NATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY

FACULTY OF COMMERCE
DEPARTMENT OF INSURANCE AND ACTUARIAL SCIENCE
QUANTITATIVE ANALYSIS FOR BUSINESS

TUTORIAL: CALCULUS

1. Find the first derivatives of the following w.r.t: x

2
a)𝑓(𝑥) = 1000 b)𝑓(𝑥) = −𝑥 𝑎+1 + 𝑥 3 c)𝑓(𝑥) = −2𝑥 4 + 15𝑥 2 − 36𝑥
3

d)𝑓(𝑥) = (2𝑥 2 + 4𝑥)(𝑥 2 + 5𝑥)

2𝑥 2 +4𝑥 4x3 −20 1

e)𝑓 = (3x + 1)(∛x) f)𝑓(𝑥) = g)𝑓(𝑥) = h)𝑓(𝑥) =
𝑥 2 +5𝑥 2√x+7 𝑥 4 +1

√3𝑥+1 3
i)𝑓(𝑥) = (4𝑥 2 + 6𝑥)5 j) 𝑓(𝑥) = 9𝑥 −3
k)𝑓(𝑥) = (5√𝑥 2 )(7𝑥 5 − 3𝑥 −2 )4

2. Sketch the following graphs:

(a) 𝑓(𝑥) = 𝑥 + 1

(b) 𝑓(𝑥) = 𝑥 2

(c) 𝑓(𝑥) = 𝑥 3 − 18𝑥 + 12

3. A company has determined that the total cost (in dollars) of manufacturing 𝑥 units of a
product is given by: C(x) = x 2 + 0.5x + 5000 0 ≤ x ≤ 150

Determine the ARC in total cost as the number of units of products manufactured is increased
from 60 to 65.

4. Find the equation describing the ARC from 𝒙 = 𝒂 to 𝒙 = 𝒂 + 𝜟𝒙 for each of the following
functions and evaluate the ARC over the intervals specified.

(a) 𝑓(𝑥) = 4𝑥 2 + 7 𝑓𝑟𝑜𝑚 𝑥 = 3 𝑡𝑜 𝑥 = 5

(b) 𝑓(𝑥) = 𝑒 𝑥 𝑓𝑟𝑜𝑚 𝑥 = 1 𝑡𝑜 𝑥 = 3

5. Use the delta process to derive the function of the instantaneous rate of change for the
following:

(a) 𝑓(𝑥) = 2𝑥 2 + 5

(b) 3𝑥 2

(c) 𝑥 2 − 1

6.Roberts Hair Salon offers a basic haircut and a deluxe haircut. Let X represent the demand for
basic haircuts and y represent for deluxe haircuts. The price demand equations are given by

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𝑝 = 12 − 0.3𝑥 + 0.1𝑦 gives the price in dollars of a basic haircut and 𝑞 = 20 + 0.1𝑥 − 0.2𝑦 the
price of a deluxe haircut.

(a) Determine the Revenue function R(x,y).

(b) How many of each hair cut should be given to maximize revenue?
(c) What is the maximum revenue
1 3
7. A small company manufactures bikes. The cost function is 𝐶(𝑥) = 10 + 5𝑥 + 𝑥 and the revenue
60
𝑅(𝑥) = 90𝑥 − 𝑥 2 where x is the bikes produced each week and R(x) and C(x) are in dollars

(a) Find the maximum Revenue and when it occurs

(b) Find the profit function P(x)

(c) Where is the profit increasing?

