for the function f(t) =( t^3 + 2)/t find the instantaneous rate of change of f(t) when t= 2
To find the instantaneous rate of change of f(t) when t = 2, we need to find the derivative of f(t) with respect to t and evaluate it at t = 2.
Let's find the derivative of f(t) using the quotient rule:
f'(t) = [(t)(d/dt(t^3 + 2)) - (t^3 + 2)(d/dt(t))]/(t^2)
Simplifying this expression, we have:
f'(t) = [(t)(3t^2) - (t^3 + 2)(1)]/(t^2)
= (3t^3 - t^3 - 2)/(t^2)
= (2t^3 - 2)/(t^2)
Now, evaluate f'(t) at t = 2:
f'(2) = (2(2)^3 - 2)/(2^2)
= (2(8) - 2)/(4)
= (16 - 2)/4
= 14/4
= <<14/4=3.5>>3.5
Therefore, the instantaneous rate of change of f(t) when t = 2 is 3.5.
To find the instantaneous rate of change of a function at a specific point, we can calculate the derivative of the function and then evaluate it at that point. In this case, we need to find the derivative of the function f(t) = (t^3 + 2)/t.
To find the derivative, we can use the power rule, which states that if we have a function of the form f(t) = t^n, then the derivative is given by f'(t) = n*t^(n-1).
Applying the power rule, we get:
f'(t) = (3*t^2 - 0)/t^2
= 3 - 0/t^2
= 3.
Now that we have the derivative of the function f(t), we can evaluate it at t = 2 to find the instantaneous rate of change at that point:
f'(2) = 3.
Therefore, the instantaneous rate of change of f(t) when t = 2 is 3.
To find the instantaneous rate of change of a function, we need to calculate the derivative of the function and substitute the given value of t. In this case, we are given the function f(t) = (t^3 + 2)/t and we need to find the instantaneous rate of change when t = 2.
Step 1: Calculate the derivative of the function f(t).
The derivative of f(t) can be found using the quotient rule, which states that for a function g(t)/h(t), the derivative is (g'(t) * h(t) - g(t) * h'(t)) / (h(t))^2.
Let's apply the quotient rule to find the derivative of f(t):
f'(t) = ( (3t^2 * t - t^3 * 1) * t - (t^3 + 2) * 1 ) / t^2
Simplifying this expression gives:
f'(t) = (3t^3 - t^3 - t^3 - 2) / t^2
= (t^3 - 2) / t^2
Step 2: Substitute the given value of t into the derivative.
We want to find the instantaneous rate of change when t = 2. Substituting t = 2 into the derivative expression:
f'(2) = (2^3 - 2) / 2^2
= (8 - 2) / 4
= 6 / 4
= 3/2
Therefore, the instantaneous rate of change of f(t) when t = 2 is 3/2.