###### Calculus I

Exercices

Applications

Marginal analysis

# 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

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