Average rate of change:
h(x)= x^3 -0.5e^x at [0,1]
The 1/2 should be 1/3, sorry
If you mean that the 0.5 should be 1/3, then
h(x) = x^3 - 1/3 e^x
h(0) = -1/3
h(1) = 1 - e/3
so, avg rate of change is (h(1)-h(0))/(1-0) = 4/3 - e/3
To find the average rate of change of a function over an interval, you need to calculate the difference in function values between the endpoints divided by the difference in the input values of the endpoints.
In this case, the function h(x) = x^3 - 0.5e^x and we want to find the average rate of change over the interval [0, 1].
Step 1: Find the value of the function at the endpoints.
Evaluate h(0) and h(1) by plugging in the corresponding values of x into the function.
h(0) = 0^3 - 0.5e^0 = 0 - 0.5(1) = -0.5
h(1) = 1^3 - 0.5e^1 = 1 - 0.5(e) = 1 - 0.5(e) ≈ 0.065
Step 2: Find the difference in function values.
Subtract the value of the function at the lower endpoint from the value at the upper endpoint.
Difference in function values = h(1) - h(0) = 0.065 - (-0.5) = 0.565
Step 3: Find the difference in input values.
The difference in input values is simply the length of the interval.
Difference in input values = 1 - 0 = 1
Step 4: Calculate the average rate of change.
Divide the difference in function values by the difference in input values.
Average rate of change = (h(1) - h(0)) / (1 - 0) = 0.565 / 1 = 0.565
Therefore, the average rate of change of the function h(x) = x^3 - 0.5e^x over the interval [0, 1] is approximately 0.565.