Topics

Differential and integral calculus

The topic is divided into the following subtopics:
  • Limits and continuity
  • Derivative
  • Analyzing function behavior
  • Applications of derivatives
  • Primitive function
  • Definite integral
  • Applications of definite integral

Limits and continuity

Part I:
  • Calculating limits – polynomials and rational functions
  • One-sided limits
  • Finding limits of functions from graphs
Part II:
  • Calculating limits – trigonometric functions
  • Calculating limits – functions with radicals
  • Continuity, discontinuity points
Part III:
  • Theoretical aspects related to limits calculations

Derivative

Part I:
  • Geometric interpretation of the derivative
  • Derivatives of elementary functions
Part II:
  • Derivative of a product of functions
  • Derivative of a quotient function
  • Derivative of a composite function
Part III:
  • Derivative of a composite function – complex problems
  • Applications of derivatives in physics

Analyzing function behavior

Part I:
  • Function’s monotonicity
  • Local extrema
Part II:
  • Second derivative and its geometric interpretation
  • Concavity and convexity of a function
  • Inflection points
Part III:
  • Asymptotes of a graph of a function

Applications of derivatives

Part I:
  • Calculating limits using L'Hospital's rule  
Part II:
  • Tangent line to graph of a function
  • Normal line to graph of a function
Part III:
  • Global extrema
  • Optimization problems (global extrema)

Primitive function

Part I:
  • Geometric interpretation of the antiderivative (primitive function)
  • Solving simple indefinite integrals (Finding a primitive function)
Part II:
  • Solving integrals requiring simplification of expressions
  • Solving integrals by substitution
  • Solving integrals by Parts
Part III:
  • Integrals solved by substitution – complex problems
  • Integrals solved by Parts – complex problems
  • Solving integrals requiring partial fraction decomposition

Definite integral

Part I:
  • Evaluation of simple definite integrals
Part II:
  • Evaluating integrals requiring simplification of expressions
  • Evaluating integrals using substitution
  • Evaluating integrals by Parts
Part III:
  • Evaluating integrals using substitution – complex problems
  • Evaluating integrals by Parts – complex problems
  • Evaluating integrals requiring partial fraction decomposition

Applications of definite integral

Part I:
  • The area of a plane region
Part II:
  • The volume of a solid
Part III:
  • The area of a plane region – complex problems
  • The volume of a solid – complex problems
  • Applications to physics