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 the function f(t) when t = 2, we need to find the derivative of the function and evaluate it at t = 2.
First, let's find the derivative of f(t):
f(t) = t^3 + 2/t
To find the derivative, we can use the power rule and the quotient rule:
f'(t) = 3t^2 - 2/t^2
Now, let's evaluate the derivative at t = 2:
f'(2) = 3(2)^2 - 2/(2)^2
= 3(4) - 2/4
= 12 - 1/2
= 12 - 0.5
= 11.5
Therefore, the instantaneous rate of change of f(t) when t = 2 is 11.5.
To find the instantaneous rate of change of a function, you need to calculate its derivative. In this case, you're given the function f(t) = t^3 + 2/t.
Step 1: Take the derivative of the function.
To find the derivative of f(t), you can use the power rule for differentiation and the quotient rule. Let's break down the steps:
First, differentiate t^3 using the power rule:
d/dt (t^3) = 3t^2
Next, differentiate 2/t using the quotient rule:
d/dt (2/t) = (2 * d/dt(t) - t* d/dt(2)) / t^2 = -2/t^2
Step 2: Combine the derivatives.
Now, combine the derivatives of t^3 and 2/t. Since we have a sum, we can add them together:
f'(t) = 3t^2 - 2/t^2
Step 3: Evaluate the derivative when t = 2.
Now that we have the derivative function f'(t), we can substitute t = 2 into it to find the instantaneous rate of change:
f'(2) = 3(2)^2 - 2/(2)^2
= 3(4) - 2/4
= 12 - 0.5
= 11.5
Therefore, the instantaneous rate of change of the function f(t) = t^3 + 2/t when t = 2 is 11.5.
To find the instantaneous rate of change of the function f(t) = t^3 + 2/t when t = 2, you need to find the derivative of the function and then evaluate it at t = 2.
First, let's find the derivative of f(t).
Step 1: Take the derivative of the first term, t^3, using the power rule. The power rule states that if you have a function of the form x^n, the derivative is nx^(n-1).
Derivative of t^3 = 3t^(3-1) = 3t^2
Step 2: Take the derivative of the second term, 2/t, using the quotient rule. The quotient rule states that if you have a function of the form h(x) = f(x)/g(x), the derivative is (f'(x)*g(x) - f(x)*g'(x))/g(x)^2.
Derivative of 2/t = (0*t - 2*1)/t^2 = -2/t^2
Now, let's combine the derivatives of both terms to get the derivative of f(t):
f'(t) = 3t^2 - 2/t^2
Finally, let's evaluate the derivative at t = 2 to find the instantaneous rate of change:
f'(2) = 3*(2^2) - 2/(2^2) = 3*4 - 2/4 = 12 - 1/2 = 11.5
Therefore, the instantaneous rate of change of f(t) when t = 2 is 11.5.