AP Calculus
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Calendar
Summer
Pre-Calculus
Limits and Continuity
Derivatives
Applications
Integral
Integration Techniques
Volumes
L'Hopital
Infinite Series
Parametric
Vectors Unit
Marks                   

MCB4UW

COURSE OUTLINE   Sept 2008

 

CURRICULUM:     The Ontario Curriculum Grades 11 and 12: Mathematics 2007

[click logo to see curriculum guide]  

 

AP Calculus AB program.

TEXT:                    Calculus Graphical, Numerical, Algebraic; Finney, Demana, Waits, Kennedy

COURSE

DESCRIPTION:     This is a university-level course in Differential and Integral Calculus, equivalent to one semester of Calculus at most universities. The AP Calculus course is designed to develop the student’s understanding of the concepts of Calculus and to provide experience with its methods and applications. The course emphasizes a multi-representational approach to Calculus with concepts, results, and problems expressed geometrically, algebraically, numerically, analytically, and verbally. Successful completion of the AP Calculus Course also provides the student with an MCB4UI credit.   Math is a subject that builds upon itself; attending class, being on time, and participating in class are not only essential for good progress, but are sometimes essential for survival of the class.  Unexcused absences result in no make up for the work.

 

 

 

EVALUATION:    70% is based on tests, quizzes, assignments.

                         30% is based on the summative assessment, which includes the June exam.

                          AP Credit is solely based on a successful 3.25 hour exam.

 

 AP Books            KAPLAN AP Calculus  AB and BC 2008   (required Chapters)

                              Harcourt Brace Calculus  (ISBN: 0-15-601556-0)  (optional)
 

                               

OUTLINE

 

 

1.     Title and course info:

Waterloo Collegiate Institute – Science Department

MCV4UW-01, Period A&B, Room 410, Mr. Burns

Contact Info: burns@mjburns.net, Science Teacher Prep Room Rm 202 Period C/G/H

 


2.        Course Description:

 

This course builds on your previous experience with functions and your developing understanding of rates of change. You will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three dimensional space; broaden your 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 realworld relationships. You will also refine your 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.

 

This course extends students' experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs.

The Advanced Functions course (MHF4UW) must be taken prior to or concurrently with Calculus and Vectors (MCV4UW).

 

Prerequisite: Science, Grade 11, Academic AP

From: The Ontario Curriculum Grade 11 and 12 Science (Revised 2008), p. 194

Textbooks:

                    Calculus Graphical, Numerical, Algebraic; Finney, Demana, Waits, Kennedy                

3.       Overall Expectations (MCV4UI& MHF4U)  

 

A: Rate of Change

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 concepts of the limit

2.   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

3.   verify graphically and algebraically the rules for determining derivatives; apply these new 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

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

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

C: Geometry and Algebra of Vectors

1.   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

2. distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and there-space, and determine the different geometric configurations of lines and planes in three-space represent line and planes using scalar, vector, and parametric equations, and solve problems involving distance and intersections

A. Exponential and Logarithmic Functions

1.   Demonstrate an understanding of the relationship between exponential expressions and logarithmic expressions, evaluate logarithms, and apply the laws of logarithms to simplify numeric expressions;

2.   Identify and describe some key features of the graphs of logarithmic functions, make connections among the numeric, graphical, and algebraic representations of logarithmic functions, and solve related problems graphically;

3.   Solve exponential and simple logarithmic equations in one variable algebraically, including those in problems arising from real-world applications.

B. Trigonometric Functions

1.   Demonstrate an understanding of the meaning and application of radian measure;

2.   Make connections between trigonometric ratios and the graphical and algebraic representations of the corresponding trigonometric functions and between trigonometric functions and their reciprocals, and use these connections to solve problems;

 C. Polynomial and Rational Functions

1.   Identify and describe some key features of polynomial functions, and make connections between the numeric, graphical, and algebraic representations of polynomial functions;

2.   Identify and describe some key features of the graphs of rational functions, and represent rational functions graphically;

3.   Solve problems involving polynomial and simple rational equations graphically and algebraically;

4.   Demonstrate an understanding of solving polynomial and simple rational inequalities.

 D. Characteristics of Functions

1.   Demonstrate an understanding of average and instantaneous rate of change, and determine, numerically and graphically, and interpret the average rate of change of a function over a given interval and the instantaneous rate of change of a function at a given point;

2.   Determine functions that result from the addition, subtraction, multiplication, and division of two functions and from the composition of two functions, describe some properties of the resulting functions, and solve related problems;

3.   Compare the characteristics of functions, and solve problems by modeling and reasoning with functions, including problems with solutions that are not accessible by standard algebraic techniques.  

