Rate Change Average Ff1568

1. **State the problem:** We are asked to find the rate of change of the average rates of change for the functions given over consecutive equal-length intervals. 2. **Recall the average rate of change formula:** For a function $f(x)$ over an interval $[a, b]$, the average rate of change is $$\frac{f(b) - f(a)}{b - a}$$ 3. **Function 1: $y = 3x + 6$** - This is a linear function. - The average rate of change over any interval is constant and equals the slope, which is $3$. 4. **Function 2: $f(x) = 2x - 5x^2$** - This is a quadratic function. - We calculate the average rate of change over an interval $[x, x+h]$: $$\frac{f(x+h) - f(x)}{h} = \frac{(2(x+h) - 5(x+h)^2) - (2x - 5x^2)}{h}$$ 5. **Expand and simplify:** $$f(x+h) = 2x + 2h - 5(x^2 + 2xh + h^2) = 2x + 2h - 5x^2 - 10xh - 5h^2$$ So, $$\frac{f(x+h) - f(x)}{h} = \frac{(2x + 2h - 5x^2 - 10xh - 5h^2) - (2x - 5x^2)}{h}$$ $$= \frac{2h - 10xh - 5h^2}{h}$$ 6. **Cancel $h$ in numerator and denominator:** $$= \frac{\cancel{h}(2 - 10x - 5h)}{\cancel{h}} = 2 - 10x - 5h$$ 7. **Rate of change of the average rates of change:** - The average rate of change itself depends on $x$ and $h$. - To find the rate of change of the average rate of change over consecutive equal-length intervals, consider the difference between average rates over intervals $[x, x+h]$ and $[x+h, x+2h]$ divided by $h$. 8. **Calculate average rate over $[x+h, x+2h]$ similarly:** $$\frac{f(x+2h) - f(x+h)}{h}$$ Compute $f(x+2h)$: $$f(x+2h) = 2(x+2h) - 5(x+2h)^2 = 2x + 4h - 5(x^2 + 4xh + 4h^2) = 2x + 4h - 5x^2 - 20xh - 20h^2$$ So, $$\frac{f(x+2h) - f(x+h)}{h} = \frac{(2x + 4h - 5x^2 - 20xh - 20h^2) - (2x + 2h - 5x^2 - 10xh - 5h^2)}{h}$$ $$= \frac{2h - 10xh - 15h^2}{h}$$ 9. **Cancel $h$ again:** $$= \frac{\cancel{h}(2 - 10x - 15h)}{\cancel{h}} = 2 - 10x - 15h$$ 10. **Rate of change of average rates of change:** $$\frac{(2 - 10x - 15h) - (2 - 10x - 5h)}{h} = \frac{-15h + 5h}{h} = \frac{-10h}{h} = -10$$ 11. **Interpretation:** - For the quadratic function, the rate of change of the average rates of change over consecutive equal-length intervals is constant and equals $-10$. - For the linear function, this rate is $0$ because the average rate of change is constant. **Final answers:** - For $y = 3x + 6$, rate of change of average rates of change = $0$. - For $f(x) = 2x - 5x^2$, rate of change of average rates of change = $-10$.