# Calculus and Vectors – Grade 12

This course builds on students’ 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 coursse.

From: $500.00

### Overall curriculum expectations

**Rate of Change**

By the end of this course, students will:

1. 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.

2. 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.

3. 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**

By the end of this course, students will:

1. 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.

2. 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**

By the end of this course, students will:

1. Demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications.

2. 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.

3. 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.

4. Represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.

### 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 |

### Course Details

Course Code | MCV4U |

Course Type | University Preperation |

Format | Online School Course |

Prerequisite | MCR3U or MCT4C |

Tuition Fee | 650 – 1300 |

Department | Mathematics |

Course Title | Advanced Functions |

Grade | Grade 12 |

Credit Value | 1.0 |

### Overall Curriculum Expectations

**Exponential and logarithmic functions **

By the end of this course, students will:

- Demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions.
- Identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically.
- Solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

**Trigonometric Functions**

By the end of this course, students will:

- Demonstrate an understanding of the meaning and application of radian measure.
- Make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals and use these connections to solve problems.
- Solve problems involving trigonometric equations and prove trigonometric identities.

**Polynomial and Rational Functions**

By the end of this course, students will:

- Identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions.
- Identify and describe some key features of the graphs of rational functions and represent rational functions graphically.
- Solve problems involving polynomial and simple rational equations graphically and algebraically.

Demonstrate an understanding of solving polynomial and simple rational inequalities.

**Characteristics of Functions**

By the end of this course, students will:

- Demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point.
- Determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems.
- Compare the characteristics of functions, and solve problems by modelling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.

### Outline of Course Content

Unit | Title | Time |

1 | Polynomial Functions | 15 hours |

2 | Polynomial Equations and Inequalities | 15 hours |

3 | Rational Functions | 13 hours |

4 | Trigonometry | 13 hours |

5 | Trigonometric Functions | 13 hours |

6 | Exponential and Logarithmic Functions | 13 hours |

7 | Tools and Strategies for Solving Exponential and Logarithmic Equations | 13 hours |

8 | Combining Functions | 13 hours |

Final Examination | 2 hours | |

Total | 110 hours |