Question: Problems 13 - 16, What is the rate of change of the average rates of change for each function over consecutive equal-length input intervals?

1. **State the problem:** We need to find the rate of change of the average rates of change for each function over consecutive equal-length input intervals. 2. **Recall the concept:** The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by $$\frac{f(b)-f(a)}{b-a}.$$ The rate of change of these average rates over consecutive intervals measures how the average rate itself changes as we move along the $x$-axis. 3. **Analyze each function:** **Problem 13: $f(x) = 5x - 12$** - This is a linear function. - The average rate of change over any interval is constant and equals the slope $5$. - Therefore, the rate of change of the average rates of change is $0$. **Problem 14: $g(x) = x^2 - 4x + 3$** - This is a quadratic function. - The average rate of change over $[x, x+h]$ is $$\frac{g(x+h)-g(x)}{h} = \frac{(x+h)^2 - 4(x+h) + 3 - (x^2 - 4x + 3)}{h}.$$ - Simplify numerator: $$ (x^2 + 2xh + h^2 - 4x - 4h + 3) - (x^2 - 4x + 3) = 2xh + h^2 - 4h $$ - So average rate of change is $$\frac{2xh + h^2 - 4h}{h} = 2x + h - 4.$$ - The average rate of change depends linearly on $x$. - The rate of change of the average rates over consecutive intervals of length $h$ is the difference: $$ (2(x+h) + h - 4) - (2x + h - 4) = 2h.$$ - Dividing by $h$ (to get rate per unit $x$), we get $$\frac{2h}{h} = 2.$$ - So the rate of change of average rates of change is constant $2$. **Problem 15: $h(x) = 3x - 2x^2 + 1$** - Quadratic function. - Average rate of change over $[x, x+h]$: $$\frac{h(x+h) - h(x)}{h} = \frac{3(x+h) - 2(x+h)^2 + 1 - (3x - 2x^2 + 1)}{h}.$$ - Simplify numerator: $$3x + 3h - 2(x^2 + 2xh + h^2) + 1 - 3x + 2x^2 - 1 = 3h - 2x^2 - 4xh - 2h^2 + 2x^2 = 3h - 4xh - 2h^2.$$ - So average rate of change is: $$\frac{3h - 4xh - 2h^2}{h} = 3 - 4x - 2h.$$ - Rate of change of average rates over consecutive intervals: $$[3 - 4(x+h) - 2h] - [3 - 4x - 2h] = -4h.$$ - Dividing by $h$: $$\frac{-4h}{h} = -4.$$ - So the rate of change of average rates of change is constant $-4$. **Problem 16: $k(x) = 7x$** - Linear function. - Average rate of change is constant $7$. - Rate of change of average rates is $0$. 4. **Summary:** - For linear functions $f(x) = mx + b$, rate of change of average rates is $0$. - For quadratic functions $ax^2 + bx + c$, rate of change of average rates is $2a$. 5. **Final answers:** - Problem 13: $0$ - Problem 14: $2$ - Problem 15: $-4$ - Problem 16: $0$