Mathematical Economics Lecture PDF

MATHEMATICS FOR ECONOMISTS  MATHEMATICS FOR ECONOMISTS 1. Types of Functions 2. Differentiation 3. Maxima and Minima 4. Integration 5. Differential Equations 6. Matrices 7. Input – Output analysis 8. Introduction to Linear Programming 9. Linear Programming 1-Types of Fu...

MATHEMATICS FOR ECONOMISTS  MATHEMATICS FOR ECONOMISTS 1. Types of Functions 2. Differentiation 3. Maxima and Minima 4. Integration 5. Differential Equations 6. Matrices 7. Input – Output analysis 8. Introduction to Linear Programming 9. Linear Programming 1-Types of Functions In mathematics, a function is a relationship between an input (independent variable) and an output (dependent variable), where each input is associated with exactly one output. There are several types of functions based on their properties and behavior:  1. Linear Functions General form: y = mx + b Graph: A straight line Characteristics: Constant rate of change (slope), intercepts the y-axis at a point (y-intercept). Example: The relationship between temperature in Celsius and Fahrenheit.  2. Quadratic Functions General form: y = ax² + bx + c Graph: A parabola Characteristics: Symmetrical, has a maximum or minimum point (vertex), can have two, one, or no real roots. Example: The path of a projectile.  3. Polynomial Functions General form: y = aₙxⁿ + aₙ₋₁xⁿ⁻¹ +... + a₁x + a₀ Graph: Can have various shapes depending on the degree of the polynomial Characteristics: Can have multiple roots, end behavior depends on the leading coefficient. Example: The volume of a sphere as a function of its radius.  4. Exponential Functions General form: y = a * b^x Graph: A curve that either grows or decays rapidly Characteristics: Base b is a positive constant, growth or decay rate depends on b. Example: Population growth, radioactive decay.  5-Logarithmic Functions General form: y = logₐ(x) Graph: The inverse of an exponential function Characteristics: Base a is a positive constant, undefined for x ≤ 0. Example: Sound intensity measurement (decibels).  6-Trigonometric Functions. General form: Sine (sin), cosine (cos), tangent (tan), etc. Graph: Periodic functions, repeat their patterns Characteristics: Used to model periodic phenomena like waves. Example: The height of a pendulum as a function of time.  7. Rational Functions. General form: y = p(x) / q(x), where p(x) and q(x) are polynomial functions Graph: Can have vertical and horizontal asymptotes, holes Characteristics: Undefined at points where q(x) = 0. Example: The relationship between voltage and current in an electrical circuit.  8- Piecewise Functions. Definition: A function defined by different rules for different intervals of its domain Graph: Can have discontinuous points or jumps Example: The cost of a taxi ride as a function of distance.