Final answer:
The marginal cost, C'(t), represents the additional cost of producing one more unit of output. To find the marginal cost, we need to calculate the derivative of the total cost function, C(t), with respect to t. C(t) = 90 - 50e^(-t). C'(t) = 50e^(-t).
Step-by-step explanation:
The marginal cost, C'(t), represents the additional cost of producing one more unit of output. To find the marginal cost, we need to calculate the derivative of the total cost function, C(t), with respect to t.
C(t) = 90 - 50e^(-t)
C'(t) = 50e^(-t)
Now, let's find the values of C'(0) and C'(5) by substituting t = 0 and t = 5 into the derivative function.
C'(0) = 50e⁻⁰ = 50
C'(5) = 50e⁻⁵
To find the limit of C(t) as t approaches positive infinity, we substitute t = infinity into the total cost function.
lim C(t) as t approaches infinity = lim (90 - 50e^(-t)) as t approaches infinity = 90 million dollars
Similarly, to find the limit of C'(t) as t approaches positive infinity, we substitute t = infinity into the derivative function.
lim C'(t) as t approaches infinity = lim (50e^(-t)) as t approaches infinity = 0 million dollars per year