Calculate the average rate of change of a function over a specified interval.

Which expression can be used to determine the average rate of change in [tex]f(x)[/tex] over the interval [tex][2,9][/tex]?

[tex]\frac{f(9) - f(2)}{9 - 2}[/tex]



Answer :

To calculate the average rate of change of a function [tex]\( f(x) \)[/tex] over a specified interval [tex]\([x_1, x_2]\)[/tex], you can use the formula:

[tex]\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]

Let's apply this formula to the interval [tex]\([2, 9]\)[/tex].

1. Identify [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
- [tex]\( x_1 = 2 \)[/tex]
- [tex]\( x_2 = 9 \)[/tex]

2. Determine [tex]\( f(x_1) \)[/tex] and [tex]\( f(x_2) \)[/tex]:
- Suppose [tex]\( f(2) = 2 \)[/tex]
- Suppose [tex]\( f(9) = 9 \)[/tex]

3. Substitute the values into the formula:
[tex]\[ \text{Average rate of change} = \frac{f(9) - f(2)}{9 - 2} \][/tex]

4. Calculate [tex]\( f(9) - f(2) \)[/tex]:
[tex]\[ f(9) - f(2) = 9 - 2 = 7 \][/tex]

5. Determine the length of the interval:
[tex]\[ 9 - 2 = 7 \][/tex]

6. Divide the difference in function values by the length of the interval:
[tex]\[ \text{Average rate of change} = \frac{7}{7} = 1 \][/tex]

So, the average rate of change of the function [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex] is [tex]\( 1 \)[/tex].

Thus, the expression that can be used to determine the average rate of change in [tex]\( f(x) \)[/tex] over the interval [tex]\([2, 9]\)[/tex] is:

[tex]\[ \frac{f(9) - f(2)}{9 - 2} \][/tex]