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.
|Unit Titles and Descriptions||Time Allocated|
|Concepts of Calculus|
A variety of mathematical operations with functions are needed in order to do the calculus of this course. This unit begins with students developing a better understanding of these essential concepts. Students will then deal with rates of change problems and the limit concept. While the concept of a limit involves getting close to a value but never getting to the value, often the limit of a function can be determined by substituting the value of interest for the variable in the function. Students will work with several examples of this concept. The indeterminate form of a limit involving factoring, rationalization, change of variables and one sided limits are all included in the exercises undertaken next in this unit. To further investigate the concept of a limit, the unit briefly looks at the relationship between a secant line and a tangent line to a curve. To this point in the course students have been given a fixed point and have been asked to find the tangent slope at that value, in this section of the unit students will determine a tangent slope function similar to what they had done with a secant slope function. Sketching the graph of a derivative function is the final skill and topic.
The concept of a derivative is, in essence, a way of creating a short cut to determine the tangent line slope function that would normally require the concept of a limit. Once patterns are seen from the evaluation of limits, rules can be established to simplify what must be done to determine this slope function. This unit begins by examining those rules including: the power rule, the product rule, the quotient rule and the chain rule followed by a study of the derivatives of composite functions. The next section is dedicated to finding the derivative of relations that cannot be written explicitly in terms of one variable. Next students will simply apply the rules they have already developed to find higher order derivatives. As students saw earlier, if given a position function, they can find the associated velocity function by determining the derivative of the position function. They can also take the second derivative of the position function and create a rate of change of velocity function that is more commonly referred to as the acceleration function which is where this unit ends.
|Derivative Applications and Related Rates|
A variety of types of problems exist in this unit and are generally grouped into the following categories: Pythagorean Theorem Problems (these include ladder and intersection problems), Volume Problems (these usually involve a 3-D shape being filled or emptied), Trough Problems, Shadow problems and General Rate Problems. During this unit students will look at each of these types of problems individually.
In previous math courses, functions were graphed by developing a table of values and smooth sketching between the values generated. This technique often hides key detail of the graph and produces a dramatically incorrect picture of the function. These missing pieces of the puzzle can be found by the techniques of calculus learned thus far in this course. The key features of a properly sketched curve are all reviewed separately before putting them all together into a full sketch of a curve.
|Derivatives of Trigonometric, Exponents and Logarithmic Functions|
A brief trigonometry review kicks off this unit. Then students turn their attention to special angles and the CAST rule which has been developed to identify which of the basic trigonometric ratios is positive and negative in the four quadrants. Students will then solve trigonometry equations using the CAST rule to locate other solutions. Two fundamental trigonometric limits are investigated for the concepts of trigonometric calculus to be fully understood.
This unit continues with examples and exercises involving exponential and logarithmic functions using Euler’s number (e). But as students have already seen, many other bases exist for exponential and logarithmic functions. Students will now look at how they can use their established rules to find the derivatives of such functions. The next topic should be familiar as the steps involved in sketching a curve that contains an exponential or logarithmic function are identical to those taken in the curve sketching unit studied earlier in the course. Because the derivatives of some functions cannot be determined using the rules established so far in the course, students will need to use a technique called logarithmic differentiation which is introduced next.
There are four main topics pursued in this initial unit of the course. These topics are: an introduction to vectors and scalars, vector properties, vector operations and plane figure properties. 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 with and without dynamic geometry software. Students will conclude the first half of the unit by proving some properties of plane figures, using vector methods and by modeling and solving problems involving force and velocity. Next students learn to represent vectors as directed line segments and to perform the operations of addition, subtraction, and scalar multiplication on geometric vectors with and without dynamic geometry software. The final topic involves students in proving some properties of plane figures using vector methods.
