2. Calculus in the FDM Process

Question 1

What are the three basic functions, and their symbols, in commerce?

Answer 1

Cost function, C (x)

Revenue function, R (x)

Profit function, P (x)

Question 2

Define the generic Cost function, C (x)

Answer 2

C (x) = F + V (x)

Where:
F = Fixed costs
V = Variable costs

Question 3

Describe the Cost function

Answer 3

The cost function relates the total cost of production, C, to the number of items produced, x

Question 4

Define the generic Revenue function, R (x)

Answer 4

R (x) = P * x

Where:
P = Price per unit
x = number of units sold

Question 5

Describe the revenue function

Answer 5

Relates the total income generated from selling ‘x’ number of units at a particular price ‘P’.

Question 6

Define the generic Profit function, P (x)

Answer 6

P (x) = R (x) - C (x)

Where:
R (x) = the revenue function
C (x) = the cost function

Question 7

Describe the profit function

Answer 7

Determined by subtracting all of the costs associated with producing an item from the revenue that is generated by selling it

Question 8

What is the break-even point?

What two methods can be used to find it?

Answer 8

The moment where total revenue generated from selling a product equals its cost, i.e. R (x) = C (x)

or

Whilst no profit has been generated, there are also no losses, i.e. P (x) = 0

Question 9

Define the revenue function for the following scenario:

A company sells 5 items for £45 each.

Answer 9

R (x) = P * x

Therefore

R (x) = 45x

Question 10

Define the cost function for the following scenario:

A company pays £50,000 to set up a new production facility, in addition to £20 for every item it produces.

Answer 10

C (x) = F + V (x)

Therefore

C (x) = 50,000 + 20x

Question 11

Define the profit function for the following scenario:

R (x) = 45x
C (x) = 50,000 + 20x

Answer 11

P (x) = R (x) - C (x)

Therefore

P (x) = 25x - 50,000

Question 12

How do you determine the loss/profit from selling a particular number of items?

Answer 12

Use the profit function where ‘x’ is equal to the number of items sold in the question. Positive values are profit, negative values are loss.

Question 13

Determine the break even point using Method 1: R (x) = C (x) based on the following scenario:

R (x) = 45x
C (x) = 50,000 + 20x

Answer 13

45x = 50,000 + 20x

25x = 50,000

x = 2000

Question 14

Determine the break even point using Method 2: P (x) = 0 based on the following scenario:

P (x) = 25x - 50,000

Answer 14

25x - 50,000 = 0

25x = 50,000

x = 2000

Question 15

Describe Actual Cost, Cact

Answer 15

The actual cost of producing one more item

Question 16

How do you calculate actual cost of, for example, the 501st unit of a batch?

Answer 16

C (501) - C (500)

Calculate the cost to produce 501 units using the cost function and subtract the cost of producing 500.

Question 17

Describe the Marginal Cost function

Answer 17

The first derivative of the cost function, C (x), an equation for estimating the cost of producing one more item

Question 18

What is the formula for the Marginal Cost function?

Answer 18

Cm (x) = d/dx C (x)

Question 19

Describe the Average Cost function

Answer 19

The ‘cost per item’

Question 20

What is the formula for the Average Cost function?

Answer 20

Cav (x) = C (x) / x

Question 21

How are the actual, marginal and average functions adapted for revenue and profit?

Answer 21

‘C’ for cost is replaced by the relevant ‘R’ or ‘P’ function. The formulas are largely the same

Question 22

How would you solve the following scenario:

A company sets a minimum amount that it must receive for each item at £10 to cover costs. Determine when this happens.

Answer 22

Use the marginal revenue equation, Rm, and set it equal to 10, then solve for ‘x’.

Question 23

What is the main difference between linear and non-linear expressions when related to marginal cost.

Answer 23

Linear equations produce straight lines when plotted (constant gradients) therefore the marginal cost, revenue or profit will be the same number irrespective of the magnitude considered. This is not the same in non-linear equations where a curve is formed.

Question 24

How do you determine how many units should be produced to maximise profits based on a limited production capacity per week and a profit function.

Answer 24

• Define production range (e.g. 0<x<20)
• Calculate the first derivative of the profit function
• Max profit occurs where the first derivative is equal to 0
• Solve for ‘x’ where the first derivative is equal to 0
• The value of x is the ‘turning point’
• Take the second derivative of the profit function, if the value is negative, it is a maximum value, and vice versa

Question 25

How do you determine the maximum or minimum profit in a system following the calculation of unit production?

Answer 25

Put the calculated value for x into the profit equation

Question 26

How would you solve the following style of question: Determine that the break even point occurs at x= (a particular value) given the following Cost function C (x) = (generic function) and that products are sold at £_

Answer 26

Break even point occurs at P(x)=0 P(x)=R(x)-C(x) - Set the equation equal to zero - Solve using the quadratic formula - Take only positive production as the answer

Question 27

How do you evaluate the Revenue function from a given Marginal Revenue function and the total revenue gained (e.g. £6000) from selling a particular number of products (e.g. 5 nr).

Answer 27

- Integrate the marginal revenue function - Set the equation equal to the revenue (£6000) and set 'x' equal to the number of products sold (5 nr) - Solve for the integration constant 'c'

Question 28

How would you calculate the price to charge for each item in order to make no profit/loss (£0 profit) based on a specified number of sales (e.g. 1000 nr)

Answer 28

P(x) = 0, therefore R(x)=C(x) R(x)=Price*x ; Price*1000 = C (1000) Therefore: Price = C(1000)/1000

Question 29

The company has not priced for delivery (£10/item) in their initial costings and have already sold 250 out of 1000 items at £225. How do you determine how much they should charge for the remaining 750 items to recover costs

Answer 29

- Determine actual cost at x=1000 using the Cost Function (e.g. £235,000) INCLUDING DELIVERY (add the delivery onto the cost of producing each item before multiplying it by x) - Determine the revenue generated from selling 250 units at £225 (250*225= £56250) - Subtract the revenue generated from the actual cost at x=1000 to find the revenue needed from the remaining sales (£235000-£56250=£178750) - Divide the revenue needed by the amount of items left to be sold (£178750/750=£238.33)