Calculus and Vectors
Grade 12, University Preparation (MCV4U)
CourseTitle: Calculus and Vectors, Grade 12, University Preparation(MCV4U)
CourseName: Calculus and Vectors
CourseType: University Preparation
Prerequisite: Advanced Functions, Grade 12, UniversityPreparation
Curriculum Policy Document: Mathematics, The Ontario Curriculum, Grades 11 and 12, 2010 (Revised)
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; broadentheirunderstandingofratesofchangetoincludethederivativesofpolynomial,sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modeling of real-worldrelationships.Studentswillalsorefinetheiruseofthemathematicalprocessesnecessary 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.
Titles and Descriptions
Time and Sequence
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 theseessentialconcepts.Studentswillthendealwithrates of change problems and the limit concept. While the conceptofalimitinvolvesgettingclosetoavaluebutnever 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 examplesofthisconcept.Theindeterminateformofalimit involving factoring, rationalization, change of variablesand onesidedlimitsareallincludedintheexercisesundertaken next in this unit. To further investigate the concept of a limit, the unit briefly looks at the relationship between a secantlineandatangentlinetoacurve.Tothispointinthe 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.Sketchingthegraphofaderivativefunctionisthe final skill and topic.
The concept of a derivative is, in essence, a way ofcreating a short cut to determine the tangent line slope function that would normally require the concept of a limit. Once patternsareseenfromtheevaluationoflimits,rulescanbe 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 bewrittenexplicitlyintermsofonevariable.Nextstudents 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 ofthepositionfunctionandcreatearateofchangeof velocity function that is more commonly referred to as the acceleration function which is where this unit ends.
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 separatelybefore
putting them all together into a full sketch of a curve.
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.
Derivative of Exponents and Log Functions-Exponential Functions
This unit begins with examples and exercises involving exponentialandlogarithmicfunctionsusingEuler’snumber (e). But as students have already seen, many other bases exist for exponential and logarithmic functions. Students willnowlookathowtheycanusetheirestablishedrulesto findthederivativesofsuchfunctions.Thenexttopicshould 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 functionscannotbedeterminedusingtherulesestablished sofarinthecourse,studentswillneedtouseatechnique
called logarithmic differentiation which is introducednext.
Trig Differentiation and Application
A brief trigonometry review kicks off this unit. Then studentsturntheirattentiontospecialanglesandtheCAST 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. The unit ends, as in all other units in the course, with an assignment and a unitquiz.
There are four main topics pursued in this initial unit ofthe 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.Studentswillconcludethefirsthalfoftheunitby proving some properties of plane figures, using vector methods and by modeling and solving problems involving forceandvelocity.Nextstudentslearntorepresentvectors as directed line segments and to perform the operationsof addition, subtraction, and scalar multiplication on geometric vectors with and without dynamicgeometry
software. The final topic involves students in proving some properties of plane figures using vector methods.
Cartesian vectors are represented in two-space and three- space as ordered pairs and triples, respectively. The addition,subtraction,andscalarmultiplicationofCartesian vectors are all investigated in this unit. 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 solutionsofsystemsoflinesandplanesintheintersections of lines andplanes.
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 intersection softwoorthreeplanes 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,andmultiplyingthem.Studentswillsolve systems of linear equations involving up to three unknowns, using row reduction of matrices, with and without the aid of technology and interpretingrow reductionofmatricesasthecreationofnewlinearsystems equivalent to the original constitute the final two new topics of this importantunit.