Calculus for Social Sciences

Chapter 1: Functions

• 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

Chapter 2: Operations for 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

Chapter 3: Introduction to differentiation

• 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

Chapter 4: Rules of differentiation

• 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

Chapter 5: Applications of differentiation

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

Chapter 6: Multivariate functions

• 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

Chapter 7: Optimization

• Extreme points
  1. Stationary points
  2. Minimum, maximum and saddle point
  3. Criteria for extrema and saddle points
  4. Convexity and concavity
  5. Criterion for a global extremum
  6. Hessian convexity criterion
• Applications
  1. Applications of optimization

Chapter 8: Constrained optimization

• 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