h(t)=cos t, [pi/4, 3pi/4]
find the average rate of change of the function over the given interval or intervals
4.6
To find the average rate of change of a function over a given interval, you need to calculate the difference in function values at the endpoints of the interval and divide it by the difference in inputs (or "x-values"). In this case, the function h(t) = cos t is given over the interval [π/4, 3π/4].
1. Find the function values at the endpoints of the interval:
- Plug in the value of the lower endpoint, t = π/4, into the function h(t) = cos t:
h(π/4) = cos(π/4)
- Plug in the value of the upper endpoint, t = 3π/4, into the function h(t) = cos t:
h(3π/4) = cos(3π/4)
2. Calculate the difference in function values:
- Subtract the function value at the lower endpoint from the function value at the upper endpoint:
h(3π/4) - h(π/4)
3. Calculate the difference in inputs (x-values):
- Subtract the lower endpoint from the upper endpoint:
3π/4 - π/4
4. Divide the difference in function values by the difference in inputs:
- Divide the result from step 2 by the result from step 3:
(h(3π/4) - h(π/4)) / (3π/4 - π/4)
After following these steps, you will have the average rate of change of the function h(t) = cos t over the interval [π/4, 3π/4].