Unit 3 Applications of derivatives in Finance and in Economics
Key Unit competence: Apply differentiation in solving Mathematical problems
that involve financial context such as marginal cost,
revenues and profits, elasticity of demand and supply
Introductory activityA can company produces open cans, in cylindrical shape, each with
constant volume of 300 cm3 . The base of the can is made from a
material that costs 50 FRW per cm2 , and the remaining part is made of
material that costs 20 FRW per cm2 .
a) Express the height of one can as function of the base radius x of the can.
b) Express the total cost of the material to make a can, as function
of the base radius x of the can.
c) Find the dimensions of the can that will minimize the total costof the material to make a can.
3.1. Marginal quantities
3.1.1. Marginal cost
Learning Activity 3.1.1
A company found that the total cost y of producing x items is given byy = 3x2 + 7x +12 .
a) Find the instantaneous rate of change in the total cost, when x = 3
b) How is the instantaneous rate of change in the total cost called?
CONTENT SUMMARY
The marginal cost is the instantaneous rate of change of the cost.
It represents the change in the total cost for each additional unit of production.
Suppose a manufacturer produces and sells a product. Denote C(q) to be the
total cost for producing and marketing q units of the product. Thus, C is a
function of q and it is called the (total) cost function. The rate of change of Cwith respect to q is called the marginal cost, that is,
Application activity3.1.1
3.1.2. Marginal revenue
Learning Activity 3.1.2
A firm has the following demand function: P =100 −Q .
Find: a) in terms of Q, the total revenue function
b) The instantaneous rate of change of the total revenue when Q =11.
CONTENT SUMMARY
Application activity 3.1.2
3.2. Minimization and maximization of functions
3.2.1. Minimization of the total cost function
Learning Activity 3.2.1
Consider the following problem: A can company produces open cans, in
cylindrical shape, each with constant volume of 300 cm3 . The base of the
can is made from a material that costs 50 FRW per cm2 , and the remaining
part is made of material that costs 20 FRW per cm2 . Assume you are the
manager of the company, and you have to buy the material for constructing
the can. Which question do you ask yourself regarding the dimensions ofthe can and the money to use for buying the material?
CONTENT SUMMARY
Application activity 3.2.1
Find the value of Q for which the total cost is minimum, and find theminimum total cost in each of the following cases:
3.2.2. Maximization of the total revenue function
Learning Activity 3.2.1
Consider the following problem: A company has to buy a plot for the building
of its factory. The plot must have a rectangular shape with a constant
perimeter of 400 meters, and the cost of the plot is constant. Assume you
are the manager of the company, and you have to choose the dimensions of
the rectangular plot located in a flat uniform area. Which question do youask yourself regarding the dimensions of the plot and the area of the plot?
CONTENT SUMMARY
Application activity 3.2.2
Given the demand function, P = 24 − 3Q, find the value of Q at which thetotal revenue is maximum, and find the maximum revenue.
3.3. Price elasticity
3.3.1. Elasticity of demand
Learning Activity 3.3.1
CONTENT SUMMARY
Application activity 3.3.1
Find the price elasticity of the demand if the quantity demanded d Q andthe price P are related by:
3.3.2. Elasticity of supply
Learning Activity 3.3.2
CONTENT SUMMARY
Application activity 3.3.1
Find the price elasticity of the supply if the quantity supplied s Q and theprice P are related by:
End of unit assessment 3
