Question

1. what is the average rate of change of ( g(x)=x^{2} ) on the interval ( 3 leq x leq 9 )?

(1) 8
(3) 15
(2) 12
(4) 18

1. given the graph of the function ( f(x) ) shown below, find the average rate of change over each of the following intervals. express in simplest form. show the calculations you use to find your answers.

(a) ( -3 leq x leq 1 )
(b) ( -1 leq x leq 8 )
(c) ( 1 leq x leq 9 )
(d) ( -6 leq x leq 2 )

1. consider the function given by ( f(x)=6 x+5 ). find its average rate of change over the following intervals.

(a) ( 1 leq x leq 3 )
(b) ( 2 leq x leq 7 )

1. the average rate of change should have been the same for both (a) and (b) in the last p what is special about this function that ensures its average rate of change will always be same?

Explanation:

Step1: Calcular \( f(3) \) y \( f(1) \)

Para \( x = 3 \), \( f(3)=6\times3 + 5=18 + 5=23 \)
Para \( x = 1 \), \( f(1)=6\times1+5 = 6 + 5=11 \)

Step2: Aplicar la fórmula del promedio

La fórmula del promedio es \( \frac{f(b)-f(a)}{b - a} \), donde \( a = 1 \), \( b = 3 \)
\( \frac{f(3)-f(1)}{3 - 1}=\frac{23-11}{2}=\frac{12}{2}=6 \)

Step3: Calcular \( f(7) \) y \( f(2) \)

Para \( x = 7 \), \( f(7)=6\times7+5=42 + 5 = 47 \)
Para \( x = 2 \), \( f(2)=6\times2+5=12 + 5=17 \)

Step4: Aplicar la fórmula del promedio

La fórmula del promedio es \( \frac{f(b)-f(a)}{b - a} \), donde \( a = 2 \), \( b = 7 \)
\( \frac{f(7)-f(2)}{7 - 2}=\frac{47-17}{5}=\frac{30}{5}=6 \)

Answer:

(a) \( 6 \)
(b) \( 6 \)