Question - Comparing Average Rates of Change of Functions
Solution:
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 $$ 36 $$.
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