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Mathematical Techniques - Differentiation

Unit: Quantitative Analysis

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August 2024

 1 Questions
Question 6c
​​(i) Find the value of the following integral:

    ​\(\displaystyle \int(4x^3 + 2x + 5)\)​     dx for an interval between ​\(x = 1\)​ and ​\(x = 3\)​

(ii) A function of a curve is given as:

    ​\(Y = 3x^2 – 12x + 64\)​

Find the integral equation of the function


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April 2024

 1 Questions
Question 4c
​ ​ ​​A solar panel production firm intends to research on a launch of a new solar panel which would increase the sales of the panels to between 2,000 units to 3,000 units per week. The weekly revenue in thousands of shillings over the range of sales could be represented by: 

R=−3x2+7x

Where x is the weekly solar panel units produced and sold in thousands. Past records of the solar panel production in the firm estimates its marginal costs in thousands of shillings could be represented by the function; 

MC=2x2−3x+5

The fixed costs will be Sh.1,000 per week. 

Required: 

(i) The average cost function of the firm. 

(ii) The average revenue function of the firm. 

(iii) The profit maximising output. 

(iv) The price that should be charged to maximise profits


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December 2022

 1 Questions
Question 2a
​ ​ ​ ​​Shoetec Ltd., a manufacturer of stylish shoes, estimates that at full scale production, it would sell between 2,000 and 3,000 pairs of shoes. 

The total monthly revenue in thousands of shillings over this range is represented by the function 

TR = = 3x² + 7x. 

The firm estimates that the marginal cost (MC) in thousands of shillings could be represented by the function 

MC = 5x² - 3x - 2 and fixed cost (FC) will be Sh.1,000 per month. 

Where x is the monthly output in thousands of pairs of shoes.

Required: 

(i) Derive the average cost and average revenue functions of the firm. 

(ii) Calculate the profit maximising output. 

(iii) Calculate the price charged upon maximising profit and how much each pair of shoes would cost. 


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August 2022

 1 Questions
Question 7b
​ ​​XYZ Ltd. produces and sells a product branded “Xedo”. The product is produced in two departments; manufacturing and assembly. 

The marginal revenue (MR) of XYZ Ltd. is given by the function 

  • MR = 600 – 0.12q 
  • Where q is the number of units produced and sold. 
The total variable cost (VC) for the two departments is given as follows: 

  • Manufacturing department VC = 60q + 0.06​\(q^2\)​ 
  • Assembly department VC = 12q + 0.03​\(q^2\)​ 

The total fixed cost for each of the departments is as follows:

Sh.
Manufacturing department 
40,000
Assembly department 
120,000

Required: 

(i) The total revenue, total cost and profit functions of XYZ Ltd. 

(ii) The profit maximising level of output. 

(iii) The maximum profit of XYZ Ltd.


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Pilot September 2015

 1 Questions
Question 1c
​ ​ ​​A firm has a linear demand function for its product. When the price of the product is Sh. 220, the quantity demanded is 40 units. When the price increases to Sh. 240 the quantity demanded becomes 30 units. In addition, the firm's marginal cost function is given by:

MC = 40q-2q²+2
Fixed cost = Sh. 5million
where q = quantity demanded, MC = marginal cost (in Sh. million)

Required:
(i) The level of output that maximises profits.
(ii) The maximum profit.
(iii) The price of the product at the maximum profit.
(iv) The price elasticity of demand when the profit is at the maximum (interpret your result).


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