Calculate the average rate of change of the given function f over the intervals [a, a+h] where h=1, 0.1, 0.01, 0.001, and 0.0001.
I have no clue how to do this. Any help would be appreciated. Thanks
I know I use the formula
f=f (a+h) - f (a)/h
Then do I need to plug in
the -2x - (3x^2)
of just plug the zero in for the x?
f(x)= 3x^2 -2x ; a=0
f(a) = 3a^2-2a
f(a+h) = 3(a+h)^2 - 2(a+h)
= 3a^2 + 6ah + 3h^2 - 2a - 2h
f(a+h)-f(a) = 6ah + 3h^2 + 2h
So,
(f(a+h)-f(a))/h = 6a + 3h + 2
So, with various values of h,
f(a+1) = 6a+3+2
f(a+0.1) = 6a + 0.3 + 2 = 6a + 2.03
and so on. Nothing difficult, just algebra I.
To calculate the average rate of change of a function over an interval, you can follow these steps:
1. Start with the given function: f(x) = 3x^2 - 2x.
2. Choose a value for 'a'. In this case, a = 0.
3. Determine the value of f(a). Plug in the value of 'a' (which is 0) into the function: f(0) = 3(0)^2 - 2(0) = 0. So, f(a) = 0.
4. Set up the average rate of change formula: Average rate of change = (f(a+h) - f(a)) / h.
5. Choose different values for 'h' (the interval length) and calculate the corresponding average rate of change.
For h = 1:
f(a + h) = f(0 + 1) = f(1) = 3(1)^2 - 2(1) = 1.
Average rate of change = (f(a + h) - f(a)) / h = (1 - 0) / 1 = 1.
For h = 0.1:
f(a + h) = f(0 + 0.1) = f(0.1) = 3(0.1)^2 - 2(0.1) = 0.29.
Average rate of change = (f(a + h) - f(a)) / h = (0.29 - 0) / 0.1 = 2.9.
Similarly, for h = 0.01, 0.001, and 0.0001, you can calculate the corresponding average rate of change.
By following these steps, you can find the average rate of change of the given function over the given intervals.