G12 Calculus and Vectors (MCV4U)

Chapter 1 - Introduction to Calculus

What you'll learn
- Rationalize denominators of radical expressions, simplifying them by removing radicals from the denominator of a fraction.
- Determine the slope of a tangent line using the limit of the slope of secant lines, connecting geometry to calculus.
- Calculate average and instantaneous rates of change, interpreting them as the slope of a secant and tangent line.
- Evaluate the limit of a function and determine function continuity at a point using the formal definition of a limit.

Requirements
- G11 Functions (MCR3U) and G12 Advanced Functions (MHF4U)

Description
G12 Calculus and Vectors (MCV4U) is designed for students planning to qualify for college or university.

This course builds on students’ foundational knowledge of functions from previous courses to explore the fundamental concepts of calculus and vectors. Students will extend their understanding of rates of change to develop the formal concept of the derivative, investigating the properties and applications of derivatives for various classes of functions, including polynomial, rational, exponential, logarithmic, and sinusoidal. The course emphasizes a conceptual understanding of limits and how they are used to define the derivative as an instantaneous rate of change and the slope of a tangent line. This provides the essential mathematical framework for analyzing dynamic systems and change.

A significant portion of the course is dedicated to the practical and theoretical applications of derivatives. Students will master techniques for curve sketching, using derivatives to determine key features such as intervals of increase and decrease, critical points, local extrema, and points of inflection. They will apply these skills to solve a wide array of optimization problems, finding the most efficient or effective solutions in contexts ranging from geometric max-min problems to real-world scenarios in business and economics. Furthermore, students will explore related rates problems, learning to model situations where multiple variables change in relation to one another.

The final major unit introduces the world of vectors and three-dimensional space. Students will represent and operate on vectors algebraically and geometrically, understanding concepts of dot product, cross product, and projections. They will then apply vectors to describe lines and planes in three-space, solving problems involving distances and intersections. This geometric application of vectors is crucial for fields like engineering and physics. By the conclusion of MCV4U, students will have a strong foundation in the principles of differential calculus and vector algebra, preparing them for the rigorous demands of university programs in science, engineering, computer science, economics, and mathematics.

Who this course is for:
- The expected applicants for STEM majors in the G12 cohort, along with some advanced G11 students, are also suitable for candidates preparing for the AP Calculus exam.
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