MCV4U


 * MCV4U - Calculus & Vectors** Grade 12, University Preparation

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**Overall 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.

=Prerequisites= Advanced Functions (MHF4U) must be taken prior to or concurrently with Calculus and Vectors.

=Overall Curriculum Expectations= __The Geometry and Algebra of Vectors__
 * Demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;
 * 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;
 * 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;
 * Represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.

__Understanding Concepts of Calculus__
 * 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;
 * 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;
 * 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.

__Understanding the Derivative and Its Applications__
 * 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;
 * Solve rate of change and optimization problems, including those arising from real world applications and that involve investigation and mathematical modeling and require the use of the concepts and procedures associated with the derivative.

=Units= As part of each unit, mathematical processes are incorporated across all strands: Problem Solving, Reasoning and Proving, Reflecting, Selecting Tools and Computational Strategies, Connecting, Representing, and Communicating.

Adapted from OAME MCV4U Course Outline = Unit 1 =

Exploring rate of "flow" problems using non-algebraic means

 * Explore contexts and solve problems where one needs to know rate of change at a specific point,using verbal and graphical representations of the function. Include examples where mechanical tools are not readily available (e.g.,income flow, garbage accumulation rate).
 * Analyse rates of change and provide qualitative solutions to problems (e.g.,increase,decrease, tend towards something).

Standardize the process of finding instantaneous rate of change at a particular point.
= Unit 2 =
 * Apply a standard process for determining instantaneous rate of change of a function at a specific point on its graph.
 * For polynomial, simple rational, and radical functions, form, evaluate, and interpret the first principles definition of the derivative, using a fixed (numerical) value.

Derivative functions from first principles.

 * Recognize numerical and graphical representations of increasing and decreasing rates of change.
 * Use patterning and reasoning to determine thatthere 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.

Application of derivatives of polynomial functions.

 * Graph, without technology, the derivative of polynomials with given equations.
 * Given the graph of the derivative, sketch the original polynomial.

Derivative Functions Through Investigation.
= Unit 3 =
 * Through investigation, determine the algebraic representation of the derivative at any point for exponential, logarithmic and sine/cosine functions.

Derivative functions: properties and their applications.
= Unit 4 =
 * Investigate properties of derivatives (power rule, chain rule as change of scale and as patterning, no quotient rule use product rule, Sample Problem: Examine the relationship between the derivative of a function and the derivative of its inverse. Generalize the power rule for all rational powers).
 * Apply these properties to form derivatives of functions and simple combinations of functions (no simplification of derivatives formed outside of problem-solving contexts).

Applications of derivatives in rate of change and optimization problems, including those requiring modelling.
= Unit 5 =
 * 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.

Representing vectors.
= Unit 6 =
 * Introduce vectors in 2-D and 3-D.
 * Represent vectors geometrically and algebraically.
 * Operate with vectors.
 * Solve problems involving vectors.

Representing lines and planes.
=Resources= OAME MCV4U Resources Careers in Math Calculus Lessons The Geometer's Sketchpad Resource Centre Links to more Calculus Resources AP Calculus Resources Calculus Manipulatives Karl's Calculus Tutor Visual Calculus Tutorials Khan Academy Video Tutorials =Texts= //Calculus and Advanced Functions
 * Parametric equations of functions.
 * Represent lines and planes in a variety of ways.
 * Find intersections of two planes.
 * Find intersections of three planes.
 * Course Texts**

Advanced Functions and Introductory Calculus// //Harcourt Advanced Functions and Introductory Calculus 12// //Advanced Functions 12//
 * Supplementary Texts**