This comprehensive Calculus course introduces students to the fundamental concepts and techniques of differential and integral calculus, with emphasis on both theoretical understanding and practical applications. The course begins with an exploration of limits and continuity as the foundation of calculus, followed by an in-depth study of differentiation techniques, including gradients, first principles, and advanced rules for differentiation such as the quotient rule, chain rule, and product rule. Students will master the differentiation of various function types, including trigonometric, exponential, logarithmic, and inverse functions, as well as techniques for implicit and parametric differentiation.
The latter portion of the course focuses on applications of derivatives, particularly the analysis of stationary points and function behavior, followed by a comprehensive study of integration techniques from basic indefinite and definite integrals to more advanced methods. Students will engage in extensive calculation exercises to develop proficiency and confidence in applying calculus methods to solve theoretical and applied problems. Students will develop mathematical reasoning skills essential for advanced study in science, engineering, economics, and other quantitative disciplines.
Learning objectives
After the completion of this course, you will be able to:
- Apply principles of limits and continuity to analyze functions and explain the foundation of differential calculus
- Calculate derivatives using first principles and interpret their meaning as instantaneous rates of change and gradients
- Apply differentiation rules (power rule, quotient rule, chain rule, product rule) to differentiate complex functions efficiently
- Evaluate derivatives of trigonometric, exponential, logarithmic, and inverse functions using appropriate techniques
- Solve problems involving implicit differentiation and parametric differentiation in various mathematical and applied contexts
- Analyze functions to identify and classify stationary points, determine intervals of increase and decrease, and sketch accurate function graphs
- Evaluate indefinite and definite integrals using basic integration techniques, including the power rule, substitution method, and integration by parts
- Apply advanced integration methods to evaluate integrals involving various function types, including trigonometric, exponential, and rational functions
- Interpret the definite integral as area under a curve and apply integration to solve problems involving areas, volumes, and other physical quantities
Course outline
- Calculus Methods: Gradients and First Principles
- Calculus Methods: Differentiation
- Quotient Rule, Chain Rule, and Product Rule
- Differentiation of Trigonometric Functions
- Differentiation of Exponential and Logarithmic Functions
- Implicit Differentiation
- Parametric Differentiation
- Differentiation of Inverse Functions
- Stationary Points
- Basic Integration
- Advanced Integration
- Calculation Methods: Exercises