 

Overall Expectations (MCV4UW)

 

Titles and Descriptions

Advanced Functions

Several types of functions needed in this course will be reviewed along with their characteristics including: differences in polynomials, absolute value functions, polynomial in equalities and division, remainder theorem and factor theorem, and factoring polynomials.

Concepts of Calculus

A variety of mathematical operations with functions will be investigated including: rationalization, rates of change, the limit concept, indeterminate form, finding the slope of a curve, tangent slope function, derivatives and graphs.

Derivatives

In this unit students will see the power of the slope function and the applications of derivatives in a variety of style problems.

Curve Sketching

The key features of a properly sketched curve including x and y intercepts, vertical, horizontal and oblique asymptotes, maximum and minimum values, points of inflection, undefined tangent slope points will be examined separately before putting them all together into a full sketch of a curve.

Derivative Applications

A variety of types of problems will be presented in this unit and can generally be grouped into the following categories: Pythagorean problems, volume problems, trough problems, shadow problems, general rate problems. Each type will be examined separately.

Exponents and Log Functions

As a review of previous courses, the unit will begin with the rules associated with exponents. Then the exponential functions, applications and logarithms and log functions and applications will be covered.

Derivatives of Exponents and Log Functions

Exponential functions, logarithmic functions, curve sketching and logarithmic differentials are all topics of this unit.

Trigonometry Differentials and Applications

The unit begins with a review of the three basic trig functions (sine, cosine, tangent). Angles, the CAST rule, sums and differences for sine/cosine form the second major topic. Finally solving trigonometric equations are pursued with a focus on limits, derivatives and applications of trigometric functions.

Antiderivatives and Applications

The topics covered in this unit include the concept of antiderivatives, acceleration, velocity, differential equations, Riemann’s sums and areas, area function, definite integral and integration and area between curves.

Final Evaluation

Total

   

4.       Units:

          C1        Pre-Calculus (Summer)                      8 lessons                    

                C1.1        Functions

                C1.2        Linear Equations

                C1.3        Parametric Equations

                C1.4        Factor and Remainder Theorem

                C1.5        Trigonometric Functions

                C1.6        Exponential Equations

                C1.7        Logarithmic Functions

                C1.8        Geometry and Similar Triangle

          C2        Limits and Continuity                         7 lessons        

                C2.1        Rate of Change and Limits

                C2.2        Indeterminate Forms

                C2.3        Limits involving Infinity

                C2.4        Continuity

                C2.5        Rates of Change and Tangent Lines

                C2.6        Epsilon and Delta

                C2.7

          C3        Derivatives                                           16 lessons     

                C3.1        Limit Definition of Derivative

                C3.2        Differentiability

                C3.3        Power Rule

                C3.4        Proof of Power Rule

                C3.5        Chain Rule

                C3.6        Product Rule

                C3.7        Proof of Product Rule

                C3.8        Quotient Rule

                C3.9        Proof of Quotient Rule

                C3.10      Velocity and Acceleration

                C3.11      Derivatives of Trigonometric Functions

                C3.12      Derivatives of Inverse Trigonometric Functions

                C3.13      Derivatives of Exponential and Logarithmic Functions

    C3.14      Implicit Differentiation

                C3.15      Review

                C3.16      Review                  

 

C4        Applications                                       14 lessons      

                C4.1        Extreme Values of Functions

                C4.2        Mean Value Theorem

                C4.3        Connecting  and  with and graphing (1) – Large Lesson

                C4.4        Connecting  and  with and graphing (2)