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. These vector products will be revisited to predict characteristics of the solutions of systems of lines and planes in the intersections of lines and planes.
|Linear Dependence and Coplanarity|
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 co-linearity and co-planarity of vectors.
|Intersection of Lines and Planes|
This unit begins with students determining the vector, parametric and symmetric equations of lines in R2 and R3. Students will go on to determine the vector, parametric, symmetric and scalar equations of planes in 3-space. The intersections of lines in 3-space and the intersections of a line and a plane in 3-space are then taught. Students will learn to determine the intersections of two or three planes by setting up and solving a system of linear equations in three unknowns. Students will interpret a system of two linear equations in two unknowns geometrically, and relate the geometrical properties to the type of solution set the system of equations possesses. Solving problems involving the intersections of lines and planes, and presenting the solutions with clarity and justification forms the next challenge. As work with matrices continues students will define the terms related to matrices while adding, subtracting, and multiplying them. Students will solve systems of linear equations involving up to three unknowns, using row reduction of matrices, with and without the aid of technology and interpreting row reduction of matrices as the creation of new linear systems equivalent to the original constitute the final two new topics of this important unit.
This is a proctored exam worth 30% of your final grade.
Overall Curriculum Expectations
|A. Rate of Change|
|A1||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;|
|A2||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;|
|A3||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.|
|B. Derivatives and their Applications|
|B1||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;|
|B2||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.|
|C. Geometry and Algebra of Vectors|
|C1||demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;|
|C2||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;|
|C3||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;|
|C4||represent lines and planes using scalar, vector, and parametric equations, and solve problems involving distances and intersections.|
Teaching and Learning Strategies:
The over-riding aim of this course is to help students use the language of mathematics skillfully, confidently and flexibly, a wide variety of instructional strategies are used to provide learning opportunities to accommodate a variety of learning styles, interests, and ability levels. The following mathematical processes are used throughout the course as strategies for teaching and learning the concepts presented:
- Problem solving: This course scaffolds learning by introducing concepts and relating them back to concepts that were taught in prior units and mathematics courses. The course guides students toward recognizing opportunities to apply knowledge they have gained to solve real-world problems.
- Selecting Tools and Computational Strategies: This course models the use of 3D graphing software to help solve problems and to familiarize students with technologies that can help make solving problems faster and more accurately.
- Connecting: Students will connect concepts learned in this course to real-world applications of Vectors and Calculus through assignments, examples, and practice problems.
- Reflecting: This course models the reflective process. Through the use of examples and practice exercises, the course demonstrates proper communication to explain intermediate steps and reflect on solutions to determine if they make sense in the given context.
- Representing: Through the use of examples, practice problems, and solution videos, the course models various ways to demonstrate understanding, poses questions that require students to use different representations as they are working at each level of conceptual development – concrete, visual or symbolic, and allows individual students the time they need to solidify their understanding at each conceptual stage.
Assessment, Evaluation and Reporting Strategies of Student Performance:
Our theory of assessment and evaluation follows the Ministry of Education’s Growing Success document, and it is our firm belief that doing so is in the best interests of students. We seek to design assessment in such a way as to make it possible to gather and show evidence of learning in a variety of ways to gradually release responsibility to the students, and to give multiple and varied opportunities to reflect on learning and receive detailed feedback.
Growing Success articulates the vision the Ministry has for the purpose and structure of assessment and evaluation techniques. There are seven fundamental principles that ensure best practices and procedures of assessment and evaluation by Institute of Canadian Education teachers. ICE assessments and evaluations,
- are fair, transparent, and equitable for all students;
- support all students, including those with special education needs, those who are learning the language of instruction (English or French), and those who are First Nation, Métis, or Inuit;
- are carefully planned to relate to the curriculum expectations and learning goals and, as much as possible, to the interests, learning styles and preferences, needs, and experiences of all students;
- are communicated clearly to students and parents at the beginning of the course and at other points throughout the school year or course;
- are ongoing, varied in nature, and administered over a period of time to provide multiple opportunities for students to demonstrate the full range of their learning;
- provide ongoing descriptive feedback that is clear, specific, meaningful, and timely to support improved learning and achievement;
- Develop students’ self-assessment skills to enable them to assess their own learning, set specific goals, and plan next steps for their learning.
The Final Grade:
The evaluation for this course is based on the student’s achievement of curriculum expectations and the demonstrated skills required for effective learning. The final percentage grade represents the quality of the student’s overall achievement of the expectations for the course and reflects the corresponding level of achievement as described in the achievement chart for the discipline. A credit is granted and recorded for this course if the student’s grade is 50% or higher. The final grade will be determined as follows:
- 70% of the grade will be based upon evaluations conducted throughout the course. This portion of the grade will reflect the student’s most consistent level of achievement throughout the course, although special consideration will be given to more recent evidence of achievement.
