for the function f(t) = t^3 + 2/t find the average rate of change of f(t) at interval[1, 4]
To find the average rate of change of the function f(t) = t^3 + 2/t over the interval [1, 4], we need to calculate the difference in the function values at the endpoints of the interval, and divide it by the difference in the values of t at the endpoints.
The function values at the endpoints are:
f(1) = 1^3 + 2/1 = 1 + 2 = 3
f(4) = 4^3 + 2/4 = 64 + 1/2 = 128 + 1/2 = 129/2
The values of t at the endpoints are:
t1 = 1
t2 = 4
Now, we can calculate the difference in function values and divide it by the difference in t values:
Average rate of change = (f(4) - f(1))/(4 - 1) = (129/2 - 3)/(4 - 1) = (129/2 - 6/2)/3 = (123/2)/3 = 123/6 = 41/2
Therefore, the average rate of change of f(t) at the interval [1, 4] is 41/2.
To find the average rate of change of a function on a given interval, we need to calculate the change in the function's value divided by the change in the input variable. In this case, we want to find the average rate of change of the function f(t) = t^3 + 2/t over the interval [1, 4].
First, calculate the function value at the beginning and end of the interval:
f(1) = 1^3 + 2/1 = 1 + 2 = 3
f(4) = 4^3 + 2/4 = 64 + 0.5 = 64.5
Next, calculate the change in the function value and the change in the input variable:
Change in function value = f(4) - f(1) = 64.5 - 3 = 61.5
Change in input variable = 4 - 1 = 3
Finally, divide the change in function value by the change in input variable to get the average rate of change:
Average rate of change = Change in function value / Change in input variable = 61.5 / 3 = 20.5
Therefore, the average rate of change of f(t) over the interval [1, 4] is 20.5.
To find the average rate of change of a function f(t) over an interval [a, b], you can use the formula:
Average rate of change = (f(b) - f(a)) / (b - a)
In this case, the interval is [1, 4], so a = 1 and b = 4. The function is f(t) = t^3 + 2/t.
Now, let's calculate the average rate of change:
f(4) = (4)^3 + 2/(4) = 64 + 1/2 = 129/2
f(1) = (1)^3 + 2/(1) = 1 + 2 = 3
Average rate of change = (f(4) - f(1)) / (4 - 1)
Average rate of change = ((129/2) - 3) / 3
Average rate of change = (129/2 - 6/2) / 3
Average rate of change = (123/2) / 3
Average rate of change = 123/6
Therefore, the average rate of change of f(t) over the interval [1, 4] is 123/6 or 61/3.