Course Syllabus

AP CALCULUS (BC) COURSE SYLLABUS 2021-2022

 

TEXT: Calculus: Graphical, Numeric, Algebraic by Finney, Demana, Waits and Kennedy. Publisher : Scott Foresman Addison Wesley. ISBN 0-201-32445-8 (1999) 

CALCULATOR: TI 83 (or equivalent) – allowed in parts of class tests and final exam.

Every member of the class is expected to have his own TI-83 or TI-84. 

INTRODUCTION: The course will follow the guidelines established by the College Board in the United States and covers both differential and integral calculus with a significant emphasis on applications of this material. “An AP course in Calculus consists of a full high-school academic year of work that is comparable to calculus courses in universities and colleges. It is expected that students who take an AP course in calculus will seek college credit or placement”. This is a college level mathematics course. It is expected that students successful in this course will be able to gain a college credit in both differential and integral calculus. Students taking this course are expected to write the AP Calculus BC exam in May. 

The marks on this course are now accepted by UBC for admissions purposes and good results will lead to credits at UBC and SFU. Also, taking Calculus AB may be an advantage for those applying for admission to American universities.

Past results have the majority of students earning good grades on this course, but it has also been obvious that a good one hour’s study for each class is essential if a student is to gain a creditable result. This is a university level course and it is expected that students be both well prepared for class and highly motivated.

After the exam in May students will be required to submit a course project in order to complete the course. Students who do not write the exam will be required to write the BC Universities Calculus Challenge exam, or an internal exam.

 

TOPICAL OUTLINE FOR AP CALCULUS (BC)

The course closely follows the text; fondly known as FDWK! Questions assigned from the text are given to the student at the beginning of each unit.

Brackets at the end of each Unit suggest the number of 75-minute lessons necessary to cover this unit.

The symbol * denotes a topic that is solely on the BC syllabus, any other topic is on both the AB and BC syllabus.

 

TERM 1         25 classes

UNIT I: PREVIEW FOR CALCULUS

• Lines

            Increments; Slope; Parallel and Perpendicular; Equations; Applications;

• Functions and Their Graphs

            Domain / Range; Viewing / Interpreting Graphs; Even, Odd, Symmetry; Piecewise Functions;         Absolute Value; Composition of Fucntions;

• Exponential Functions

            Exponential growth and decay’ Applications; Significance and background of e;

• Parametric Functions   *

The analysis of planar curves in parametric form; relations, circles, ellipses, lines and other curves

• Functions and Logarithms

            One-One Functions; Finding Inverses; Properties of Logarithms, Graphing Logarithms;     Applications;

1.6 Trigonometric Functions

            Radians; Graphs of Trig Functions; Periodicity, Even, Odd; Transformations; Applications;             Inverse Trig Functions;

 

Review and Unit Test                                                                       [4]

 

UNIT II: LIMITS AND CONTINUITY AND THE DERIVATIVE

 

2.1 Rates of Change and Limits

            Average / Instantaneous Speed; Definition of a Limit; Properties of Limits; One-sided and Two-    sided limits; Sandwich theorem;

2.2 Limits Involving Infinity

            Finite limits; infinite limits; end behaviour models; ‘seeing’ limits;

2.3 Continuity

            Continuity at a Point; Continuous Functions; Algebraic combinations; Composites; Intermediate Value Theorem for Continuous Functions;

2.4 Rates of Change and Tangent Lines

            Average rate of change; Tangent to a curve; Slope of a curve; Normal to a curve; Speed;

3.1 Derivative of a function

            Definition; Notation; Relationship between graphs of f and f’; Graphing derivative from data;       One-sided derivatives;

3.2 Differentiability                                                                          

            Existence of f’?; Differentiability => local linearity; Using a calculator; Differentiability =>             Continuity; Intermediate Value Theorem for Derivatives;

 

Review and Unit Test                                                                       [5]

 

UNIT III: DERIVATIVES

3.3 Rules for Differentiation

            +ve integer powers; multiples, sums and differences; Products and Quotient Rule; Negative            Powers; Higher Order Derivatives;

3.4 Velocity and Other Rates of Change

            Instantaneous Rate of Change; Motion along a line; Sensitivity to change; Economics

3.5 Derivatives of Trigonometric Functions

            Sine; Cosine; SHM; Jerks; derivation of derivatives to other trig functions;

3.6 The Chain Rule

            Composite Functions; Chain Rule; Power Chain Rule;

3.7 Implicit Differentiation

            Implicit functions; Rational Powers;

3.8 Derivatives of Inverse Trigonometric Functions

            Differentiation of Inverse Functions; Inverse trig developed through dy/dx = 1/(dx/dy);

 

3.9 Derivatives of exponential and Logarithmic functions

            Analysis of slope of a^x; Development to importance of e; Derivative of e^x and a^x. Derivative of ln(x) and log(x).

