Question

1. $0 < x < 3$

find the average rate of change over the given interval.
a) $3/7$
b) $7$
c) $4$
d) $7/3$

Explanation:

Step1: Identify points on the interval

From the graph, when \( x = 0 \), the \( y \)-value (let's call it \( f(0) \)) is 1 (since the graph passes through \( (0,1) \)). When \( x = 3 \), the \( y \)-value (\( f(3) \)) is 8 (from the plotted point at \( (3,8) \)).

Step2: Apply average rate of change formula

The formula for the average rate of change of a function \( f(x) \) over the interval \( [a,b] \) is \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 0 \), \( b = 3 \), \( f(a)=1 \), and \( f(b)=8 \).
So, substitute the values into the formula: \( \frac{f(3) - f(0)}{3 - 0}=\frac{8 - 1}{3 - 0}=\frac{7}{3} \).

Answer:

d) \( \frac{7}{3} \)