Apex Academy Courses

Calculus & Vectors, Grade 12

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Calculus & 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 course.

Corequisite: Advanced Functions, Grade 12 University Preparation*

*Advanced Functions and Calculus and Vectors may be taken at the same time.

Course Outline

Module 1 Overview: Rates of change

Guiding Question:
How do rates of change and limits describe the shape of a graph?

In this module, you will consider the limit of a function at various points on a graph. You will consider types of discontinuities when the limit does not exist. You will compare the average and instantaneous rates of change. You will use limits and difference quotient to calculate instantaneous rates of change to find slopes and equations of a tangent line to a graph. 

Module 2 Overview: Derivatives

Guiding Question:
What patterns exist in calculating derivatives of functions?

In this module, you will consider the patterns that exist when you calculate the instantaneous rate of change for a polynomial function or a rational function. You will call these results derivative rules and you can use these to find equations of tangent lines.

Module 3 Overview: Curve Sketching and optimization

Guiding Question:
How can we use derivatives to sketch graphs and solve optimization problems?

In this module, you will use the first and second derivative to determine the intervals of increase/decrease and the intervals of concavity for polynomial and rational functions, in order to sketch their graphs. You will also apply derivative rules and properties to solve optimization problems about business, geometry and others. You will consider the derivatives of exponential and trigonometric functions. 

Module 4 Overview: vector operations

Guiding Question:
How can combinations of vectors model forces and movement?

In this module, you will represent vectors both geometrically and algebraically. You will carry out operations on vectors and explore their characteristics.

Module 5 Overview: equations of lines in 2-space

Guiding Question:
How can equations of lines in two spaces be expressed to determine their relationship?

In this module, you  will examine the vector, parametric and scalar equations of lines in 2-space. You will practice writing equations in these different forms, using given information about vectors and points. You will also determine the intersection of two lines.

Module 6 Overview: equations of lines in 3-space

Guiding Question:
How can solutions of systems of equations show the geometry of lines and planes?

In this module, you will practice writing equations of lines and planes in 3-space in different forms. You can use these equations to find intersection of lines, planes, and lines with planes in 3-space.

Assessment Breakdown

Course Work, Assignments, and Proctored Tests: 70%
Culminating Independent Study Project: 10%
Final Exam: 20%