find the average rate of change of the function f(x)=x^3+1 over the interval [-1,1]
average rate=(f(1)-f(-1))/2
= (4-0)/2=2
To find the average rate of change of a function over an interval, you need to calculate the difference in the function's values at the endpoints of the interval and divide it by the difference in the x-values of those endpoints.
In this case, the function is f(x) = x^3 + 1, and the interval is [-1, 1].
Step 1: Find the value of the function at the endpoint x = -1
Plug x = -1 into the function:
f(-1) = (-1)^3 + 1 = -1 + 1 = 0
Step 2: Find the value of the function at the endpoint x = 1
Plug x = 1 into the function:
f(1) = (1)^3 + 1 = 1 + 1 = 2
Step 3: Find the difference in the function values:
Difference = f(1) - f(-1) = 2 - 0 = 2
Step 4: Find the difference in x-values:
Difference in x-values = 1 - (-1) = 2
Step 5: Calculate the average rate of change:
Average Rate of Change = Difference in function values / Difference in x-values = 2/2 = 1
Therefore, the average rate of change of the function f(x) = x^3 + 1 over the interval [-1, 1] is 1.