Sure, let's solve the problem step-by-step.
Given the function:
[tex]\[ f(x) = x^3 - 5x^2 \][/tex]
and the values of [tex]\( x \)[/tex] are [tex]\( x_0 = 0 \)[/tex] and [tex]\( x_1 = 10 \)[/tex].
### (a) Net Change
First, we need to determine the net change in the function values between [tex]\( x = 0 \)[/tex] and [tex]\( x = 10 \)[/tex].
1. Calculate [tex]\( f(x_0) \)[/tex]:
[tex]\[ f(0) = 0^3 - 5(0^2) = 0 \][/tex]
2. Calculate [tex]\( f(x_1) \)[/tex]:
[tex]\[ f(10) = 10^3 - 5(10^2) = 1000 - 500 = 500 \][/tex]
3. Determine the net change:
[tex]\[ \text{Net Change} = f(x_1) - f(x_0) = 500 - 0 = 500 \][/tex]
So, the net change is:
[tex]\[ \boxed{500} \][/tex]
### (b) Average Rate of Change
Next, we need to determine the average rate of change of the function between [tex]\( x = 0 \)[/tex] and [tex]\( x = 10 \)[/tex].
The average rate of change is given by:
[tex]\[ \text{Average Rate of Change} = \frac{\text{Net Change}}{x_1 - x_0} \][/tex]
From part (a), we know that the net change is 500 and the change in [tex]\( x \)[/tex] values is:
[tex]\[ x_1 - x_0 = 10 - 0 = 10 \][/tex]
So, the average rate of change is:
[tex]\[ \text{Average Rate of Change} = \frac{500}{10} = 50 \][/tex]
Therefore, the average rate of change is:
[tex]\[ \boxed{50} \][/tex]
In summary:
(a) The net change is [tex]\( 500 \)[/tex].
(b) The average rate of change is [tex]\( 50 \)[/tex].