- 30% of the grade will be based on final evaluations administered at the end of the course. The final assessment may be a final exam, a final project, or a combination of both an exam and a project.
The Report Card:
Student achievement will be communicated formally to students via an official report card. Report cards are issued at the midterm point in the course, as well as upon completion of the course. Each report card will focus on two distinct, but related aspects of student achievement. First, the achievement of curriculum expectations is reported as a percentage grade. Additionally, the course median is reported as a percentage. The teacher will also provide written comments concerning the student’s strengths, areas for improvement, and next steps. Second, the learning skills are reported as a letter grade, representing one of four levels of accomplishment. The report card also indicates whether an OSSD credit has been earned. Upon completion of a course, ICE will send a copy of the report card back to the student’s home school (if in Ontario) where the course will be added to the ongoing list of courses on the student’s Ontario Student Transcript. The report card will also be sent to the student’s home address.
Program Planning Considerations:
Teachers who are planning a program in this subject will make an effort to take into account considerations for program planning that align with the Ontario Ministry of Education policy and initiatives in a number of important areas.
Planning Programs for Students with Special Education Needs, Program Considerations for, English Language Learners, Environmental Education, Healthy Relationships, Equity and, Inclusive Education, Financial Literacy Education, Literacy, Mathematical Literacy, and Inquiry Skills, Critical Thinking and Critical Literacy, The Role of the School Library, The Role of Information and Communications Technology, The Ontario Skills Passport: Making Learning Relevant and Building Skills, Education and Career/Life Planning, Cooperative Education and Other Forms of Experiential Learning, Planning Program Pathways and Programs Leading to a Specialist High Skills Major, Health and Safety, Ethics
Calculus and Vectors 12, Nelson Canada ELHI; 1 edition (Aug. 15 2008)
|1. Concepts of Calculus|
|Concepts of Calculus- 1.1 Radical Expressions||00:00:00|
|Concepts of Calculus- 1.2 Rate of Changes||00:00:00|
|Concepts of Calculus- 1.3 Rate of Changes Con’td||00:00:00|
|Concepts of Calculus- 1.4 Limits||00:00:00|
|Concepts of Calculus- 1.5a Properties of Limits||00:00:00|
|Concepts of Calculus- 1.5b Properties of Limits Con’td II||00:00:00|
|Concepts of Calculus- 1.6 Continuity-Supplementry||00:00:00|
|Concepts of Calculus- 1.6 Continuity||00:00:00|
|Concepts of Calculus- Review of Factor and Rational Express||00:00:00|
|Concepts of Calculus- Unit1 Review||00:00:00|
|Concept of Calculus- Assignment 1||10, 00:00|
|CH1 M- Test Unit 1: Rate of Change, Limits & Continuity||02:00:00|
|Quiz Limit 1||02:00:00|
|Quiz Limit 2||02:00:00|
|Unit 1 TEST||02:00:00|
|Derivatives- 2.1 Derivatives First Principle||00:00:00|
|Derivatives- 2.2 Derivative of Poly. Function||00:00:00|
|Derivatives- 2.3 Product Rule||00:00:00|
|Derivatives- 2.3-2.4 More Practice on Differentiation||00:00:00|
|Derivatives- 2.4 Quotient Rule||00:00:00|
|Derivatives- 2.5 Derivative of Composite Function||00:00:00|
|Derivatives- Homework and Review Problems||00:00:00|
|Derivatives- Assignment 1||10, 00:00|
|Derivatives- Assignment 2015||10, 00:00|
|Test Unit 2: Derivatives||02:00:00|
|3. Derivatives Applications and Related Rates|
