Question

1. the average monthly rainfall, in inches, for fresno, california can be modeled by a quadratic function $c$, where $c(m)$ represents the average rainfall in the mth month of the year. if $c(1)=2.16,c(5)=0.39$ and $c(10)=0.65$, is the rate of change of the average rates of change of $c$ positive or negative? how do you know?

Explanation:

Step1: Calculate average rate of change between first - two intervals

The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is $\frac{f(b)-f(a)}{b - a}$.
For the interval from $m = 1$ to $m = 5$, the average rate of change $r_1$ of $C(m)$ is $\frac{C(5)-C(1)}{5 - 1}=\frac{0.39 - 2.16}{4}=\frac{-1.77}{4}=- 0.4425$.

Step2: Calculate average rate of change between second - two intervals

For the interval from $m = 5$ to $m = 10$, the average rate of change $r_2$ of $C(m)$ is $\frac{C(10)-C(5)}{10 - 5}=\frac{0.65 - 0.39}{5}=\frac{0.26}{5}=0.052$.

Step3: Calculate the rate of change of the average rates of change

The rate of change of the average rates of change is $\frac{r_2-r_1}{10 - 1}=\frac{0.052-(-0.4425)}{5}=\frac{0.052 + 0.4425}{5}=\frac{0.4945}{5}=0.0989$.

Answer:

Positive, because the rate of change of the average rates of change calculated as $\frac{\frac{C(10)-C(5)}{10 - 5}-\frac{C(5)-C(1)}{5 - 1}}{10 - 1}=0.0989>0$.