8) If total cost 𝑇𝐶 = 0.75𝑞3 − 6𝑞 2 + 50𝑞 + 20, derive functions for

a) Marginal Cost (MC) b) Average Cost (AC) c)Slope of AC

𝜕𝑓 𝜕𝑓
9) Find , of the following functions
𝜕𝑥 𝜕𝑦

3 1
i)𝑓(𝑥; 𝑦) = x 6 y 4 + 6(x√y) + x 2 y 2 − 12 iv) )𝑓(𝑥; 𝑦) = (16𝑥 4 𝑦 8 )2 − 2𝑥 3 𝑦 2

x2 +y
ii)𝑓(𝑥; 𝑦) = x−2y3 − x 4 y

4𝑥 𝑦2
iii)𝑓(𝑥; 𝑦) = 𝑦2 + 𝑥
−5

10) A manufacturer estimates that when q units of a certain commodity are produced, the profit obtained
is P(q) thousand dollars, where:

2
𝑃(𝑞) = − 𝑞 2 + 78𝑞 − 128
9

i. Find the marginal profit function.

ii. At what level of production is profit maximised?
iii. Calculate maximum profit.

11) The total revenue in dollars for selling q books is given by the function
𝑇𝑅(𝑞) = −0.05𝑞2 + 600𝑞

The variable cost in dollars for selling q books is given by the function

𝑉𝐶(𝑞) = 0.000002𝑞3 − 0.03𝑞2 + 400𝑞

i. If you sell 1000 books, your profit is $103,000. What is the value of your fixed cost?
ii. Find the largest interval on which the total revenue function is positive.
iii. How many books must you sell in order to maximize profit?
iv. What is the smallest value of average variable cost?

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12) Obtain the first partial derivatives for the following:
a) F(x,y) = x2 – 11xy4 + 1 d) G(t,v) = (t2+ v2)1/2
𝑥𝑦+1
b) H(r,s) = (r – s)(r + 7s) e) U = 1−2𝑥𝑦

13) An entrepreneur finds that revenue generated by his 2 vending machines, a soft-drink machine
and a fruit-juice machine, is given by the following function:
R (x,y) = 4x2 + 2y2 + 6xy
Where: x is the number of soft-drink machines and y is the number of fruit-juice machines.

i) Find the revenue generate if 8 soft drinks and 7 fruit juice vending machines are installed
at a college
ii) What is the effect on revenue generated, if the number of soft-drink vending machines
was increased by 1 to 9, while holding the number of fruit juice vending machines
constant at 7?
iii) What is the effect on revenue generated if the number of fruit-juice vending machines
was increased by 1 to 8 while holding the number of soft-drink vending machines
constant at 8?
iv) Calculate the percentage change in revenue if 2 soft-drink machines and 1 fruit-juice
machine were sent or servicing at the same time.

14) A large clothing manufacturer has found that revenue generated by 2 lines of business, denim
and cotton, is given by the function:
R (d,c) = 20d2 + 3c2 + 8dc

Where: d is the number of denim items produced (in thousands) and c is the number of cotton
items produced (in thousands).

i) What is the revenue generated if 10 000 denim items are produced and 17 000 cotton
items are produced?
ii) Estimate the effect on revenue if an additional 1000 denim items are produced while the
number of cotton items is held constant, using partial derivatives.
iii) Estimate the effect on revenue if an additional 1000 cotton items are produced while the
number of denim items is held constant, using partial derivatives.
iv) What would be the percentage change in revenue if production in each line of business
was increased by 5000 items simultaneously?

15) Find the first derivatives of the following:

a) 𝐻(𝑥, 𝑦) = 35000 − 19
1
b) 𝑦(𝑡) = 300𝑡 + 10(3𝑡 3 − 5𝑡)2
c) ℎ(𝑥) = (2𝑥 3 − 4𝑥 + 1)−4
d) ℎ(𝑥) = (7𝑥 5 − 𝑥 4 + 2)10
5
e) ℎ(𝑥) = (3𝑥 − 1)2 (−2𝑥 + 9)5

16) The demand functions for pairs of commodities are given below. Use partial derivatives to
determine whether these commodities are substitute, complementary, or neither.
i). 𝐷1 = 500 − 6𝑝1 + 5𝑝2 and 𝐷2 = 200 + 2𝑝1 − 5𝑝2

