Course content

Introduction to functions
- The notion of function
- Arithmetic perations for functions
- The range of a function
- Functions and graphs
- The notion of limit
- Continuity
- Arithmetic operations for continuity
Lines and linear functions
- Linear functions with a single unknown
- The general solution of a linear equation
- Systems of equations
- The equation of a line
- Solving systems of equations by addition
- Equations and lines
Quadratic functions
- Completing the square
- The quadratic formula
- Factorization
- Solving equations with factorization
- The notion of polynomial
- Calculating with polynomials
Rational functions
- The notion of a rational function
Power functions
- Power functions
- Equations of power functions
- Applications of functions
Inverse functions
- The notion of inverse function
- Injective functions
- Characterizing invertible functions
Exponential and logarithmic functions
- Exponential functions
- Properties of exponential functions
- Growth of an exponential function
- Logarithmic functions
- Properties of logarithms
- Growth of a logarithmic function
New functions from old
- Translating functions
- Scaling functions
- Symmetry of functions
- Composing functions
- Applications of operations for functions
Chapter 3: Introduction to differentiation
Definition of differentiation
- The notion of difference quotient
- The notion of derivative
Calculating derivatives
- Derivatives of polynomials and power functions
Derivatives of exponential functions and logarithms
- The natural exponential function and logarithm
- Rules of calculation for exponential functions and logarithms
- Derivatives of exponential functions and logarithms
- Applications
Chapter 4: Rules of differentiation
Rules of computation for the derivative
- The sum rule for differentiation
- The product rule for differentiation
- The quotient rule for differentiation
- The chain rule for differentiation
- Exponential functions and logarithmic derivatives revised
- The derivative of an inverse function
Applications of derivatives
- Tangent lines revisited
- Approximation
- Elasticity
Chapter 5: Applications of differentiation
Analysis of functions
- Monotonicity
- Local minima and maxima
- Analysis of functions
Higher derivatives
- Higher derivatives
- Applications of differentiation
Chapter 6: Multivariate functions
Basic notions
- Functions of two variables
- Functions and relations
- Visualizing bivariate functions
- Multivariate functions
Partial derivatives
- Partial derivatives of the first order
- Chain rules for partial differentiation
- Higher partial derivatives
- Elasticity in two variables
- Applications of multivariate functions
Chapter 7: Optimization
Extreme points
- Stationary points
- Minimum, maximum and saddle point
- Criteria for extrema and saddle points
- Convexity and concavity
- Criterion for a global extremum
- Hessian convexity criterion
- Applications of optimization
Chapter 8: Constrained Optimization
The Lagrange multiplier method
- Lagrange multipliers
- Lagrange multiplier interpretation
- Lagrange's theorem
Sufficient conditions for optimality
- Convexity conditions for global optimality
- Second-order conditions for local optimality

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