Question

a function is shown in the table below. on which interval of x is the average rate of change of the function the smallest?answerx=7 to x=17x=17 to x=41x=41 to x=48x=48 to x=65

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is given by \(\frac{\Delta y}{\Delta x}=\frac{f(x_2)-f(x_1)}{x_2 - x_1}\)

Step2: Calculate the average rate of change for \(x = 7\) to \(x = 17\)

Here \(x_1 = 7,y_1=6,x_2 = 17,y_2 = 18\)
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{18 - 6}{17 - 7}=\frac{12}{10}=1.2\)

Step3: Calculate the average rate of change for \(x = 17\) to \(x = 41\)

Here \(x_1 = 17,y_1 = 18,x_2=41,y_2 = 22\)
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{22 - 18}{41 - 17}=\frac{4}{24}=\frac{1}{6}\approx0.17\)

Step4: Calculate the average rate of change for \(x = 41\) to \(x = 48\)

Here \(x_1 = 41,y_1 = 22,x_2 = 48,y_2 = 27\)
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{27 - 22}{48 - 41}=\frac{5}{7}\approx0.71\)

Step5: Calculate the average rate of change for \(x = 48\) to \(x = 65\)

Here \(x_1 = 48,y_1 = 27,x_2 = 65,y_2 = 32\)
\(\frac{y_2 - y_1}{x_2 - x_1}=\frac{32 - 27}{65 - 48}=\frac{5}{17}\approx0.29\)

Answer:

\(x = 17\) to \(x = 41\)