Review and Unit Test                                                                       [7]

 

 

UNIT IV: APPLICATIONS OF DERIVATIVES

 

4.1 Extreme Values of Functions

            Absolute extreme value; Local extrema; Finding extrema;

4.2 Mean Value Theorem

            MVT; Physical Interpretation; Increasing and Decreasing functions; Consequences;

4.3 First and Second Derivatives

            First derivative test; Concavity; Second derivative test; Understanding behaviour of functions       from their derivatives;

4.4 Modeling and Optimization

            Cylinders / Boxes; Examples form Business; industry; mathematics; economics. Setting up            models; Modelling discrete phenomena;

4.5 Linearization

            Linear Approximation; Estimating change with differentials; Errors;

4.6 Related Rates

            Related rates; Solution Strategy; Simulation;

Review and Unit Test                                                                       [7]

 

Term 2            26 classes

 

UNIT V: THE DEFINITE INTEGRAL

 

5.1 Estimating With Finite Sums

            Distance traveled / acceleration; LRAM / RRAM / MRAM ; Application to Volumes;

5.2 Definite Integrals

            Riemann Sums; Notation; Developing deeper understanding; Terminology; Definite Integrals and                 Areas; Using a calculator; Dealing with discontinuity?;

5.3 definite Integrals and Antiderivatives

            Average Value of a Function; Mean Value Theorem for Definite Integrals; Connecting     Differential and Integral Calculus;

5.4 Fundamental Theorem of Calculus

            FTC Part I; FTC Part II; Connection with bounded area; Applications;

5.5 Trapezoidal Rule

            Trapezoidal Approximation; Brief discussion of Error Analysis;

Review and Unit Test                                                                       [6]

 

 

UNIT VII: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING

 

6.1 Antiderivatives

            Solving Initial Value problems; Slope Fields; Properties of Indefinite Integrals;

6.2 Integration By Substitution

            Power Rule; Trigonometric Integrands; Substitution; Separable Differential Equations;

6.3 Integration By Parts   *

            Product Rule In Integral Form; Repeated Use; Solving For The Unknown Integral

8.4 Integration With Partial Fractions   *

            Partial Fractions; Integrating Using Partial Fractions; Solving Differential equations Using

                Partial Fractions

6.4 Exponential Growth and Decay

            Exponential Growth and Decay; Continuous Compounding; Half-Life; Newton’s Law of Cooling;

6.5 Population Growth

            Exponential Model;

             Logistic Model; Logistic Regression   *

Review and Unit Test                                                                      [7]

 

 

UNIT VII: APPLICATIONS OF DEFINITE INTEGRALS

 

7.1 Integral as Net Change

            Linear Motion; General Strategy; Consumption over Time; Net Change from Data;

7.2 Areas in The Plane

            Areas Between Curves; Areas Enclosed by Intersecting Curves; Areas Bounded by Different           Functions; Areas to the y-axis, Integrating wrt y.

7.3 Volumes

            Volume as an integral; Square and Cylindrical Cross Sections; Cylindrical Shells; Other Cross

                Sections.

7.4 Lengths Of Curves   *

            Length of a smooth curve; Vertical tangents; Corners and Cusps.

10.2-3 Vectors In A Plane and 2-Dimensional Motion   *

            Component Form, Examples With Speed, Velocity and Acceleration

 

            Review and Unit Test                                                                       [8]

 

UNIT VIII: L’HOPITAL’S RULE, IMPROPER INTEGRALS AND POLAR FUNCTIONS 

8.1 L’Hopital’s Rule   *

            Indeterminate Forms 0/0, ∞/∞, ∞/0, ∞ - ∞

8.2 Relative Rates of Growth   *

            Comparing Rates of Growth of Functions Transitivity of Growing Rates 

8.3 Improper Integrals   *

            Infinite Limits of Integration; The Integral ; Integrands With Infinite Discontinuities;

                Tests For Convergence and Divergence.

10.5 Polar Coordinates and Polar Curves   *

            Polar Coordinates; Polar Graphing; Relating Polar and Cartesian Coordinates

10.6 Calculus of Polar Curves   *

Slope; Area in a Plane; Length of a Curve 

Review and Unit Test          [8]

 

UNIT IX: POLYNOMIAL APPROXIMATIONS AND SERIES 

9.1 Power Series   *

            Geometric Series; Representing Functions By Series, Differentiation and Integration,

                Identifying Series,

9.2 Taylor Series   *

Constructing a Series; Series For Sin x and Cos x; The ‘Beauty’ of a Series to Represent a Function; Maclaurin and Taylor Series; Combining Taylor Series; Table of Macclaurin Series

9.3 Taylors Theorem   *

            Taylor Polynomials; The Remainder and Taylors Theorem.

9.4 Radius of Convergence   *

            Convergence; nth Term Test; Comparing Nonnegative Series; Ratio Test; Endpoint Convergence.

9.5 Testing Convergence at End Points   *

            Integral Test; Harmonic Series and ‘p’ Series; Comparison Tests; Alternating Series; Absolute and

                Conditional Convergence; Intervals of Convergence. 

