# MCV4U, Grade 12, Calculus and Vectors, University Preparation

Course Title/ Grade/ Course Type: Calculus and Vectors, Grade 12, University Preparation

Ministry Course Code: MCV4U

Curriculum Document: Mathematics, the Ontario Curriculum, Grades 11 and 12, 2007

SKU: N/A Category: Tags: ,

## Description

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, 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.

Outline of Course Content

Unit 1:  Introduction to Calculus

In this chapter, the students will review the concepts of average and instantaneous rate of change to determine the rate of change over an interval or at a single point. Students are introduced to the concept of limit which involves getting close to a value but never getting to that value. They solve limit using variety of method like factoring, rationalization and change of variables. Students determine if the function is continuous or not by using the concepts of one-sided limits.

Unit 2:  Derivatives and their Application

Students use the first principle definition of derivative to determine the derivative of simple polynomial functions. They use the power rule, the product rule, the quotient rule and the chain rule to determine the derivative of more complex polynomial and rational functions. In this chapter, student use derivatives to solve optimization problems and problems involving position, velocity and acceleration. Students use the first derivative to determine the interval of increase and decrease of a function. They use that information to identify if a function has a local maximum or minimum. Students also use the concept of limit to determine the horizontal asymptote of a rational function. Student use the second derivative to identify the interval of concavity and use the second derivative test to determine if a function has a local maximum or minimum at the critical points. In the end of the unit, students are able to use the curve sketching algorithm (x-intercepts, y-intercepts, critical points, interval of increase and decrease, point of infection, and interval of concavity) to sketch a proper graph of a polynomial and rational functions.

Unit 3:  Curve Sketching

In this chapter, student use the second derivative to identify the interval of concavity and use the second derivative test to determine if a function has a local maximum or minimum at the critical points.

In the end of the unit, students are able to use the curve sketching algorithm (x-intercepts, y-intercepts, critical points, interval of increase and decrease, point of infection, and interval of concavity) to sketch a proper graph of a polynomial and rational functions.

Unit 4: Derivatives of Exponential and Trigonometric Functions

This chapter starts with an introduction to the Euler’s number (e). Students determine the derivative of Euler’s number and other exponential functions. They use the concepts used in other units like, product rule, quotient rule, chain rule and curve sketching for the exponential functions. Students are also determining the derivative of sinusoidal functions and they solve related problems involving trigonometric functions.

Unit 5: An Introduction to Vectors

In this chapter, students will tell the difference between a scalar and vector quantity. They will represent vectors as directed line segments and perform the operations of addition, subtraction, and scalar multiplication on geometric vectors. Cartesian vectors are represented in two-space and three-space as ordered pairs and triples, respectively. The addition, subtraction, and scalar multiplication of Cartesian vectors are all investigated in this unit. Students investigate the concepts of linear dependence and independence, and collinearity and coplanarity of vectors.

Unit 6: Application of Vectors

Applications involving work and torque are used to introduce and lend context to the dot and cross products of Cartesian vectors. The vector and scalar projections of Cartesian vectors are written in terms of the dot product. The properties of vector products are investigated and proven.

Unit 7: Equations of Lines and Planes & Relationships between Points, Lines, and Planes

In this chapter, student will begin by determining the vector, parametric and symmetric equation of lines in R2 and R3. Student will go on to determine the vector, parametric, symmetric and scalar equation of planes in 3-spaces. The intersection of lines in 3-space and the intersections of lines in 3-spaces will also be learned. Student will determine the distance between a point and a line and also between a point and a plane.

In this chapter, students are introduced to the most important idea associated with vectors, the solution of systems of equations. Geometrically, the solution of two equations in two unknowns is the point of intersection between two lines on the xy-plane. In this chapter, the students are going to extend these ideas and consider systems of equations in �� and interpret their meaning.

Unit 8: Final Culminating Evaluation & Final Exam

Using the knowledge and skills gained from previous units, the students will complete an assignment that shows what they know. The students will choose from various format for the final product or they may consult with the teacher to suggest a format they want to do. The assignment will have to be submitted in class. The final evaluation will take the form of an assignment that will be divided into two parts, Part A and Part B, and will be worth 15% and 15% respectively for a total of 30% of the final grade.

Final Mark will be determined following percentages:

70% Determined by the evaluations conducted throughout the duration of the course:

Products – Assignments, Test

Observations – Presentations, Problem Solving

Conversations – Oral Test, Conference with the Teacher

30% Final examination of the students and/or a Culmination Assignment