TEAM HOMMIES

CPA ONLINE DISCUSSION

QUANTITATIVE TECHNIQUES
TOPIC: CALCULUS
INTRODUCTION
In day to day life we are often interested in the extent to which a change in one variable affects a
change in another related variable. This is called a rate of change. For example, if you own a motor car you
might be interested in how much a change far you have travelled affects the amount of fuel used. This rate
of change is called fuel consumption. If your car has high fuel consumption then a large change in the amount
of fuel in your tank is accompanied by a small change in the distance you have travelled. Sprinters are
interested in how a change in time is related to a change in their position. This rate of change is called
velocity. Other rates of change may not have special names like fuel consumption or velocity, but are
nonetheless important. For example, an agronomist might be interested in the extent to which a change in
the amount of fertilizer used on a particular crop affects the yield of the crop. Economists want to know how
a change in the price of a product affects the demand for that product. Consider the increase in consumption
resulting from an increase in one’s disposable income (marginal propensity to consume. You may also get
interested in the change in the quantity demanded due to a change in the price of a commodity and you recall
this is the price elasticity of demand. There are a host of such examples in all disciplines: biology,
engineering, economics and all fields of business management (finance, operations, human resource, etc).

Therefore, calculus is the mathematical study concerning measurement of change. It deals with rate and
direction of change (increase or decrease). Calculus is primly applied in optimization problems. It has two
major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus
(concerning accumulation of quantities. these two branches are related to each other by the fundamental
theorem of calculus.

DIFFERENTIAL CALCULUS (DIFFERENTIATION)

Differential calculus is about describing in a precise fashion the ways in which related quantities change.
It measures the rate of change in one variable e.g. y resulting from a change in another variable e.g. x at a
particular point. Consider: distance y changing as a result of change in time as a particle moves from one
point to another, this implies change in distance with respect to time. Change in revenue resulting from
change in the quantity sold or change in price. This is change in revenue R with respect to from change in
quantity Q or Change in price P.

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 Use the chain rule to find the derivative of the following functions

Use the product rule to find the derivative of the following functions

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 Use the quotient rule to find the derivative of the following functions

Combination of both rules

Here both the quotient and product rules could be required to solve a particular business or economics
problem.

Example
Find the derivative of the function: f (x) = 4x (3x - 1) / (2x-5)

Gradient of A Curve
Differentiation measures the rate of change and this is expressed in terms the gradient of a curve at a
point. The first derivative of y with respect to x is a function that gives the gradient the curve at a given
point.
Consider the curve defined by the function y = x 2 +5

The gradient of y = x2+5 at x=2 is found by differentiating y with respect to x and evaluating the gradient
𝑑𝑦
function i.e: = 2x
𝑑𝑥
At the point where x = 2, the gradient is 2(2) = 4
The gradient of the curve at a point is equal to the gradient of the tangent to the curve at this point.

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Question
1. Find the gradient of the curve y = x2+5 where: (i) x=3, (ii) x= -1, (iii) x=0
2. Find the gradient of the curve defined by y = 2x3 - x2 - 10 at x = 1.

STATIONARY OR TURNING POINTS

A relative extremum is a point at which a function is at a relative maximum or relative minimum. To
be a relative extremum, the point must be a critical pointie a point in the domain of a function, where the
derivative equals zero or is undefined.

Illustration of maxima, minimum and inflexion points

A-maximum point (local maximum) has the highest value of y in a particular region while B-minimum
point (local minimum) has the lowest value of y in a particular region. C is the point of inflexion (neither
maxim nor minimum). At the points A, B and C the curve has zero gradient and therefore the points are
called stationery points.

Optimization of Business Functions

People in business and economists are frequently asked to help firms solve such problems such as;
Maximizing profits, levels of physical output or productivity and minimizing costs. Levels of pollutions or
noise, the use of scare natural resources.
All these problems can be solved by use of optimization process.
Optimization is the process of finding the relative maximum or minimum of a function.
Application of differential calculus begins by Locating and identifying stationery pointsie points on
function’s curve where gradient is zero.
The following steps are followed:
𝑑𝑦
1. Take the first derivative ie differentiate the function ie f’(X) - 𝑑𝑥 for y=f(x) dx
2. Equate the derivative to zero since the derivative has the value of zero at stationary points. This
step is known as first-order condition and necessary condition. (Finding the values that that
satisfy the equation)
3. Solve the equation in (2) above for the critical point (s).

