Question
For which type of function does the average rate of change produce the same value for any interval of the domain on the function? Why does this occur for this type of function? dooodoooodooooo
Answers
GPT-4o mini
The type of function for which the average rate of change produces the same value for any interval of the domain is a **linear function**. Linear functions can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
### Why Does This Occur?
The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
For a linear function, the value of \( f(b) - f(a) \) can be computed as:
\[
f(b) = mb + b
\]
\[
f(a) = ma + b
\]
Thus,
\[
f(b) - f(a) = (mb + b) - (ma + b) = m(b - a)
\]
When you substitute this back into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{m(b - a)}{b - a} = m
\]
This shows that the average rate of change over any interval \([a, b]\) for a linear function is equal to the slope \( m \) of the line, which is a constant value.
For non-linear functions, the slope will change depending on the values of \( a \) and \( b \), leading to different average rates of change for different intervals. Therefore, the property that holds for linear functions is the constancy of their slope, leading to a consistent average rate of change across any interval of their domain.
### Why Does This Occur?
The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is calculated using the formula:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
For a linear function, the value of \( f(b) - f(a) \) can be computed as:
\[
f(b) = mb + b
\]
\[
f(a) = ma + b
\]
Thus,
\[
f(b) - f(a) = (mb + b) - (ma + b) = m(b - a)
\]
When you substitute this back into the average rate of change formula:
\[
\text{Average Rate of Change} = \frac{m(b - a)}{b - a} = m
\]
This shows that the average rate of change over any interval \([a, b]\) for a linear function is equal to the slope \( m \) of the line, which is a constant value.
For non-linear functions, the slope will change depending on the values of \( a \) and \( b \), leading to different average rates of change for different intervals. Therefore, the property that holds for linear functions is the constancy of their slope, leading to a consistent average rate of change across any interval of their domain.