Question

ange using graphs
$f(x)=2|x - 4|+2$

1. using the graph calculate the average rate of change in the interval 5 ≤ x ≤ 7
2. using the graph calculate the instantaneous rate of change at the point where x = 7

Explanation:

Step1: Recall average rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 5$, $b = 7$, and $f(x)=2|x - 4|+2$.

Step2: Calculate $f(5)$ and $f(7)$

For $x = 5$, $f(5)=2|5 - 4|+2=2\times1 + 2=4$. For $x = 7$, $f(7)=2|7 - 4|+2=2\times3+2 = 8$.

Step3: Compute average rate of change

$\frac{f(7)-f(5)}{7 - 5}=\frac{8 - 4}{2}=2$.

Step4: Recall instantaneous rate - of change concept

The instantaneous rate of change of a function at a point is the slope of the tangent line at that point. For the absolute - value function $y = 2|x - 4|+2$, when $x\geq4$, $y=2(x - 4)+2=2x-8 + 2=2x-6$.

Step5: Find the slope for $x\geq4$

The derivative of $y = 2x-6$ with respect to $x$ is $y^\prime=2$. So the instantaneous rate of change at $x = 7$ is 2.

Answer:

1. 2
2. 2