standard deviation~20 points
Suppose Mr. Slugworth produces chocolate bars and knows that the derivative of his cost function is given by"
where x is the number of chocolate bars and C'(x) is the rate that his cost increases (in dollars per bar).
a. Compute C'(100) and explain what it means in terms of cost for Mr. Slugworth.
b. Estimate how much it costs Mr. Slugworth to produce his 500th chocolate bar.
c. Estimate how much it costs Mr. Slugworth to produce his 2000th chocolate bar.
d. Find a function C(x) that could represent Mr. Slugworth’s cost function by “undoing” the derivative.
e. Use your answer to part (d) and the fact that Mr. Slugworth’s fixed costs are $2500 to find his total cost to produce 500 chocolate bars.
f. Repeat part (e) to find the total cost of producing 2000 chocolate bars.
- wyLv 78 年 前最佳解答
(a) C'(100) = - 0.0001(100) + 0.865 = 0.855, its exact meaning is the rate of cost increase when x = 100. It roughly means the incremental cost increase when x changes from 99 to 100, that is it is the estimated cost of making the 100th items.
2012-07-16 18:00:00 補充：
(b) The estimated cost of making the 500th bar = C'(500) = - 0.0001(500) + 0.865 = 0.815.
(c) The estimated cost of making the 2000th bar = C'(2000) = -.0.0001(2000) + 0.865 = 0.665.
2012-07-16 18:07:25 補充：
(d) Integrating C'(x) gives C(x) = - 0.0001(x^2/2) + 0.865 x + A = - 0.00005x^2 + 0.865x + A.
(e) Since fixed cost is $2500, that means when x = 0, C = $2500, so C(x) = -0.00005x^2 + 0.865x + 2500. So total cost for x = 500 = C(500) = -0.00005(500^2) + 0.865(500) + 2500 = $2920
2012-07-16 18:09:32 補充：
(f) Total cost for 2000 bars = C(2000) = -0.00005(2000^2) + 0.865(2000) + 2500 = -200 + 1730 + 2500 = $4030.