Review and Unit Test           [6] 

Final Exam Review              [2]

 

Post-Exam, there will be a major project which everyone is required to submit, in order to complete the course. This will take approximately 8 classes, and the mark from this will contribute approximately 30% for the mark for term 3.

 

ASSESSMENT

Homework / Classwork             ~10%

Quizzes / major assignments   ~15-20%

Unit Tests                                   ~70-75%

HOMEWORK will be checked for quality and completion and any problems will be dealt with in class, or in tutorials if necessary.

CLASSWORK will be assessed by contributions students make to a lesson and / or completing a written assignment. As part of this mark students will be expected to illustrate and explain solutions to various problems to the class.

QUIZZES will be every three or four classes, and will be based on homework assigned and in-class examples. They are intended to be diagnostic, and hence it is important to review mistakes prior to a unit test.

UNIT TESTS will be similar to College Board questions, but will also reflect strongly on questions detailed for study from the text. The tests will follow the format used on the final College Board exam and will have both a calculator and non-calculator section students will also be asked to “justify or explain” their answers.

Tutorial help is available throughout the year. Asking for a tutorial session is both acceptable and expected! 

 

PHILOSOPHY 

The course is aimed at developing the students understanding of the concepts of calculus and providing experience with its methods and applications. The concepts are presented in many different forms, and the oft used phrase is “ how does what we have just learnt compare with our experiences and / or what we know”.

A simple example would be how we arrive at the formula for the volume of a sphere; we use

• A visualization of a semi circle rotating about the x axis
• The physical vision of ‘summing the cross sectional area x ∆x’
• Integrating form –r to r to obtain the formula

When dealing with position time functions the support provided by the graphing calculator is invaluable when it comes to discussing such concepts as “total distance traveled, speed and position”. Many students come to the course with knowledge of this from their Physics course, but little understanding…..this course develops understanding.

For many students it will be their first experience with mathematical modeling. The concept of “integral as net change” is often cited when dealing with such topics as pollution or consumption of limited reserves.

The course provides the students with the skills to answer a question analytically, but it also makes many of those questions relevant to the world in which we live. It is this progression from the theoretical to the real world that makes Calculus BC a challenging but satisfying course to study.

The importance of estimating a function through linearization and slope fields allows the students to gain some grasp of how empirical mathematics can be, and when tables of data are given to a student to represent a function the pooling of all current mathematical knowledge to solve the problem is very immediate, and gives an effective perception of the power of math.

Problem solving is an important course skill and the fact that this requires knowledge of several topics to effectively answer a question just emphasizes how many of the concepts on this course are interrelated. 

Mathematical communication is often emphasized, and in class discussions will often start with the question “can you tell me why, or how”, “the calculator will do the rest for us once we know what to enter”. Students will be assessed in such a way so that they are required to communicate in complete sentences certain ideas and solutions. For example, they will be asked to clearly explain if the mean value theorem can be used in a specified example, and exactly what conclusions can be made if it can be used. Or they can be required to explain clearly why the calculator gives a faulty solution to a derivative evaluated at a cusp or a corner. It is important that a student is properly able to communicate what a rate of change, or an integral / accumulation actually “means’ in the context of the question, and the units involved in that specific scenario. Students will also be assigned to research a mathematician who contributed to the development of calculus, and to explain their contributions in essay format and in an oral presentation. 

The graphing calculator will be used in a number of ways. One of the most effective ways to explain subtleties is to take advantage of the inadequacies of the calculator, and to exploit these problems in order to deepen the student’s understanding of calculus. For example, we use the symmetric definition of the derivative to explain why the calculator erroneously gives us a derivative of zero at an absolute-value corner or a symmetric cusp. We can also show that the calculator’s limitations cause it to deceivingly imply there are endpoints to a function that actually proves to have vertical asymptotes when analyzed analytically.

The calculator is also used in order to experimentally arrive at apparent conclusions that can be verified analytically, and vice versa. For example, the limit as x approaches zero of (sinx)/x is found numerically using tables with small increments or by zooming into the graph near zero. Afterwards, the result can be proven using L’Hopital’s Rule. Also, once it is analytically proven that the derivative of sinx is cosx, we can use the calculator’s numerical differentiation function in the graphing display to show consistency with the result. Consider a function such as a radical algebraic expression with common factors in the numerator and denominator - at a particular value we can use the table of values from a graph-plot, to confirm the existence of a ‘hole’ in the function– a point discontinuity, and show that this function everywhere else, is equivalent to another (simplified) function. 

One really good way to teach the notion of local linearity is to zoom into a function with the derivative graphed at a point, and to see how the graph approaches the shape of the tangent line as the magnification is increased. This also can lead to a reasonable explanation of L’Hopital’s rule using linear approximations extremely close to the 0-limit functions in a given quotient.

Students will be expected to know when they can and when they cannot rely on results based on work they have done using a calculator given their knowledge of the advantages and limitations of the technology.