Calculus and Vectors - 12

Course Description:

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 course.

Prerequisite: MHF4U - Advanced Functions (co-requisite)

Course Outline

Unit 1: Rate of Change (40 hours)

This unit will focus on rate of change, limits and derivatives. Imagine a car driver speeding down a highway at 150km/hour. He passed a police car and was pulled over. The officer told him that he was speeding and was over the limit by 50km/hour. The driver argued that because he travelled 100 km from his office in one hour, therefore his average speed is within the 100km/hour limit. The driver's argument fails because he was charged based on instantaneous speed, not his average speed. There are many other situations where rate of change is very important. In this unit, you will learn how the derivatives and limits can be determined and applied in a great variety of ways. 

1-1 Instantaneous Rate of Change at a Point
1-2 Concepts of Limits                        
1-3 Applying Limits                                                   
1-4 Exploring Derivatives
1-5 The Chain Rule
1-6 Derivatives of Trigonometric Functions
1-7 Derivatives of Exponential Functions


Unit 2: Derivatives and their Applications (30 hours)

In this unit you will extend your knowledge and understanding of applications of derivatives. In the previous unit, you learned about the derivative rules and derivatives of various functions. You will make the connections between motion and derivatives and solve rate of change in natural/social sciences and optimization problems in a wide variety of contexts. You will also investigate the properties and applications of second derivatives and its applications. Lastly you will learn how to draw the graph of a function using the methods of calculus, including the first and second derivatives of function.

2-1 Higher Order Derivatives
2-2 Extreme Values
2-3 Optimization
2-4 Concavity
2-5 Curve Sketching


Unit 3: Geometry and Algebra of Vectors (40 hours)

How you ever tried to swim across a river with a strong current? Have you ever tried sailing across a windy lake? These experiences are all related to vector quantities. Vectors are essential tools, which are used in mathematics and social sciences. For example, pilots need to know what effect a crosswind will have in terms of navigation. Engineers need to know what load a particular bridge design will support. In this unit, we will focus on vectors and how they are presented algebraically or graphically. We will discover how vectors enable calculations in situations involving the velocity and forces and how vectors can be applied in many ways. We will also investigate the geometric configurations of lines and planes and their applications.

3-1 Geometric Vectors
3-2 Cartesian Vectors
3-3 Cross Product and its Properties                           
3-4 Equations of Lines
3-5 Equations of Planes
3-6 Relationships Between Points, Lines and Planes