Comparing Average Rates of Change of Functions
The question is asking to compare the average rates of change for the pair of functions over the given interval [1, 5].
The given functions are:
- \( f(x) = 9x^2 \)
- \( g(x) = 1 + 3x^2 \)
The average rate of change of a function over an interval \([a, b]\) can be found using the formula:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
Let's compute this for each function over the interval [1, 5].
For \( f(x) = 9x^2 \):
- \( f(1) = 9(1)^2 = 9 \)
- \( f(5) = 9(5)^2 = 225 \)
- \( \text{Average rate of change of} \ f(x) \text{ over} \ [1, 5] = \frac{f(5) - f(1)}{5 - 1} = \frac{225 - 9}{4} = \frac{216}{4} = 54 \)
For \( g(x) = 1 + 3x^2 \):
- \( g(1) = 1 + 3(1)^2 = 4 \)
- \( g(5) = 1 + 3(5)^2 = 1 + 75 = 76 \)
- \( \text{Average rate of change of} \ g(x) \text{ over} \ [1, 5] = \frac{g(5) - g(1)}{5 - 1} = \frac{76 - 4}{4} = \frac{72}{4} = 18 \)
Therefore:
- The average rate of change of \( f(x) \) over [1, 5] is \( 54 \).
- The average rate of change of \( g(x) \) over [1, 5] is \( 18 \).
- The average rate of change of \( f(x) - g(x) \) over [1, 5] is \( 54 - 18 = 36 \).
So, the average rate of change of \( f \) over [1, 5] is \( 54 \), of \( g \) over [1, 5] is \( 18 \), and of \( f - g \) over [1, 5] is \( 36 \).
