Question

1. the graph of y = f(x) is shown. order the following from least to greatest. a. average rate of change of f on 0,4. b. average rate of change of f on 0,2. c. average rate of change of f on 2,4. d. average rate of change of f on 2,5.

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b - a}$, which is the slope of the secant line connecting the points $(a,f(a))$ and $(b,f(b))$.

Step2: Analyze interval $[0,4]$

From the graph, let $a = 0$ and $b = 4$. The secant line from $(0,f(0))$ to $(4,f(4))$ has a positive slope.

Step3: Analyze interval $[0,2]$

Let $a = 0$ and $b = 2$. The secant line from $(0,f(0))$ to $(2,f(2))$ has a positive slope. The secant line on $[0,2]$ is steeper than the secant line on $[0,4]$, so the average rate of change on $[0,2]$ is greater than on $[0,4]$.

Step4: Analyze interval $[2,4]$

Let $a = 2$ and $b = 4$. The secant line from $(2,f(2))$ to $(4,f(4))$ has a negative slope since $f(2)>f(4)$.

Step5: Analyze interval $[2,5]$

Let $a = 2$ and $b = 5$. The secant line from $(2,f(2))$ to $(5,f(5))$ has a positive slope. The secant line on $[2,5]$ is less steep than the secant line on $[0,2]$ and $[0,4]$.

Step6: Order the average rates of change

Since the average rate of change on $[2,4]$ is negative and the others are positive, the order from least to greatest is: C, A, D, B.

Answer:

C, A, D, B