f(x) = 4x2; x = 2, x = 2 + h
this is telling me to find the average rate of change and the net change? I found one but my answer is disapproved from a friend.
the average rate of change from x=2 to x=2+h is
[f(x+h)-f(x)]/h
= (4(x+h)^2-4x^2)/h
= (4x^2+8xh+4h^2-4x^2)/h
= (8xh+4h^2)/h
= 8x+4h
The net change is just f(x+h)-f(x) = 8xh+4h^2
At x=2, the net change is 16h+4h^2
and the average change is 16+4h
To find the average rate of change of a function, you need to calculate the difference in the function's values at two given points, and then divide that difference by the difference in the input values. In this case, you are given the function f(x) = 4x^2 and two x-values, x = 2 and x = 2 + h.
To find the average rate of change, you can follow these steps:
1. Find the value of the function f(x) at x = 2 and x = 2 + h:
- Substitute x = 2 into the function f(x) = 4x^2: f(2) = 4(2)^2 = 4(4) = 16.
- Substitute x = 2 + h into the function f(x) = 4x^2: f(2 + h) = 4(2 + h)^2.
2. Calculate the difference in the function's values:
- Subtract f(2) from f(2 + h): f(2 + h) - f(2).
3. Calculate the difference in the input values:
- Subtract 2 from 2 + h: (2 + h) - 2 = h.
4. Divide the difference in the function's values by the difference in the input values:
- Average rate of change = (f(2 + h) - f(2)) / (2 + h - 2) = (f(2 + h) - f(2)) / h.
To find the net change, you need to calculate the difference in the function's values at two points. In this case, you are finding the net change between x = 2 and x = 2 + h.
To find the net change, you can use the same steps as above, but without dividing by the difference in the input values (h) in step 4.
Make sure to carefully follow these steps and double-check your calculations to ensure accurate answers. If your friend's answer is different, it is possible that there was an error either in your calculations or in the interpretation of the problem.