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 𝑑2 𝑦
4. Take the second derivative ie f’’(X) that is for y=f(x)
𝑑𝑥 2
5. Evaluate it at the critical point (s) and check the direction ie sign of the value.
This is the second-order condition and the sufficient condition. If at a critical
point(s) a is:-

Example1
3 2
Given the following function: y = x - 3x +10. Obtain the nature of the stationery points

Example 2
Given the following total revenue and total cost functions for different firms:
. Obtain the profit function of the firm and hence the
maximum profit:
Example 3
Given that a firm has a total revenue function of TR=3300q-26q2 and the firms total costs are represented
by q3-2q2+420q+750. Assuming that q > 0; find the profit maximizing function and also compute the
maximum profit.
Example 4
A manufacturer knows that if x hundred products are demanded in a particular week; the total cost function
would be; TC = 14 + 3x and the corresponding revenue function would be; TR = 19x – 2x2
a) Derive the total profit function.
b) Find the profit break even points
c) Calculate the level of demand that maximizes the profit of the company and hence the maximum profit.

Example 5

The revenue of a firm is defined by the function R=400x–4x2 and cost C=x2+10x + 30 where x is the number of
units of the commodity produced. If the firm wishes to maximize profit:-

a) What quantity should it produce?

b) Derive the price function and determine the price level at maximum output
c) Calculate the maximum profit
d) Determine the level of output at which the firm would break even.

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 INTEGRAL CALCULUS
It is the reverse of differential calculus. It studies the accumulation of quantities
E.g. linear distance travelled, area under a curve, volume displaced, etc. It is the mirror image of
differential calculus. An integral is an anti-derivative ie the reverse of a derivative.

Application of integration to business scenarios

While as differentiation measures the rate of change, integration measures quantities ie reverses the
differentiation process.

Example 6

A manufacturer has found that his marginal cost is (x + 3) shillings per unit when x have been produced.
The total cost of producing the first four (4) items is 5,020 Shillings.
Find:
a) The total cost of producing the first eight (8) units
b) The overall fixed costs

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 CPA QUESTIONS

Question 5 (Dec 2021)

(a) Kuka Ltd are producers of empty plastic bottles used for packaging 350 ml of Uganda Waragi. The
total cost function of the bottles production is given C (x) = X3 – 315x2 + 27,000x + 2,000. Where x
is the number of bottles produced.
Required:
i. Find the number of bottles that should be produced in order to minimize total cost. (6 marks)
ii. Determine the cost per bottle at minimum price. (3 marks)

Question 5 (March 2021)

(a) XYZ is the leading distributor of Tampako drink. Their fixed costs are Shs.660 per unit. The variable
costs are given by the function V (x) = 250 -2x2, where x is the number of units produced per day.
Required:
i. Derive a function to determine the total costs for a day. (2 marks)
ii. Find the minimum costs per day. (4 marks)
(b) ABIB Ltd specializes in the production of hedges (x) and doors (y).The raw material absorption of
the two products per day follows a production function given by y = 4x – 2x2.
Required:
i. Sketch a graph of y = 4x – 2x2. (8 marks)
ii. From the graph, find the maximum number of hedges and doors that can be produced in the
above factory per day. (1 mark)

Question 4 (August 2019)

(a) Distinguish between the terms ‘fixed costs’ and ‘variable costs’. (4 marks)
(b) A bike company has invested in a new sports bike assembling plant and it has been estimated that
after assembling y sports bikes, the profit, p in shs’ 000’ is given by the function
P(y) = 4y2 – 600y + 12,500. Required:
(i) Find the values of y for which the profit is zero. (4 marks)
(ii) Determine the maximum profit. (3 marks)

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 Question 5 (DEC 2020)
a) Distinguish between marginal revenue and marginal cost. (2 marks)
b) BAK executive bus company travels daily from Kampala to Mityana and accommodates three
categories of passengers: Business class, first class and economy class. Passengers are required to pay
Shs 5,000, 7,000 and 4,000 respectively on the preferred category. The bus can only accommodate 100
passengers on a single trip and on the first trip Shs 630,000 was collected as fares. On the second trip,
the business class recorded 10 more passengers; first class had 20 passengers less while the number in
economy class had doubled as compared to the first trip.
Required:
i. Determine the number of passengers who travelled in each category on the first bus
trip. (7 marks)
ii. Determine the total income earned from the fares the bus company got on the second
trip. (2 marks)
−𝒒𝟑
c) A firm has found out from past experience that its profit is given by P (q) = + 729q – 2,500
𝟑
where 0≤ q ≤ 35.
Required: Find the;
i. Value of q that maximizes profit. (6 marks).
ii. Profit per unit product when this maximum is achieved. (3 marks)

Question 4 (June 2019)

(a) Explain the term ‘marginal cost’. (2 marks)
(b) JJ Metal Factory Ltd fabricates iron rods for sale. Their total daily revenue is given by the
R (q) = (50-2q) (2q-2)2 where q is the number of iron rods sold.
Required:
i. Determine the number of sales that will maximize total revenue. (11 marks).
ii. Calculate the maximum total daily revenue earned. (2 marks)

