Course 1- MAE

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Mathematics Applied to Economics
2023-2024
Prof. dr. Paula Curt
Faculty of Economics and Business Administration, UBB Cluj-Napoca
Statistics, Forecasts and Mathematics Department
paula.curt@econ.ubbcluj.ro
Course 1
Linear Functions
Real Functions (one variable)
Definition
Let D ⊂ R, f : D −→ R.
I f function= a rule which assigns to each number x (from the domain
of definition) a unique number y=f(x) (in the range of the function)
I x = independent variable; y = dependent variable
Linear functions: y=f(x)=
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 2 / 11
Linear Functions
Linear functions
Equations of a straight line:
I slope-intercept form
I point-slope form; the line passes through the point (x1, y1)
I point-point form; is the equation of the line that passes through the
points (x1, y1) and (x2, y2)
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 3 / 11
Linear Functions
Linear functions: example
Example 1. Find the relationship between Celsius (C) and Fahrenheit (F)
scales, by knowing that: the relationship is linear; water boils at 100 C and
212F; water freezes at 0C and 32F. Is there any temperature measured by
the same number in both scales?
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 4 / 11
Derivatives
Derivatives
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 5 / 11
Derivatives
Derivatives. Linear Approximation
Definition
Let D ⊂ R, a ∈ R and f : D −→ R. The derivative of the function f at the
point a is
f ′(a) = lim
x→a
f(x)− f(a)
x− a
.
Remark
1. f ′(a) ∼=
f(x)− f(a)
x− a
, x close to a
2. f(x) ∼= f(a) + f ′(a)(x− a), x close to a;
linear (1st order) approximation formula
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 6 / 11
Derivatives
Derivatives. Linear Approximation
Example 2.
I Use linear approximation (2) to show that ln(1 + x) ∼= x for x close to
0
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 7 / 11
Derivatives
Derivatives. Linear and Quadratic Approximation
Example 3.
I Find an approximate value for 1.00150
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 8 / 11
Economic applications
Derivatives. Economic Applications.
Example 4. It is determined that a certain country’s gross domestic
product (GDP) can be approximated by f(x) = 432x1/4 where f(x) is
measured in billions of dollars and x is the capital expenditure in billions of
dollars. Approximate the change in GDP if the country’s capital
expenditure changes from $81 billion to $83 billion.
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 9 / 11
Economic applications
Derivatives. Economic Applications
Example 5.
Suppose C(x) = 8000 + 200x− 0.2x2, (0 ≤ x ≤ 400) is the total cost that a
company incurs in producing x units of a certain commodity.
I What is the cost of producing the 251st unit of that product?
I C ′(250) =?
I Compare the results obtained at the previous parts.
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 10 / 11
Economic applications
Derivatives. Economic Applications. Marginal cost
Remark Marginal cost is the derivative of the cost function (can be
approximated by the cost of producing one aditional unit)
C ′(x) ∼= C(x + 1)− C(x)
Example 6.
Let C(Q) denote the cost of producing Q units per month of a commodity.
What is the interpretation of C ′(1000) = 25? Suppose the price obtained per
unit is fixed at 30 and that the current output per month is 1000. Is it
profitable to increase production?
Prof. dr. Paula Curt (UBB FSEGA) Mathematics Applied to Economics Course 1 11 / 11
	Linear Functions
	Derivatives
	Economic applications