|Derivatives Applications and Related Rates- 3.1 Higher Order Derivatives,vel,acc||00:00:00|
|Derivatives Applications and Related Rates- 3.2||00:00:00|
|Derivatives Applications and Related Rates- 3.3||00:00:00|
|Derivatives Applications and Related Rates- 3.4||00:00:00|
|Derivatives Applications and Related Rates- CH 3 Problems||00:00:00|
|Derivatives Applications and Related Rates- MCV4U1 Extreme Value Problem-2016||00:00:00|
|Derivatives Applications and Related Rates- Assignment 1||10, 00:00|
|CH3 Problems TEST||02:00:00|
|CH3- 2015 Problems TEST||02:00:00|
|MCV4U CH 3 Test AM||02:00:00|
|TEST BANK MCGREW – CH3-2015||02:00:00|
|4. Curve Sketching|
|Curve Sketching- 4.1 Increasing and Decreasing Function||00:00:00|
|Curve Sketching- 4.2 Local and Absolute Extrema||00:00:00|
|Curve Sketching- 4.3 Asymptotes||00:00:00|
|Curve Sketching- 4.4 Concavity & Inflection Point||00:00:00|
|Curve Sketching- 4.5 Using Second Derivative Test to Find Local Extrema||00:00:00|
|Curve Sketching- 4.6 Curve Sketching||00:00:00|
|Curve Sketching- Assign Curve Sketching Solutions||00:00:00|
|Curve Sketching- Assignment Curve Sketching||10, 00:00|
|Curve Sketching- Review of Prerequisite||00:00:00|
|CH4 – PM TEST||02:00:00|
|MCV4UChap 4 Curve Sketch Skills TEST||02:00:00|
|TEST BANK MCGREW – CH4 4.4 & 4.5 -2015||02:00:00|
|5. Derivatives of Trigonometric Exponents and Logarithmic Functions|
|5.1 Instantaneous Rate of Change of Sinusoidal Functions||00:00:00|
|5.1-2015 Derivative of Natural Exponential Function||00:00:00|
|5.2 Derivative of Sine & Cosine Functions||00:00:00|
|5.2-2015 Derivative of General Exponential Functions||00:00:00|
|5.3 Differentiation of Trigonometric Functions||00:00:00|
|5.3-2015 Applications of Exponential Functions||00:00:00|
|5.4 -2015 Derivative of Sinusoidal Functions||00:00:00|
|5.4 Applications of Sinusoidal Functions and Their Derivatives||00:00:00|
|5.5 The Natural Exponential Function and Its Derivative||00:00:00|
|5.5-2015 Derivative of tan x Functions||00:00:00|
|5.6 Differentiation of Exponential Functions||00:00:00|
|5.7 Applications of Exponential Functions and Their Derivatives||00:00:00|
|5.8 Differentiation of Logarithmic Functions||00:00:00|
|Unit 5 Review||00:00:00|
|MCV4U Unit5 – Apr2015 TEST||02:00:00|
|MCV4U Unit 5 TEST||02:00:00|
|Vectors- 6.1 Introduction to Vectors||00:00:00|
|Vectors- 6.2 Vector Laws||00:00:00|
|Vectors- 6.3 Scalar Multiplication||00:00:00|
|Vectors- 6.4Properties of Vectors||00:00:00|
|Vectors- 6.5 Vectors in R2 and R3||00:00:00|
|Vectors- 6.6 Operations with Algebraic Vectors in R2||00:00:00|
|Vectors- 6.7 Operation with Algebraic Vectors in R3||00:00:00|
|Chapter 6 TEST- AM||02:00:00|
|Chapter 6- True TEST||02:00:00|
|7. Vector Application|
|Vector Application- 7.1 Force as a Vector||00:00:00|
|Vector Application- 7.2 Velocity as a Vector part I||00:00:00|
|Vector Application- 7.2 Velocity as a Vector part II||00:00:00|
|Vector Application- 7.3 & 7.4 The Dot Product of Two Vectors||00:00:00|
|Vector Application- 7.5 Scalar & Vector Projections||00:00:00|
|Vector Application- 7.7 Applications of Dot& Cross Products||00:00:00|
|Vector Application- 7.6 The Cross Product of Two Vectors||00:00:00|
|Ch7 Quiz V2 May 2011||10, 00:00|
|8. Linear Dependance and Coplanarity|
|8.1 Vector and Parametric Equations of a Lines in R2||00:00:00|
|8.2 Cartesian Equations of Lines||00:00:00|
|8.3 Vector, Parametric, and Symmetric Equations of a Line in R3||00:00:00|
|MCV4U Final ISU Calculus Exam||10, 00:00|
|MCV4U PM Chapter 7- Test, Vector Applications||02:00:00|