1.1 Mathematical Model of Production: Tornike Kadeishvili
1.1 Mathematical Model of Production: Tornike Kadeishvili
1.1 Mathematical Model of Production: Tornike Kadeishvili
WEEK 1
Reading [SB], 2.1-2.2, pp. 10-21
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1.1
Introduction
Mathematical Model of Production
Data: x number of units produced, F the fixed cost, c the production cost
of one unit, p the wished selling price of one unit, k the reduction coefficient.
The total cost of production of x units
C(x) = c x + F.
The selling price of one unit when x units are produced
p(x) = p k x.
The total revenue
R(x) = x p(x) = kx2 + px.
The total profit
P (x) = R(x) C(x) = kx2 + px cx F =
kx2 + (p c)x F.
Both functions R(x) and P (x) are concave quadratic functions, thus they
both have maximum.
> C(x) := 2 x + 5000; p(x) := 10 0.001 x;
> R(x) := x p(x); P (x) := R(x) C(x);
> plot(R(x), C(x), P (x), x = 0..10000);
20000
10000
2000
4000
6000
8000
10000
x
10000
20000
Exercises
1. The total cost of a company per month is given by C(x) = 2x + 5000
and the Price - demand function is given by p(x) = 10 0.001x. Thus the
revenue and profit functions are given by
R(x) = xp(x) = 0.001x2 + 10x,
P (x) = R(x) C(x) = 0.001x2 + 8x 5000.
A) How many units should the company manufacture each month to
maximize the revenue?
R0 (x) = 0.002x + 10 = 0, x = 5000
B) What is the maximal revenue?
R(5 000)=25 000
C) What is the selling price of one unit when the revenue is maximal?
p(5000) = 5.
D) How many units should the company manufacture each month to
maximize the profit?
P 0 (x) = 0.002x + 8 = 0, x = 4000.
E) What is the maximal profit?
P (4000) = 11000.
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F) What is the selling price of one unit when the profit is maximal?
p(4000) = 6.
G) Find the break-even points.
R(x) = C(x), P (x) = 0, 0.001x2 +8x5000 = 0, x1 = 683, x2 = 7317.
2. The total cost of a company per month is given by C(x) = 100x+1000
and the Price - demand function is given by p(x) = 300 0.1x.
A) How many TV sets should the company manufacture each month to
maximize the revenue?
B) What is the maximal revenue?
C) What is the selling price of one TV set when the revenue is maximal?
D) How many TV sets should the company manufacture each month to
maximize the profit?
E) What is the maximal profit?
F) What is the selling price of one TV set when the profit is maximal?
3. Suppose the government decides to introduce additional $ 10 tax for
each TV set. Solve the previous problem in this case.
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2.1
2.1.1
Functions
Vocabulary of Functions
The Notion of Function
2.1.2
Types of Functions
a0 + a1 x + ... + am xm
.
b0 + b1 x + ... + bn xn
Graphs
The graph of a function y = f (x) is the set of all points of the Cartesian
plane whose coordinates are (x, f (x)).
2.1.4
2.5
1.5
0.5
1.5
0.5
0.5
2
x
1
Exercises
1. Suppose x0 is a point of minimum for f (x). Then the same point x0
for the function g(x) = f (x) is .
2. Find all local and global minimums and maximums for the function
f (x) = |x2 4|.
2.1.6
For a function
f :XY
the set X is called domain, and the set Y is called target or codomain.
A number x0 belongs to the domain of a function y = f (x) if f (x0 ) is
defined.
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Example. For the function y = x3
the number x0 = 2 belongs to the
1
1
1
= 01 is not
domain: f (2) = 23 = 1 = 1 but x0 = 3 does not: f (3) = 33
defined.
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2.1.7
2.1.8
x
1
1
x
10
2.2
Linear Functions
11
x
2
5=a+b
9 = 3a + b
Recall that the slope measures how much y changes as x increases by unit.
So the slope measures the rate of change of a function.
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