Question

1. given the function ( f(x)=3x^{2}-2x + 15 ) and the function ( g(x) ) shown in the table below, which has the greater average rate of change, ( f(x) ) on the interval ( 3leq xleq7 ) or ( g(x) ) on the interval ( 4leq xleq9 )? show calculations to support your answer.

Explanation:

Step1: Calculate the average rate of change for \(f(x)\)

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\).
For \(f(x)=3x^{2}-2x + 15\) and the interval \([3,7]\):
First, find \(f(3)\) and \(f(7)\).
\(f(3)=3\times(3)^{2}-2\times(3)+15=3\times9-6 + 15=27-6 + 15=36\)
\(f(7)=3\times(7)^{2}-2\times(7)+15=3\times49-14 + 15=147-14 + 15=148\)
Then, the average rate of change of \(f(x)\) over \([3,7]\) is \(\frac{f(7)-f(3)}{7 - 3}=\frac{148-36}{4}=\frac{112}{4}=28\)

Step2: Calculate the average rate of change for \(g(x)\)

For the function \(g(x)\) and the interval \([4,9]\).
From the table, when \(x = 4\), \(g(4)=82\) and when \(x = 9\), \(g(9)=257\)
The average rate of change of \(g(x)\) over \([4,9]\) is \(\frac{g(9)-g(4)}{9 - 4}=\frac{257-82}{5}=\frac{175}{5}=35\)

Answer:

Since \(28<35\), \(g(x)\) has a greater average rate of change.