Find the average rate of change of [tex]$y=\sqrt{x-2}$[/tex] between [tex]$x=3$[/tex] and [tex]$x=6$[/tex].
The average rate of change is $\square$.
Answer (1)
Calculate y 1 at x = 3 : y 1 = 3 − 2 = 1 .
Calculate y 2 at x = 6 : y 2 = 6 − 2 = 2 .
Calculate the average rate of change using the formula: x 2 − x 1 y 2 − y 1 = 6 − 3 2 − 1 .
Simplify the expression to find the average rate of change: 3 1 . The average rate of change is 3 1 .
Explanation
Understanding Average Rate of Change We are asked to find the average rate of change of the function y = x − 2 between x = 3 and x = 6 . The average rate of change of a function between two points is the change in the y -value divided by the change in the x -value. In other words, it's the slope of the secant line connecting the two points on the curve.
Calculate y1 First, we need to find the value of the function at x = 3 . We have
y 1 = 3 − 2 = 1 = 1
Calculate y2 Next, we need to find the value of the function at x = 6 . We have
y 2 = 6 − 2 = 4 = 2
Calculate Average Rate of Change Now, we can calculate the average rate of change using the formula:
x 2 − x 1 y 2 − y 1 = 6 − 3 2 − 1 = 3 1
Final Answer Therefore, the average rate of change of the function y = x − 2 between x = 3 and x = 6 is 3 1 .
Examples
Imagine you are tracking the growth of a plant. At week 3, the plant's height is 3 − 2 = 1 inch. At week 6, the plant's height is 6 − 2 = 2 inches. The average rate of change represents the average growth rate of the plant per week over this period. In this case, the average growth rate is 3 1 inch per week. This concept is useful in various real-world scenarios, such as tracking population growth, analyzing financial investments, or monitoring the speed of a chemical reaction.
