Question

a function is graphed below. on which interval of x is the average rate of change of the function the smallest?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is given by \(\frac{f(b)-f(a)}{b - a}\).

Step2: Calculate average rate of change for \([0,1]\)

Here \(a = 0\), \(b=1\), \(f(0)=38\), \(f(1)=82\).

$$ \frac{82 - 38}{1-0}=\frac{44}{1}=44 $$

Step3: Calculate average rate of change for \([1,5]\)

Here \(a = 1\), \(b = 5\), \(f(1)=82\), \(f(5)=115\).

$$ \frac{115 - 82}{5 - 1}=\frac{33}{4}=8.25 $$

Step4: Calculate average rate of change for \([5,14]\)

Here \(a = 5\), \(b = 14\), \(f(5)=115\), \(f(14)=148\).

$$ \frac{148-115}{14 - 5}=\frac{33}{9}\approx3.67 $$

Step5: Calculate average rate of change for \([14,26]\)

Here \(a = 14\), \(b = 26\), \(f(14)=148\), \(f(26)=191\).

$$ \frac{191-148}{26 - 14}=\frac{43}{12}\approx3.58 $$

Answer:

The interval \([14,26]\) has the smallest average rate of change.