year 1), the profit \( f(1) \approx 300 \) (thousands of dollars, so \( 300,000 \)).
• At \( t = 3 \) (year 3), the profit \( f(3) \approx 800 \) (thousands of dollars, so \( 800,000 \)).

Step 3: Substitute into the Formula

Substitute \( a = 1 \), \( b = 3 \), \( f(1) = 300 \), and \( f(3) = 800 \) into the formula:
\[
\text{Average Rate of Change} = \frac{f(3) - f(1)}{3 - 1} = \frac{800 - 300}{2} = \frac{500}{2} = 250
\]
(Note: The units are thousands of dollars per year, so \( 250,000 \) dollars per year.)

Final Answer

The average rate of change of Zorana’s profit over the time interval \((1, 3)\) is \(\boldsymbol{250}\) (thousand dollars per year, or \(\boldsymbol{\$250,000}\) per year).