Question

how would you show that all linear functions have the property that their average rate of change over every interval is the same? make a table?

Answer 1

by definition a LINEAR function has the same slope at all points- as well over all intervals.

Answer 2

that is an excellent point roachie but i how would i demonstrate that?

Answer 3

The rate of change of a linear function is always a constant. The average value of a constant over every interval is that constant itself. Therefore the average rate of change of a linear funtion is constant (doesn't change) over all intervals.

Answer 4

I would show that the average rate of change over several intervals is equal to the instantaneous rate of change .

so do point slope between 2 or 3 interval of x, then compute the derivative of the function and show that the average rate are equal to the average computed from the point-slope over your intervals.

Answer 5

the derivative is the kicker. for a linear function it will be of the form Ax^0 where the slope for all x is equal to A

Answer 6

im not very far in my calc class yet we are still stalking about limits but to find the instantaneous velocity all we have to do is take the derivative of the function?