                C4.5        Connecting  and  with and graphing (3)

                C4.6        Optimization (1)

                C4.7        Optimization (2)

                C4.8        Optimization (3)

                C4.9        Linearization and Newton’s Method

                C4.10      Estimating Change and Rolle’s Theorem

                C4.11      Related Rates (1)

                C4.12      Related Rates (2)

                C4.13      Related Rates (3)

                C4.14      Review

          C5        The Definite Integral                            14 lessons     

                C5.1        Estimating with Finite Sums

                C5.2        Riemann Sums

                C5.3        Definite Integrals

                C5.4        Definite Integrals and Antiderivatives (1)

                C5.5        Definite Integrals and Antiderivatives (2)

                C5.6        Definite Integrals and Antiderivatives (3)

                C5.7        Fundamental Theorem of Calculus (1)

                C5.8        Fundamental Theorem of Calculus (2)

                C5.9        Fundamental Theorem of Calculus (3)

                C5.10      Trapezoidal Rule

                C5.11      Simpson’s Rule

                C5.12      Review (1)

                C5.13      Review (2)

                C5.14      Review (3)

 

                C6        Differential Equations and Modelling         12 lessons      

                C6.1        Slope Fields (1)

                C6.2        Slope Fields (2)

                C6.3        Integration by Substitution (1)

                C6.4        Integration by Substitution (2)

                C6.5        Integration by Parts (1)

                C6.6        Integration by Parts (2)

                C6.7        Exponential Growth and Decay

                C6.8        Population Growth

                C6.9        Numerical Methods

                C6.10      Review

                C6.11      Review

                C6.12      Review

C7a      Applications of Definite Integrals      7 Lessons      

                C7.1        Integral as Net Change

                C7.2        Areas in the Plane (1)

                C7.3        Areas in the Plane (2)

                C7.4        Volumes (1)

                C7.5        Volumes (2)

                C7.6        Volumes (3)

                C7.7        Review  

Practice exams     2 Weeks

 C7b      Applications of Definite Integrals                  4 Lessons

                C7.7        Lengths of Curves

    C7.8        Applications from Science and Statistics (1)

                C7.9        Applications from Science and Statistics (2)

                C7.10      Review

   

Achievement Chart: Mathematics, Grades 9-12

Categories

50-59% 
(Level 1)

60-69% 
(Level 2)

70-79% 
(Level 3)

80-100% 
(Level 4)

Knowledge and Understanding - Subject-specific content acquired in each course (knowledge), and the comprehension of its meaning and significance (understanding)

 

The student:

Knowledge of content (e.g., facts, terms, definitions)

demonstrates limited knowledge of content

demonstrates some knowledge of content

demonstrates considerable knowledge of content

demonstrates thorough knowledge of content

Understanding of content(e.g., concepts, ideas, theories, procedures, processes, methodologies, and/or technologies)

demonstrates limited understanding of content

demonstrates some understanding of content

demonstrates considerable understanding of content

demonstrates thorough and insightful understanding of content

Thinking - The use of critical and creative thinking skills and/or processes

 

The student:

Use of planning skills (e.g., focusing research, gathering information, organizing an inquiry, asking questions, setting goals)

uses planning skills with limited effectiveness

uses planning skills with moderate effectiveness

uses planning skills with considerable effectiveness

uses planning skills with a high degree of effectiveness

Use of processing skills (e.g., inquiry process, problem-solving process, decision-making process, research process)

uses processing skills with limited effectiveness

uses processing skills with some effectiveness

uses processing skills with considerable effectiveness

uses processing skills with a high degree of effectiveness

Use of critical/creative thinking processes (e.g., oral discourse, research, critical analysis, critical literacy, metacognition, creative process)

uses critical / creative thinking processes with limited effectiveness

uses critical / creative thinking processes with some effectiveness

uses critical / creative thinking processes with considerable effectiveness

uses critical / creative thinking processes with a high degree of effectiveness

Communication - The conveying of meaning through various forms

 

The student:

