Question

what is the rate of change of the average rates of change for each function over consecutive equal - length intervals?

1. $y = 13x - 10$
2. $f(x)=x^{2}+3x + 8$
3. $f(x)=6x - 2x^{2}+1$

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$.

Step2: Analyze $y=13x - 10$

Let the interval be $[x_1,x_1 + h]$ and $[x_1+h,x_1 + 2h]$.
For the interval $[x_1,x_1 + h]$, the average rate of change $A_1=\frac{(13(x_1 + h)-10)-(13x_1-10)}{(x_1 + h)-x_1}=\frac{13x_1+13h - 10-13x_1 + 10}{h}=13$.
For the interval $[x_1+h,x_1 + 2h]$, the average rate of change $A_2=\frac{(13(x_1 + 2h)-10)-(13(x_1+h)-10)}{(x_1 + 2h)-(x_1+h)}=\frac{13x_1+26h - 10-(13x_1+13h - 10)}{h}=13$.
The rate of change of the average - rates of change is $0$ since $A_2 - A_1=13 - 13 = 0$.

Step3: Analyze $f(x)=x^{2}+3x + 8$

For the interval $[x_1,x_1 + h]$, $f(x_1)=x_1^{2}+3x_1 + 8$ and $f(x_1 + h)=(x_1 + h)^{2}+3(x_1 + h)+8=x_1^{2}+2hx_1+h^{2}+3x_1+3h + 8$.
The average rate of change $A_1=\frac{f(x_1 + h)-f(x_1)}{(x_1 + h)-x_1}=\frac{(x_1^{2}+2hx_1+h^{2}+3x_1+3h + 8)-(x_1^{2}+3x_1 + 8)}{h}=2x_1+h + 3$.
For the interval $[x_1+h,x_1 + 2h]$, $f(x_1 + 2h)=(x_1 + 2h)^{2}+3(x_1 + 2h)+8=x_1^{2}+4hx_1+4h^{2}+3x_1+6h + 8$.
The average rate of change $A_2=\frac{f(x_1 + 2h)-f(x_1+h)}{(x_1 + 2h)-(x_1+h)}=\frac{(x_1^{2}+4hx_1+4h^{2}+3x_1+6h + 8)-(x_1^{2}+2hx_1+h^{2}+3x_1+3h + 8)}{h}=2x_1+3h + 3$.
The rate of change of the average - rates of change is $A_2 - A_1=(2x_1+3h + 3)-(2x_1+h + 3)=2h$. Since we are interested in the non - variable part with respect to the intervals, when considering the general case (treating $h$ as a non - zero constant for the interval length), the rate of change of the average rates of change is $2h$ (or just $2$ if we consider the unit interval i.e., $h = 1$).

Step4: Analyze $f(x)=6x-2x^{2}+1$

For the interval $[x_1,x_1 + h]$, $f(x_1)=6x_1-2x_1^{2}+1$ and $f(x_1 + h)=6(x_1 + h)-2(x_1 + h)^{2}+1=6x_1+6h-2(x_1^{2}+2hx_1+h^{2})+1=6x_1+6h-2x_1^{2}-4hx_1-2h^{2}+1$.
The average rate of change $A_1=\frac{f(x_1 + h)-f(x_1)}{(x_1 + h)-x_1}=\frac{(6x_1+6h-2x_1^{2}-4hx_1-2h^{2}+1)-(6x_1-2x_1^{2}+1)}{h}=6 - 4x_1-2h$.
For the interval $[x_1+h,x_1 + 2h]$, $f(x_1 + 2h)=6(x_1 + 2h)-2(x_1 + 2h)^{2}+1=6x_1+12h-2(x_1^{2}+4hx_1+4h^{2})+1=6x_1+12h-2x_1^{2}-8hx_1-8h^{2}+1$.
The average rate of change $A_2=\frac{f(x_1 + 2h)-f(x_1+h)}{(x_1 + 2h)-(x_1+h)}=\frac{(6x_1+12h-2x_1^{2}-8hx_1-8h^{2}+1)-(6x_1+6h-2x_1^{2}-4hx_1-2h^{2}+1)}{h}=6 - 4x_1-6h$.
The rate of change of the average - rates of change is $A_2 - A_1=(6 - 4x_1-6h)-(6 - 4x_1-2h)=-4h$ (or $- 4$ if $h = 1$).

Answer:

1. $0$
2. $2$
3. $-4$