- X X 100 2 X Hundred Dollars A Monopolist S Cost Function Is C X 2500 For Items Produced Their Price Or Demand F 1 (49.64 KiB) Viewed 74 times
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x (x-100)2 + x hundred dollars A monopolist's cost function is C(x) = 2500 for items produced. Their price (or demand) function is given by p(x) = 20- hundred dollars for x items sold. x 25 1. In the CAS Perspective, type C(x):=(x/2500)*(x-100)^2 + x (the cost function) on an empty Input line and hit enter. On the In- put line below, find the derivative of C(x) by either typing C'(x) or Derivative (C(x)). This is the marginal cost function. Write it below exactly as it appears in the CAS panel. MC(x) = C'(x) = 1/2500(3^x2-400x+12500) 2. Find the the company's revenue function. In the CAS Perspective, type x/25 on an empty Input line. On the line below, type p(x). This will be your revenue function. Write it below exactly as it appears in the CAS panel. p(x): 20 R(x):= x *
4. Profit decreases over intervals where the cost of producing an extra item exceeds the revenue generated from producing an extra item. De- termine open intervals (x-intervals) where marginal cost exceeds marginal revenue. In CAS Perspective, type Solve(C'(x)>R'(x)). Write your answer as an interval below. Since z represents items produced and sold, you can eliminate all negative values of x from consideration. For infinite intervals, just type inf or infinity to indicate the interval has no upper bound or upper limit. C'(x) > R'(x) on the interval(s): On this interval(s), profit is creasing (increasing or de-
5. Profit increases over intervals where the revenue generated from pro- ducing an extra item exceeds the cost of producing an extra item. Determine the open interval(s) (x-intervals) where marginal revenue ex- ceeds marginal cost. In CAS Perspective, type Solve (R'(x)>C'(x)). Write your answer (as an interval) below. Since a represents items pro- duced and sold, you can eliminate all negative values of a from consid- eration. For infinite intervals, just type inf or infinity to indicate the interval has no upper bound. R'(x) > C'(r) on the interval(s):_ On this interval(s), profit is creasing (increasing or de-
6. If profit is to be maximized, it occurs when the marginal cost equals marginal revenue. Determine the number of items produced and sold for which the marginal functions agree. In the CAS Perspective, type Solve(R'(x)=C'(x)). Write your answers below. Again, consider only x ≥ 0. C'(x) = R'(x) when z = items 7. Determine if your answer in #6 maximizes or minimizes profit by filling in all of the blanks below? Since C'(x)= R(x) when z = C'(x) > R'(a) on the interval interval profit is minized when = items, are produced and sold, and R'(x) > C'(x) on the (marimized or items are produced and sold.
8. Now let's confirm our results using the profit function. In the CAS Perspective, type P(x):=R(x)-C(x). This is the profit function. Write it below as it appears in the CAS panel, with like terms collected. P(x) = 9. In the CAS Perspective find the marginal profit function by typing P'(x) or Derivative(P(x)). Write your answer below as it appears in the CAS panel. MP(x) = P'(x) =_ 10
10. Recall that local extreme values of a function can occur only at the crit- ical numbers of the function (domain values at which the derivative is equal to zero). Find the positive critical numbers of P(x) by typing Solve(P'(x)=0) in the CAS Perspective. Write your answer(s) in the space provided. Explain how these answers relate to your work with the marginal functions. Recall we are only considering x ≥ 0. Critical Number(s) of P(x): x= Type a sentence or two explaining how your answer here relates to your answers above with the marginal functions. Look at your answer for #6. items
11. Recall that if f'(x) > 0 on an interval I, then f(x) increases on the interval I. Similarly if f'(x) < 0 on an interval I, then f(x) decreases on the interval I. Because the sign of the derivative tells us where functions increase and decrease, we use the sign of the derivative to determine if the critical numbers give us a relative maximum or relative minimum (or neither) value of f(x). Working in the CAS Perspective type Solve(P'(x)>0) and Solve(P'(x) <0) (of course, this should be done on two separate input lines). Write your answers below in interval notation. Remember we are only considering postive values of a since this represents items produced and sold. Explain how these answers relate to your work with the marginal functions by filling in the blanks.
Interval(s) on which P'(x) > 0: On this interval(s), the profit funciton P(x) is (increasing or decreasing) Interval(s) on which P'(x) < 0: On this interval(s), the profit funciton P(x) is. (increasing or decreasing) Comparing my answers with #4 and #5 above, I see that: • P'(x) > 0 when marginal cost is ● less) than marginal revenue. On this interval(s), profit is (increasing or decreasing). P'(x) < 0 when marginal cost is (greater or (greater or
12. Fill in the sign chart for P'(r) below. Type your critical number under the hash mark on the number line next to "c.n.". The type either ">0" or "<0" next to each P' above the appropriate intervals on the number line, and finally type either increasing or decreasing next to each "P is" below the appropriate intervals on the number line. Then fill in the blanks to determine if this monopoly has a maximum or minimum profit.
P' 0 Pis This monopoly has a of items. c.n. = P' P is (maximum or minimum) profit (include units) when they produce and sell