Question

average rate of change 1<= x <= 5.
x y
1 5
3 3
5 9
7 23

Answer 1

To find the average rate of change of \( y \) with respect to \( x \) over the interval \( 1 \leq x \leq 5 \), we will use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \( a \) and \( b \) are the endpoints of the interval. In this case:

- \( a = 1 \) (corresponding to \( x = 1 \), where \( y = 5 \))
- \( b = 5 \) (corresponding to \( x = 5 \), where \( y = 9 \))

Now, we can find \( f(a) \) and \( f(b) \):

- \( f(1) = 5 \)
- \( f(5) = 9 \)

Now, substituting these values into the formula:

\[
\text{Average Rate of Change} = \frac{f(5) - f(1)}{5 - 1} = \frac{9 - 5}{5 - 1} = \frac{4}{4} = 1
\]

Thus, the average rate of change of \( y \) with respect to \( x \) over the interval \( 1 \leq x \leq 5 \) is \( 1 \).