Estimate the relative rate of change of
f(t)=5t^2 at t=4. Use change of 0.01.
To estimate the relative rate of change of a function at a specific point, we need to calculate the average rate of change of the function over a small interval around that point. In this case, we are asked to estimate the relative rate of change of the function f(t) = 5t^2 at t = 4 using a change of 0.01.
To begin, we need to find the average rate of change over the interval [4, 4 + 0.01].
Step 1: Calculate f(4)
Evaluate the function at t = 4:
f(4) = 5(4)^2 = 5(16) = 80
Step 2: Calculate f(4 + 0.01)
Evaluate the function at t = 4 + 0.01:
f(4 + 0.01) = 5(4.01)^2 ≈5(16.0801) ≈ 80.40
Step 3: Calculate the average rate of change
The average rate of change is defined as the change in the function's value divided by the change in the input (t).
Average Rate of Change = (f(4 + 0.01) - f(4)) / (4 + 0.01 - 4)
Substituting the values we found:
Average Rate of Change = (80.40 - 80) / (4.01 - 4)
Average Rate of Change = 0.40 / 0.01
Average Rate of Change = 40
Therefore, the estimated relative rate of change of f(t) = 5t^2 at t = 4 using a change of 0.01 is 40.
df = 10t dt
so,
df = 10(4)(0.01) = 0.4