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Front

What is the average rate of change of a function?

Back

The average rate of change of a function over an interval is the change in the function's value divided by the change in the input value. It is calculated as \( \frac{f(b) - f(a)}{b - a} \) for the interval [a, b].

Front

How do you calculate the average rate of change for the function \( f(x) = 6(2)^x \) from x = -1 to x = 2?

Back

1. Calculate \( f(-1) = 6(2)^{-1} = 3 \) and \( f(2) = 6(2)^{2} = 24 \). 2. Average rate of change = \( \frac{24 - 3}{2 - (-1)} = \frac{21}{3} = 7 \).

Front

What is the formula for the average rate of change of a linear function?

Back

For a linear function \( f(x) = mx + b \), the average rate of change is constant and equal to the slope \( m \).

Front

How do you find the average rate of change of \( f(x) = 5x - 7 \) from x = -3 to x = 7?

Back

1. Calculate \( f(-3) = 5(-3) - 7 = -22 \) and \( f(7) = 5(7) - 7 = 28 \). 2. Average rate of change = \( \frac{28 - (-22)}{7 - (-3)} = \frac{50}{10} = 5 \).

Front

What is the average rate of change in a real-world context?

Back

In real-world scenarios, the average rate of change can represent speed, growth rate, or any change over time, such as the rate of descent of a diver.

Front

How do you calculate the average rate of change of Maya's dive from 1 to 2.5 seconds?

Back

1. Determine the height at 1 second and 2.5 seconds. 2. Average rate of change = \( \frac{height_{2.5} - height_{1}}{2.5 - 1} \).

Front

What is the average rate of change of fruit flies between Day 3 and Day 5 if there were 1900 flies on Day 3 and 7600 on Day 5?

Back

Average rate of change = \( \frac{7600 - 1900}{5 - 3} = \frac{5700}{2} = 2850 \) fruit flies per day.

Front

How do you find the average rate of change from a table of values?

Back

Identify the values at the two points of interest, then use the formula \( \frac{value_{2} - value_{1}}{time_{2} - time_{1}} \).

Front

What is the average rate of change in the number of lemons harvested between Day 4 and Day 6 if 20 lemons were harvested on Day 4 and 40 on Day 6?

Back

Average rate of change = \( \frac{40 - 20}{6 - 4} = \frac{20}{2} = 10 \) lemons per day.

Front

What does a negative average rate of change indicate?

Back

A negative average rate of change indicates that the function's value is decreasing over the interval.