Question

A company's total cost, in millions of dollars, is given by C(t) 90-50e where t is the time in years since the start-up date. The graph of C(t) is shown to the right. Find each of the following. a) The marginal cost, C'(t) b) C'(0) c) C'(5) d) Find lim C(t) and lim C'(t). AY 150- 75- 0+ too t-oo 30 15 a) C'(t) (Do not include the $ symbol in your answer.) b) C'(0) $ (Simplify your answer. Do not include the $ symbol in your answer.) million per year c) C'(5)= $ (Simplify your answer. Round to the nearest thousand per year as needed. Do not include the $ symbol in your answer.) d) lim C(t) t-o0 million (Simplify your answer. Do not include the $ symbol in your answer.) lim C'(t)-S per year t+00 (Simplify your answer. Do not include the $ symbol in your answer.) To

Answer 1

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

Answer 2

a) The marginal cost, C'(t), is -50e^(-t). b) C'(0) = -50. c) C'(5) ≈ -0.0124 million per year. d) lim C(t) = 90 million, lim C'(t) = 0.

Step-by-step explanation:

a) The marginal cost, C'(t), is the derivative of the total cost function, C(t). In this case, the derivative is given by:

C'(t) = -50e-t

b) To find C'(0), substitute t = 0 into the derivative equation:

C'(0) = -50e0 = -50

c) To find C'(5), substitute t = 5 into the derivative equation:

C'(5) = -50e-5 ≈ -0.0124 million per year

d) To find the limit of C(t) as t approaches infinity (lim C(t)), evaluate:

lim C(t) = lim (90 - 50e-t) = 90 million

To find the limit of C'(t) as t approaches infinity (lim C'(t)), evaluate:

lim C'(t) = lim (-50e-t) = 0