Differential and Integral Calculus

The topic is divided into the following subtopics:

Limits and Continuity of Functions
Derivatives of Functions
Function Behavior
Applications of Derivatives
Primitive Function
Definite Integral
Applications of Definite Integrals

Limits and Continuity of Functions

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

Derivatives of Functions

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

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 Integrals

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