Expression and organization of ideas and information (e.g., clear expression, logical organization) in oral, graphic, and written forms, including media forms

expresses and organizes ideas and information with limited effectiveness

expresses and organizes ideas and information with some effectiveness

expresses and organizes ideas and information with considerable effectiveness

expresses and organizes ideas and information with a high degree of effectiveness

Communication for different audiences (e.g., peers, adults)and purposes (e.g., to inform, to persuade) in oral, written, and visual forms

communicates for different audiences and purposes with limited effectiveness

communicates for different audiences and purposes with some effectiveness

communicates for different audiences and purposes with considerable effectiveness

communicates for different audiences and purposes with a high degree of effectiveness

Use of conventions (e.g., conventions of form, map conventions), vocabulary, and terminology of the discipline in oral, written, and visual forms

uses conventions, vocabulary, and terminology of the discipline with limited effectiveness

uses conventions, vocabulary, and terminology of the discipline with some effectiveness

uses conventions, vocabulary, and terminology of the discipline with considerable effectiveness

uses conventions, vocabulary, and terminology of the discipline with a high degree of effectiveness

Application - The use of knowledge and skills to make connections within and between various contexts

 

The student:

Application of knowledge and skills (e.g., concepts, procedures, processes, and/or technologies) in familiar contexts

applies knowledge and skills in familiar contexts with limited effectiveness

applies knowledge and skills in familiar contexts with some effectiveness

applies knowledge and skills in familiar contexts with considerable effectiveness

applies knowledge and skills in familiar contexts with a high degree of effectiveness

Transfer of knowledge and skills (e.g., concepts, procedures, methodologies, technologies) to new contexts

transfers knowledge and skills to new contexts with limited effectiveness

transfers knowledge and skills to new contexts with some effectiveness

transfers knowledge and skills to new contexts with considerable effectiveness

transfers knowledge and skills to new contexts with a high degree of effectiveness

Making connections within and between various contexts(e.g., past, present, and future; environmental; social; cultural; spatial; personal; multidisciplinary)

makes connections within and between various contexts with limited effectiveness

makes connections within and between various contexts with some effectiveness

makes connections within and between various contexts with considerable effectiveness

makes connections within and between various contexts with a high degree of effectiveness

   

5.       Course Evaluation:


 “The SPH4UI course will be evaluated based on term work worth 70% of your final report grade and the components of the final evaluation are worth 30% of your final grade.  Term work includes; tests, quizzes, assignments, lab reports.  Our final evaluation will be composed of a final exam.”

 

The final grade will be determined as follows:

Assessment

Percentage

Knowledge and Understanding

Communication                                              

Application                                                   

Thinking and Investigation                         

June Examination                                         

Total

              25%

              15%

              15%

              15%

              30%

            100%

 

   

6.       Late Work Policy:  

At WCI is the expectation that students will submit all required work by the assigned due date as evidence of their learning.  Students who fail to meet a due date for an essential course component will be subject to the completion policy found the student planner. Failure to submit this work, despite these interventions, will be recorded as incomplete and may result in a loss of credit.”

 

 

7.       Cheating/Plagiarism Policy :

 

At WCI it is the expectation that students will submit their own original work for the purpose of demonstrating their learning.  In the event that cheating or plagiarism occurs, the following              consequences may be implemented, in consultation with administration, depending on the     situation:

·         The student may be required to redo all or part of the assignment or assessment.

·         The student may be required to complete an alternate assignment of assessment.

·         The student’s work may be treated as a missed assignment.

·         There may be other consequences that are determined to be appropriate, including disciplinary consequences as outlined in the Cheating/Plagiarism section of the student planner.”

 

8.       Learning Skills :

 

The development of learning skills and work habits is an integral part of a student’s learning.  The achievement of these skills is officially reported on the Provincial Report Card.  The evaluation of learning skills and work habits is reported as follows: E-Excellent, G-Good, S-Satisfactory, and N-Needs Improvement.  For a full description of the 6 Learning Skills; Responsibility, Organization, Independent Work, Collaboration, Initiative, and Self-Regulation, please see the WCI Student Planner.”