Calculus for Social Sciences

Introduction to functions
1. The notion of function
2. Arithmetic operations for functions
3. The range of a function
4. Functions and graphs
5. The notion of limit
6. Continuity
7. Arithmetic operations for continuity
Lines and linear functions
1. Linear functions with a single unknown
2. The general solution of a linear equation
3. Systems of equations
4. The equation of a line
5. Solving systems of equations by addition
6. Equations and lines
Quadratic functions
1. Completing the square
2. The quadratic formula
3. Factorization
4. Solving equations with factorization
Polynomials
1. The notion of polynomial
2. Calculating with polynomials
Rational functions
1. The notion of a rational function
Power functions
1. Power functions
2. Equations of power functions
Applications
1. Applications of functions

Inverse functions
1. The notion of inverse function
2. Injective functions
3. Characterizing invertible functions
Exponential and logarithmic functions
1. Exponential functions
2. Properties of exponential functions
3. Growth of an exponential function
4. Logarithmic functions
5. Properties of logarithms
6. Growth of a logarithmic function
New functions from old
1. Translating functions
2. Scaling functions
3. Symmetry of functions
4. Composing functions
Applications
1. Applications of operations for functions

Definition of differentiation
1. The notion of difference quotient
2. The notion of derivative
Calculating derivatives
1. Derivatives of polynomials and power functions
Derivatives of exponential functions and logarithms
1. The natural exponential function and logarithm
2. Rules of calculation for exponential functions and logarithms
3. Derivatives of exponential functions and logarithms
Applications
1. Applications

Rules of computation for the derivative
1. The sum rule for differentiation
2. The product rule for differentiation
3. The quotient rule for differentiation
4. The chain rule for differentiation
5. Exponential functions and logarithmic derivatives revisited
6. The derivative of an inverse function
Applications of derivatives
1. Tangent lines revisited
2. Approximation
3. Elasticity

Analysis of functions
1. Monotonicity
2. Local minima and maxima
3. Analysis of functions
Higher derivatives
1. Higher derivatives
Applications
1. Applications of differentiation

Basic notions
1. Functions of two variables
2. Functions and relations
3. Visualizing bivariate functions
4. Multivariate functions
Partial derivatives
1. Partial derivatives of the first order
2. Chain rules for partial differentiation
3. Higher partial derivatives
4. Elasticity in two variables
Applications
1. Applications of multivariate functions

The Lagrange multiplier method
1. Lagrange multipliers
2. Lagrange multiplier interpretation
3. Lagrange’s theorem
Sufficient conditions for optimality
1. Convexity conditions for global optimality
2. Second-order conditions for local optimality

Course content