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 17) Find the equation describing the ARC from 𝑥 = 𝑎 to 𝑥 = 𝑎 + 𝛥𝑥 for each of the following functions
and evaluate the ARC over the intervals specified.

i. 𝑓(𝑥) = 4 − 3𝑥 2 a) from x=3 to x=5 b) from x=6 to x=10

2
ii. 𝑓(𝑥) = 11– 2𝑥– 3𝑥 a) from x=0 to x=2 b) from x=1 to x=3

18. Integrate the following functions w.r.t.

2
𝑎) ∫ 2𝑥 4 𝑏) ∫ 𝑥 3 + 𝑥 2 𝑐) ∫(2𝑥 + 3)(3𝑥 − 2) 𝑑) ∫(2𝑥 + 3)7 𝑒) ∫(1 − 3𝑥)5 𝑑𝑥

1
𝑓) ∫ 𝑒 4𝑥 𝑑𝑥 𝑔) ∫ (5𝑥+1) 𝑑𝑥

3
h) ∫2 2𝑥 4 + 2

19) Given that𝑅(𝑥) = 2000𝑥 − 25𝑥 2 𝐶(𝑥) = 12𝑥 2 + 65𝑥 + 10. Find:

a) P(x)
b) R(3) ,C(3), P(3)
c) Find the marginal cost, marginal revenue, marginal profit, if x=3
d) Find the value of x that maximises profit
e) Find the value of x, which minimises the cost

20) Given the following functions:

𝑅(𝑥) = 9.99𝑥 and 𝐶(𝑥) = 45000 + 6.99𝑥, find:

a𝑃(𝑥)
b) Find the marginal cost, marginal revenue, marginal profit.
c) Comment on each of your answers in (b).
d) Find the value of x that maximises profit.
e) Find the value of x, which minimises the cost.

21) A firm has analysed its sales and probability and found that its profit can be expressed as:

𝑃(𝑄) = −100 + 100𝑄 − 5𝑄 2

where 𝜋 is profit ($000s) and 𝑄 is the number of units sold (in 000s)
It has also been found that its costs are: 𝐶(𝑄) = 100 + 2𝑄 2
where 𝐶 is costs ($000s)
a) Obtain a third equation showing the firm’s revenue, TR.
b) Find the marginal cost, marginal revenue, marginal profit.
c) Comment on each of your answers in (b).
d) Find the value of x that maximises profit.
e) Find the value of x, which minimises the cost.

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22) Use the Delta Process to derive the function of the Instantaneous Rate of Change (IRC). Also sketch
the graphs of each of the following, stating the ranges where the function is increasing or decreasing.

a) 𝑓(𝑥) = 𝑥 2 + 5𝑥 + 6

b) 𝑓(𝑥) = 2𝑥 2 + 7𝑥 − 15

c) 𝑓(𝑥) = 14𝑥 − 𝑥 2 − 27

23)

a)Find the Average Rate of Change (ARC) function for the following:

c) Sketch the graphs of each of the following curves, stating the ranges where the function is
increasing or decreasing.

i) f(x) = 3x2 - 5x + 1

ii) 𝑓(𝑥) = 9𝑥 − 𝑥 2 − 3

iii) 𝑓(𝑥) = 𝑥 2 − 11𝑥 + 28

iv) 𝑓(𝑥) = −(𝑥 − 4)2

24) Draw the graph of 𝑓(𝑡) = 𝑠𝑡𝑒𝑝 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 = {0 𝑓𝑜𝑟 𝑡 < 0, 1 𝑓𝑜𝑟 𝑡 ≥ 0} then draw

i) f(𝑡) + 2 𝑖𝑖) 𝑓(3𝑡) 𝑖𝑖𝑖)𝑓(𝑡 + 2)

25) Draw the graph of 𝑓(𝑡) = 1/2𝑡 + 3 with domain of 0 ≤ 𝑡 ≤ 2. Then give a formula and graph
for

a) 𝑓(𝑡) + 1 b) 𝑓(𝑡 + 1) c) 𝑓(4𝑡)

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