Find the profit function find the marginal profit function

Question

QUESTION 1:

a) Suppose the monthly revenue and cost functions (in dollars) for x units of a commodity produced and sold are: R(x) = 400x - x2/20 and C(x) = 5000 + 70x respectively.

i) Find the profit function.

ii) Find the marginal profit function.

iii) Use your answer in ii) to estimate the profit from selling the 42^nd unit and interpret the result.

b) Find the first partial derivatives of:

f(x, y) = xy/x²-y²)² 

QUESTION 2:

a) Given the Average Revenue Function:

A.R = 54Q² + 13Q -12 where Q is the quantity of units produced and sold. 

i) Determine the Total Revenue function.

Find:

ii) The total revenue if 100 units were sold.

iii) The revenue attributable to the sale of the 81^st unit, using the marginal approach.

b) A firm has determined that its weekly profit function is given by:

P(x) = 95x - 0.05x² - 5000 for 0 ≤ x≤ 1000.

Find the volume of production that maximises profit.

QUESTION 3

The following is a table of revenue (in $) and quantities of a certain commodity.

RAW MATERIAL	TOTAL REVENUE ($)		QUANTITIES
	1997	1998	1997	1998
I	1300	1000	50	60
II	2200	1500	40	50
III	2000	1200	30	40
IV	1200	1600	20	30
V	1150	1150	10	20

Calculate:

a) The Price Relative for Raw Material V.

b) The Quantity Relative for Raw Material II.

c) The Unweighted Aggregative Price Index.

d) The Fischer's Ideal Index.

QUESTION 4

a) Identify the intervals over which the function f(x) = 3x² + 4 is increasing and decreasing.

b) Sketch the graph of f(x).

c) Mrs Joseph finds that profit (in $) for her shop is given by:

f(x) = 12x -0.25x²

Where 0 ≤ x ≤ 50

i) Is profit increasing at x = 20? and at x = 30?

ii) Over what intervals of x is profit increasing and decreasing?