Calculus I: Marginal analysis

The marginal analysis is the examination of the situation when we add one unit as a independent variable to the system.

In other words, it is a technique that allows to study the effect on a function f(x) (cost, production, income, ...) by a unit increase in its independent variable (x).

In economics, the marginal analysis identifies the effect (benefits or costs) on total revenue or cost caused by a marginal or incremental change (a unit change) to given resources.

This technique is well explained by derivative.

1. Definitions

The marginal analysis involves the marginal cost which is the the derivative of the function of the related situation.

The marginal cost for the function f(x) is

(f(x+1) - f(x))/((x + 1) - x) =
(f(x+1) - f(x)) ≈ f '(x) at the point x

2. Example

A student spends: E (x) = x + (x/2)² kilo dollars for x terms for the related school fees.

At the present time, it has been taken 10 semesters.

The cost for these ten terms is: E(10) = 10 + 5² = 35 k$

The cost of the fees, if the student takes an extra term (11th term) will be: E(10 + 1) - E(10).

E(x + 1) - E(x) = x + 1 + ((x+1)/2)² - x - (x/2)² =
1 + (x/2)² + x/2 + (1/2)² - (x/2)² =
1 + x/2 + 1/4 = (5/4) + x/2

E(x + 1) - E(x) = (5/4) + x/2

Therefore

The cost for the 11th term is:

E(0 + 1) - E(10) = (5/4) + 10/2 = 25/4 = 6.25 k$

If we use the definition of the marginal cost we will have :

(E(x+1) - E(x))/((x + 1) - x) =
(E(x+1) - E(x)) ≈ E'(x) at the point x = 10.

E'(x) = (x + (x/2)²)' = 1 + (1/2)2(x/2) = 1 + x/2.

At the point x = 10, we obtain the marginal cost:
E'(10) = 1 + 10/2 = 6.00 k$



From the following graph, we understand the difference between the two values.
The first expression: (E(x+1) - E(x)) represents the slope of the secant, and the second expression which is the derivative E'(x) of the function E(x) represents the slope of the tangent at the point x = 10.

3. Marginal analysis formulas