To find the average rate of change of the function [tex]\( f(x) = 9 - x^2 \)[/tex] over given intervals, we will use the formula for the average rate of change, which is:
[tex]\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \][/tex]
Let's apply this to each set of points provided:
### (a) For the interval [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex]:
1. Calculate [tex]\( f(x) \)[/tex] at the given points:
- At [tex]\( x_1 = 0 \)[/tex]:
[tex]\[ f(0) = 9 - 0^2 = 9 \][/tex]
- At [tex]\( x_2 = 3 \)[/tex]:
[tex]\[ f(3) = 9 - 3^2 = 9 - 9 = 0 \][/tex]
2. Calculate the change in [tex]\( f(x) \)[/tex] and [tex]\( x \)[/tex]:
- Change in [tex]\( f(x) \)[/tex]:
[tex]\[ \Delta f = f(3) - f(0) = 0 - 9 = -9 \][/tex]
- Change in [tex]\( x \)[/tex]:
[tex]\[ \Delta x = 3 - 0 = 3 \][/tex]
3. Compute the average rate of change:
[tex]\[ \text{Average Rate of Change} = \frac{\Delta f}{\Delta x} = \frac{-9}{3} = -3.0 \][/tex]
So, the average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex] is -3.0.
### (b) For the interval [tex]\( x = -1 \)[/tex] to [tex]\( x = 5 \)[/tex]:
1. Calculate [tex]\( f(x) \)[/tex] at the given points:
- At [tex]\( x_1 = -1 \)[/tex]:
[tex]\[ f(-1) = 9 - (-1)^2 = 9 - 1 = 8 \][/tex]
- At [tex]\( x_2 = 5 \)[/tex]:
[tex]\[ f(5) = 9 - 5^2 = 9 - 25 = -16 \][/tex]
2. Calculate the change in [tex]\( f(x) \)[/tex] and [tex]\( x \)[/tex]:
- Change in [tex]\( f(x) \)[/tex]:
[tex]\[ \Delta f = f(5) - f(-1) = -16 - 8 = -24 \][/tex]
- Change in [tex]\( x \)[/tex]:
[tex]\[ \Delta x = 5 - (-1) = 5 + 1 = 6 \][/tex]
3. Compute the average rate of change:
[tex]\[ \text{Average Rate of Change} = \frac{\Delta f}{\Delta x} = \frac{-24}{6} = -4.0 \][/tex]
So, the average rate of change from [tex]\( x = -1 \)[/tex] to [tex]\( x = 5 \)[/tex] is -4.0.
### Summary:
- The average rate of change from [tex]\( x = 0 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(-3.0\)[/tex].
- The average rate of change from [tex]\( x = -1 \)[/tex] to [tex]\( x = 5 \)[/tex] is [tex]\(-4.0\)[/tex].
