we know that
The Average Rate of Change is the slope of a line or a curve on a given range. It is defined as the ratio of the difference in the function f(x) as it changes from 'a' to 'b' to the difference between 'a' and 'b'.
So
\(A=\frac{(f(b)-f(a))}{(b-a)}\)
case a) \(f(x) = x^{2} + 3x\)
in the interval \([-2, 3]\)
\(a=-2\\ b=3\\ f(a)=(-2)^{2} +3*(-2)\\ f(a)=-2\\ f(b)=(3)^{2} +3*(3)\\ f(b)=18\)
Find the Average Rate of Change
\(A=\frac{(18-(-2))}{(3+2)}\)
\(A=\frac{20}{5}\)
\(A=4\)
case b) \(f(x) = 3x-8\)
in the interval \([4, 5]\)
\(a=4\\ b=5\\ f(a)=3*4-8\\ f(a)=4\\ f(b)=3*5-8\\ f(b)=7\)
Find the Average Rate of Change
\(A=\frac{(7-(4))}{(5-4)}\)
\(A=\frac{3}{1}\)
\(A=3\)
case c) \(f(x) = x^{2} -2x\)
in the interval \([-3, 4]\)
\(a=-3\\ b=4\\ f(a)=(-3)^{2} -2*(-3)\\ f(a)=15\\ f(b)=(4)^{2} -2*(4)\\ f(b)=8\)
Find the Average Rate of Change
\(A=\frac{(8-(15))}{(4+3)}\)
\(A=\frac{-7}{7}\)
\(A=-1\)
case d) \(f(x) = x^{2} -5\)
in the interval \([-1, 1]\)
\(a=-1\\ b=1\\ f(a)=(-1)^{2} -5\\ f(a)=-4\\ f(b)=(1)^{2} -5\\ f(b)=-4\)
Find the Average Rate of Change
\(A=\frac{(-4-(-4))}{(1+1)}\)
\(A=\frac{0}{2}\)
\(A=0\)
Arrange these functions from the greatest to the least value based on the average rate of change in the specified interval
so
the answer is
1) \(f(x) = x^{2} + 3x\) -------> \(A=4\)
2) \(f(x) = 3x-8\) -----> \(A=3\)
3) \(f(x) = x^{2} -5\) -----> \(A=0\)
4) \(f(x) = x^{2} -2x\) -----> \(A=-1\)
