# Calculus and Vectors MCV4U

$500.00

This course builds on studentsâ€™ previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three- dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who choose to pursue careers in fields such as science, engineering, economics, and some areas of business, including those students who will be required to take a university-level calculus, linear algebra, or physics course.

The **MCV4U Calculus and Vectors** course is integral for Grade 12 students preparing for highly analytical university programs such as engineering, physical sciences, and mathematics. This course focuses on developing student understanding of calculus and vectors, which are foundational components in many fields of study.

## Course Overview

Students will explore advanced topics in differential and integral calculus, which form the basis for many applications in the physical sciences and engineering. The vectors portion of the course covers both magnitude and direction, which are crucial for courses in physics and engineering.

## Prerequisites

Successful completion of *Advanced Functions, Grade 12, University Preparation (MHF4U)* is required.

## Key Learning Objectives

- Develop an understanding of the mathematical theories and principles underlying calculus and vectors.
- Apply derivatives to solve complex problems including those involving motion, rates of change, and optimization.
- Explore and apply vector concepts to model and solve real-world problems.
- Analyze and solve problems involving vector components, products, and geometries in two and three dimensions.

## Course Goals

This course aims to prepare students for further education in fields requiring a high level of mathematical competence and to develop the analytical skills needed for problem-solving in calculus and vector mathematics.

## Overall Expectations

By the end of this course, the student will:

## RATE OF CHANGE

- Demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit;
- Graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between the numeric, graphical, and algebraic representations of a function and its derivative;
- Verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.

## DERIVATIVES AND THEIR APPLICATIONS

- Make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching;
- Solve problems, including optimization problems, that require the use of the concepts and procedures associated with the derivative, including problems arising from real-world applications and involving the development of mathematical models.

## GEOMETRY AND ALGEBRA OF VECTORS

- Demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;
- Perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications;
- Distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space;
- Represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.

## Course Details

Course Code | MCV4U |

Course Type | University Preparation |

Format | Online School Course |

Prerequisite | MCR3U |

Co-requisite | MHF4U |

Department | Mathematics |

Course Title | Calculus and Vectors |

Grade | Grade 12 |

Credit Value | 1.0 |

## Outline of Course Content

Unit | Title | Time |

1 | Rates of Change | 12 hours |

2 | Derivatives | 14 hours |

3 | Curve Sketching | 14 hours |

4 | Derivatives of Sinusoidal Functions | 10 hours |

5 | Exponential and Logarithmic Functions | 13 hours |

6 | Geometric Vectors | 15 hours |

7 | Cartesian Vectors | 15 hours |

8 | Lines and Planes | 15 hours |

Final Examination | 2 hours | |

Total | 110 hours |