MCV4U – Calculus and Vectors – Grades 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, 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.
$450.00
*Course outline is subject to change
Overall Curriculum Expectations
By the end of this course, students will :
Rate of Change
- A.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 limit.
- A.2 – Graph the derivatives of polynomial, sinusoidal, and exponential functions, and make connections between numeric, graphical, and algebraic representations of a function and its derivative;
- A.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
- B.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;
- B.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
- C.1. – Demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their
applications; - C.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;
- C.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;
- C.4. – Represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.
Outline of Course Content
Students explore contexts and solve problems where one needs to know the rate of change at a specific point, using verbal and graphical representations of the function; analyze rates of change and provide qualitative solutions to problems; apply a standard process for determining instantaneous rate of change of a function at a specific point on its graph. Students will also form, evaluate, and interpret the first principles definition of the derivative for polynomial, simple rational and radical functions.
Students will recognize numerical and graphical representations of increasing and decreasing rates of change; use patterning and reasoning to determine that there is a function that describes the derivative at all points; for polynomial, rational and radical functions, determine, using limits, the algebraic representation of the derivative at any point; graph, without technology, the derivative of polynomials with given equations; sketch the original polynomial given the graph of the derivative, determine the algebraic representation of the derivative at any point for exponential, logarithmic and sine/cosine functions. Students investigate properties of derivatives (power rule, chain rule as change of scale and as patterning, no quotient rule use product rule, And apply these properties to form derivatives of functions and simple combinations of functions (no simplification of derivatives formed outside of problem-solving contexts).
Students solve rate of change and optimization problems given algebraic models; solve rate of change and optimization problems requiring the creation of an algebraic model (more variety in problems to get at various types of algebraic simplification and analysis); solve problems calling for the modelling of the rate of change flow problems), not necessarily finding the original function but just a property of it e.g., pt of inflection
Students are introduced to vectors in 2D and 3D; represent vectors geometrically and algebraically; operate with vectors; solve problems involving vectors.
Students are introduced to parametric equations of functions; represent lines and planes in a variety of ways; find intersection of two and three planes.
Final evaluation should be designed to provide the opportunity for students to demonstrate comprehensive learning in each of the four Achievement Chart categories. Due to the emphasis of cumulative tests and examinations in university programs, a final evaluation should play a prominent role in the final assessment of the student.