James O. answered • 01/06/15
Tutor
4.3
(4)
Patient and easy-to-understand math tutor - HMC / UCI Grad
Okay so f(x) = 2^x
Average rate of change of a function over an interval basically represents the slope of the line connecting the two endpoints of the function across that interval. Much easier to explain if I could draw, but hopefully you get what I am saying here. Slope is rise / run, and that's exactly what we need here. The rise is the change in function values f(x) and the run is the change in x.
For the interval [1,4], the value of f changes from f[1] = 2^1 = 2, to f[4] = 2^4 = 16. From 2 to 16 is a change of +14. The input value, x, changed from 1 to 4 for a total change of +3. So the average rate of change over the interval is (rise / run) = ( +14 / +3 ) = 14/3
For the interval [3,5], same calculation just different numbers. Just solve ( f(5) - f(3) ) / (5 - 3)
For the final question, you really need to draw yourself a graph of this function. Constant rate of change only happens with linear functions, i.e. straight lines. And if you graph out y = 2^x, you should quickly find that it is not a straight line but rather an exponential curve. The rate of change is increasing quickly as x increases.
Average rate of change of a function over an interval basically represents the slope of the line connecting the two endpoints of the function across that interval. Much easier to explain if I could draw, but hopefully you get what I am saying here. Slope is rise / run, and that's exactly what we need here. The rise is the change in function values f(x) and the run is the change in x.
For the interval [1,4], the value of f changes from f[1] = 2^1 = 2, to f[4] = 2^4 = 16. From 2 to 16 is a change of +14. The input value, x, changed from 1 to 4 for a total change of +3. So the average rate of change over the interval is (rise / run) = ( +14 / +3 ) = 14/3
For the interval [3,5], same calculation just different numbers. Just solve ( f(5) - f(3) ) / (5 - 3)
For the final question, you really need to draw yourself a graph of this function. Constant rate of change only happens with linear functions, i.e. straight lines. And if you graph out y = 2^x, you should quickly find that it is not a straight line but rather an exponential curve. The rate of change is increasing quickly as x increases.
