Question

the average rate of change for 0≤x≤1.5 is \boxed{}. the average rate of change for the interval 1≤x≤2 is

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $f(x)$ over $a \leq x \leq b$ is $\frac{f(b)-f(a)}{b-a}$.

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For interval $0 \leq x \leq 1.5$:

Step2: Identify $f(0)$ and $f(1.5)$

From the graph: $f(0) = -3$.
Estimate $f(1.5)$: The function goes from $(1,0)$ to $(2,6)$. The slope between these points is 6, so $f(1.5)=0 + 6(0.5)=3$.

Step3: Calculate rate of change

$\frac{f(1.5)-f(0)}{1.5-0} = \frac{3 - (-3)}{1.5} = \frac{6}{1.5} = 4$

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For interval $1 \leq x \leq 2$:

Step4: Identify $f(1)$ and $f(2)$

From the graph: $f(1)=0$, $f(2)=6$.

Step5: Calculate rate of change

$\frac{f(2)-f(1)}{2-1} = \frac{6-0}{1} = 6$

Answer:

For $0 \leq x \leq 1.5$: $4$
For $1 \leq x \leq 2$: $6$