Mathematical Techniques - Calculus

​ ​ ​​A milk production farm intends to undertake a research on a new type of feed which would increase the cow’s productivity. The monthly revenue in thousand shillings would be represented by R=−5X2+9X where X is the monthly milk produced and sold in thousand units.  

​​Explain TWO applications of calculus in business management.

​ ​​Find the area between the curve y=x+2x+x21​ , the x axis and the lines x = 1 and x = 2.

​ ​​Safi Limited manufactures and sells two types of cleaning pads; Osha and Oga. The demand functions for the two products are given by:

​ ​ ​ ​​Compu World Limited assembles and sells computers. The company estimates that if it optimally assembles computers, it could sell between 1,100 and 2,000 computers per month and the monthly revenue (in thousands of shillings) over this range of sales could be represented by the function ​\(R = 1,000x - 200x^2.\)​

Where: R is the monthly revenue.

• x is the number of computers sold per month (in thousands).

The company estimates that its marginal cost (in thousands of shillings) could be represented by the following function:

​\(MC = 200x^2 - 200x + 400\)​

Where: MC is the marginal cost

• x is the number of computers assembled.

The fixed cost of the company amounts to Sh.100,000 per month. If is assumed that all the computers

assembled in a given month are sold within the same month.

Required:

(i) The total cost function.

(ii) The profit function.

(iii) The optimal monthly output.

(iv) The maximum profit of the company.

​ ​​ The profit function (in Sh. "000") for a given company is given as: 

​​The marginal revenue and average cost functions of Biashara Limited are given as follows: 

​​ABC Limited manufactures and sells electronic calculators whose marginal cost function is given by: 

​​The unit price and total cost functions associated with the production and sale of a certain electric component are given by the following equations:

BC Limited operates two factories namely; X₁ and X2. Both factories deal in the production of a product named "Nguzo". The joint cost function for production of product "Nguzo" is given by:

​​The demand and total cost functions (in thousands of shillings) of a certain company that deals in the manufacture of a product name "Exe" are given as follows:

​​The average revenue and marginal cost functions of a certain company are given by.

​​An electronics company sells programmable calculators at a unit price of Sh.100. Studies indicate that the company can sell additional 100 calculators per year for Sh.5 decrease in unit price and 100 calculators per year less for Sh.5 increase in unit price. The company currently sells 3,000 calculators per year. The cost function of the company is assumed to be linear with a fixed cost of Sh.10,000 and variable cost of Sh.65 per calculator.

​​Shujaa Limited deals in the manufacture of a product named "Zed". The product "Zed" is produced on order and the company does not keep inventory of the product. The demand and total cost functions (in thousands of shillings) of the company are given as follows:

P = 190 - q

and

TC = q² + 10q + 500

Where:

P is the unit selling price.

q is the quantity demanded in units.

TC is the total cost.

Required:
(1) The maximum profit of the company.
(ii) The output level that would maximise total revenue.

​​Explain how differential calculus could be used in solving optimisation problems.

​​The marginal cost and demand functions for Ujenzi Limited are given as follows:

MC = 2𝑥 + 16 (in Sh.million)

and

P = 𝑥² - 24 x 117 (in Sh.million)

Where:

MC is the marginal cost function

P is the price of a building constructed

𝑥 is the number of buildings constructed in a year.

The total annual fixed costs of the company amount to Sh.39 million.

Required:
(i) The profit function.
(ii) The selling price per building constructed that will maximise profit