This is my last chance to make at least a C in my AP Calculus course, so please tell me if I got these right or wrong so that I can fix them before it's due. Also, there was one I wasn't sure how to do, so if you can help me with that, it would be much appreciated!
1)Find the maximum revenue, maximum profit, and minimum average cost. R(x)=-x^2+400, C(x)=x^2+40x+100
MY ANSWER: Max Revenue: $40,000. Max Profit: $16,100. Min Avg Cost: $60.
2)Find the maximum revenue, maximum profit, and minimum average cost. R(x)=x(-2x+60), C(x)=2x^2+12x+18
MY ANSWER: Max Revenue: $450. Max Profit: $90. Min Avg Cost: $24.
3)Given the revenue function R9x)=-x^2+80x, find the marginal revenue at x=10 and compare the result with R(11)-R(10).
MY ANSWER: Marginal revenue=$60 and R(11)-R(10)=$59.
4)A company finds that its cost for producing x units of a commodity is C9x)=3x^2+5x+10. Find the approximate cost for making the twenty-first unit.
MY ANSWER: $131
5)Let C(x)=3x^2+100 denote the cost for making x units of a product. Compare the exact cost for producing the thirty-first unit with the marginal cost at x=30 and x=31.
MY ANSWER: Exact cost= $183. Marginal cost= $180
6)Suppose R(x)=-x^2+1000x and C(x)=20x+600 are revenue and cost functions, respectively, for producing x units of a commodity. What profit is realized from the sale of 50 items? What is the approximate amount the profit changes from the sale of one more item?
MY ANSWER: Profit= $45,900. Change= $879 with the sale of one more item.
7) Given that R(x)=x(-x+300) is a revenue function, show that the marginal revenue is always decreasing.
MY ANSWER: For this one, I drew the graph of the marginal revenue function, which would be -2x+300, the derivative of the revenue function.
8) Given that C(x)=2x^3-21x^2+36x+1000 is a cost function, determine the interval(s) for which the cost is increasing. Determine whether there are any intervals on which the marginal cost increases.
MY ANSWER: Regular cost increases from "negative infinity to 1" and "6 to infinity." Marginal cost increases from "7/2 to infinity."
9) Show that the maximum profit occurs when the marginal revenue equals the marginal cost.
MY ANSWER: P(x)=R(x)-C(x), derive it and set equal to zero to get R'(x)=C'(x).
10) Show that the minimum average cost occurs when the average cost equals the marginal cost.
MY ANSWER: I had no idea how to do this one. Please help if you can!
11) Compute the elasticity of demand and state whether the demand is elastic or inelastic. D(p)=-4p+500, 0 is greater than or equal to p is greater than or equal to 125; p=50.
MY ANSWER: Pretty sure it's wrong or that I made some sort of dumb mistake, because the example we were given didn't clear things up too well for me, but I got that it's inelastic at .66666667
12) Compute the elasticity of demand and state whether the demand is elastic or inelastic. D(p)=-10p+850, 0 is greater than or equal to p is greater than or equal to 85; p=40.
MY ANSWER: I got that it's inelastic at .8888884.
Please let me know how I did, and tell me if I made any dumb mistakes. I also apologize for any typos. iPods are a pain! Thanks!
#1. ok
#2.
Max R is at (15,450) -- ok
P = R-C = -4x^2+48x-18
max P is at (6,126)
avg cost is C(x)/x = 2x+12+18/x
min avg cost is at (3,24) -- ok
#3. ok
#4. ok
#5. ok, but you did not figure the marginal cost at x=31
#6. ok
#7. ok, but you could note that the slope of R' is always negative.
#8. ok
#9. ok, but the verb is differentiate, not derive.
#10. avg cost is
A(x) = C(x)/x
A' = (xC'-C)/x^2
min A is where A'=0, or
xC' = C
C' = C/x
that is, marginal = average
#11,12 I'd have to review my understanding of elastic. Maybe another can get to it before me.
Thanks so much for pointing those things out! But I do have a question on #2? How'd you get the 6 for the maximum profit? I got the equation -4x^2+48x-18, but I couldn't get the 6. (It's probably some really stupid mistake. I make a lot of those. hahaha)
Also, for 5, is the marginal cost at 31 $186?
p(x) =-4x^2+48x-18
P'(x)= -8x +48
0 = -8x +48
8x =48
x = 6
5)C(x) = 20x+600
C(31)-C(30) = 20(31)+600-([20(30)+600])
1220- 1200= 20
C(x) 20x +600
C'(x) = 20