Question 4 (NOV 2018)

(a) A roofing materials company produces ridges (x) and iron sheets (y) using resources according to the
production function given by: x + y2 + 4y = 20.
Required:
Determine the number of ridges and iron sheets produced given that for every 4 iron sheets, 1 ridge
is produced. (5 marks)

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(b) A famous writer of project planning textbooks would like to be paid a royalty of 15% of the sales
revenue and insists the price should maximize total revenue. On the other hand the publishing
company is interested in maximizing profit. Given the total revenue and total cost functions are
R (q) = 10,000q – 10q2 and C (q) = 1,000 + 2,500q + 5q2 respectively, where q is the number of
textbooks sold.

Required: Determine the:

i. Output that maximizes total revenue and obtain the maximum royalty.(5marks)
ii. Output that maximizes profit and associated royalty at that output. (6marks)

Question 5 (NOV 2019)

a) A manufacturer estimates that when Q units of a particular commodity are produced, the total cost will
𝟏 𝟏
be C (Q) = 𝟖 Q2 + 3Q + 98, in thousand shillings and further more P (Q) = 𝟑(75-Q) thousand shillings
per unit is the price at which all Q units will be sold.
Required: Find the marginal cost and marginal revenue. (5 marks).
b) The cost function of a firm is given by C (Q) = 2Q2 + 100Q +3,600 and the revenue function is given
by R (Q) = 500Q – 2Q2.
Required:
i. Find the value of Q that yields maximum profit. (5 marks)
ii. Hence determine the maximum revenue attained when the profit level is at maximum. (2
marks)
c) A manager of a shop in a busy shopping mall on Bigo Avenue found out that his profit per annum
(million shillings) follows a relation p = 11 + 24n – 3n2 where n is hours spent on the shop per month.
Required:
i. Represent p = 11 + 24n – 3n2 on the graph (use a range of 1 - 8 months) (5marks)
ii. What is the most profitable length of time for him to be at the shopping mall? (1mark)
iii. How long will it take him to get a profit of Shs.43, 000,000? (2 marks)

Question 4 (AUG 2018)

(a) Distinguish between marginal cost and marginal revenue. (2 marks)
(b) JL Apparels produces and sells school attires to various schools within Kampala. The total cost
function C for producing and marketing q units of their products is given by
C (q) =5q3 – 20q2 – 30,000q + 300,000.
Required:
i. Find the total cost when the output is equal to 300 units. (2 marks)
ii. Find the marginal cost when the output is equal to 300 units. (5marks)

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 QUESTION 4 (JUNE 2018)
(c) A factory uses x inputs to get y units of output. If the output values are given by
𝑑𝑦
Y = (x-2)3 (x+2); Find 𝑑𝑥 in descending powers of x. (4marks)
(d) In a busy factory that uses perishable goods, all raw material delivered to the factory in a day
must be processed that very day. In case many deliveries are received in a day, more machines must be used
to avoid wastage. In respect to this, a profit function y = 12x – 2a – ax2 was designed where x is the number
of machines used, a number of deliveries received in a day and y the profit earned in million shillings.
Required:
If on a particular day, three deliveries of raw materials were received, determine:
i. The number of processing machines that should be used on that day in order maximize
profit. (4marks).
ii. The maximum profit. (2 marks)

Question 5 (OCT 2021)

a) Two factories located in Katwe (a suburb in Kampala) are engaged in the production of maize
flour. Factory A has a production function, P (q) = 2q2 + 5q and factory B has a production
function P (q) = q2 + 4q + 12 where q is the quantity of maize used (in metric tons). On 5 February
2021, the two factories produced the same quantity of maize flour.
Required: Find the amount of maize flour that was produced on that day. (5 marks).
b) Rukundo Egumeho is an enterprise specialized in the production of local wine. The cost of producing
Q units per day is given by a cost function, while the revenue is defined by the where C and R are
in thousands of shillings.

Required:
Determine
i. The output that maximizes profit. (7 marks)
ii. Cost incurred to gain this profit. (2 marks)

QUESTION 5 (Mar 2022)

Muko holdings is firm engaged in mining wolfram in Rubanda district. Muko holdings has established
that its revenue function is R (q) = 2q3 + 40q2 + 8q while its production cost function is given by
C (q) = 3q3 + 19q2 + 80q - 800 (q represents the quantity in tons, and R is the revenue in thousand
shillings).
Required:
i. Determine the quantity that maximizes profit. (7 marks)
ii. Compute the revenue per ton that maximizes profit. (2 marks)

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