This course provides a foundational understanding of differentiation, a core concept in calculus. Students will explore rates of change, limits, gradients, and the derivative function through theoretical explanations, problem-solving, and real-world applications. The course emphasizes the development of critical thinking and analytical skills needed for advanced studies in mathematics, physics, and engineering.
Course Units:
- Explore average and instantaneous rates of change and their graphical representations, laying the foundation for understanding motion and change.
- Learn the concept of limits, evaluate them for various functions, and connect limits to continuity and instantaneous rates of change.
- Understand how to calculate the gradient of a tangent to a curve and its applications in real-world problems.
- Discover the derivative as a function, apply differentiation rules, and explore its applications in motion, optimization, and concavity.
First Principles of Differentiation
- Build a deep understanding of differentiation from first principles, using limits to derive gradients and validate differentiation rules.
This course equips students with the tools to analyze and solve problems involving motion, optimization, and change, preparing them for further studies in calculus and applied mathematics.
This course provides a comprehensive understanding of the rules of differentiation, focusing on their mathematical foundations and real-world applications. Students will explore basic and advanced differentiation techniques, including exponential, logarithmic, and trigonometric functions, as well as second derivatives. By the end of the course, students will have the skills to analyze and solve complex problems in mathematics, physics, and engineering.
Course Units:
Introduction to Differentiation Rules
- Power rule, constant rule, sum and difference rules, and their applications to basic functions.
Advanced Rules: Product and Quotient
- Product and quotient rules for differentiating combined functions, with practical applications in science and engineering.
- Differentiating composite functions using the chain rule and combining it with other differentiation techniques.
Exponential, Logarithmic, and Trigonometric Functions
- Rules for differentiating exponential, logarithmic, and trigonometric functions, and their real-world applications.
- Second derivatives and higher-order derivatives, with applications in motion analysis, concavity, and optimization.
This course is ideal for students seeking to build a strong foundation in calculus and its applications in real-world problem-solving.
This course introduces students to the fascinating world of curves, exploring their geometric, differential, and applied properties. Students will learn to classify curves, analyze their behavior using calculus, and apply these concepts in real-world contexts such as physics, engineering, and design. By the end of the course, students will have a solid foundation in the mathematical and practical aspects of curves.
Course Units:
- Explore the basic definitions, classifications, and mathematical representations of curves.
- Understand tangents, normals, curvature, and other geometric properties.
Differential Properties of Curves
- Use derivatives to analyze the behavior of curves, including critical points, concavity, and inflection points.
- Represent and analyze curves using parametric and polar equations, and explore their unique applications.
- Discover how curves are used in fields like physics, engineering, and design, and solve practical problems.
This course is ideal for students looking to deepen their understanding of curves and their applications in mathematics and beyond.
This Mini takes a look at
- Rates of Change
- Optimisation
The Techniques of Integration course delves into the essential mathematical tools used to calculate areas, solve real-world problems, and explore the deeper connections between differentiation and integration. This course is ideal for students who already have a foundation in calculus and are ready to expand their knowledge of integral calculus. Through a blend of theory, problem-solving, and applications, students will master the skills needed for advanced mathematical analysis and STEM-related fields.
Course Units:
Approximating Area Under a Curve
Learn numerical methods, including the trapezoidal rule and Simpson's rule, to estimate the area under a curve and evaluate the accuracy of these techniques.Riemann Integrals
Explore the definition of Riemann sums and the properties of integrals, gaining a conceptual understanding of how integration is built from limits.Antidifferentiation
Develop skills in finding antiderivatives, using substitution, integration by parts, and partial fractions, with applications to solving differential equations.Fundamental Theorem of Calculus
Discover the powerful connection between differentiation and integration, and apply it to solve definite integrals and real-world problems.Advanced Techniques of Integration
Master complex integration techniques, including trigonometric integrals, improper integrals, and numerical methods, with applications in physics and engineering.
This course will equip students with the analytical skills necessary to tackle higher-level mathematics and apply calculus concepts to real-world challenges.
This section covers the following:
- Discovering Integrals
- Rules for Integration
- Particular Values
- Integrating f(ax + b)
- Integrating by Substitution
This section covers: