Course Syllabus

AP College Board: Course and Exam Description HERE

 

TEXT: Calculus: Graphical, Numeric, Algebraic

Finney, Demana, Waits and Kennedy 4^th edition

CALCULATOR: TI 84 (or equivalent) – allowed in parts of class tests and final the 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 be well prepared for class and highly motivated. Students successful in this course may be able to gain 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 accepted by all Canadian and 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. After the exam in May students will be required to submit a course project in order to complete the course.

 

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 70-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 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;

 

UNIT III: DERIVATIVES

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;

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;

 

UNIT IV: MORE DERIVATIVES

4.1 The Chain Rule

Composite Functions; Chain Rule;  Power Chain Rule;

4.2 Implicit Differentiation

Implicit functions; Rational Powers;

4.3 Derivatives of Inverse Trigonometric Functions

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

4.4 Derivatives of exponential and Logarithmic functions

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

 

UNIT V: APPLICATIONS OF DERIVATIVES

5.1 Extreme Values of Functions

Absolute extreme value; Local extrema; Finding extrema;

5.2 Mean Value Theorem

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

5.3 First and Second Derivatives

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

5.4 Modeling and Optimization

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

5.5 Linearization

Linear Approximation; Estimating change with differentials; Errors;

5.6 Related Rates

Related rates; Solution Strategy; Simulation;

  

Term 2                        26 classes

 

UNIT VI: THE DEFINITE INTEGRAL

6.1 Estimating With Finite Sums

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

6.2 Definite Integrals

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

6.3 definite Integrals and Antiderivatives

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

6.4 Fundamental Theorem of Calculus

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

6.5 Trapezoidal Rule

Trapezoidal Approximation; Brief discussion of Error Analysis;

 

UNIT VII: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING 

7.1 Antiderivatives

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

7.2 Integration By Substitution

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

7.3 Integration By Parts   *

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

7.4 Exponential Growth and Decay

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

7.5 Partial Fractions and Logistic Growth

Exponential Model; Logistic Model; Logistic Regression   *

  

UNIT VIII: APPLICATIONS OF DEFINITE INTEGRALS

8.1 Integral as Net Change

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

8.2 Areas in The Plane

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

8.3 Volumes

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

8.4 Lengths Of Curves   *

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

 

UNIT IX: L’HOPITAL’S RULE, IMPROPER INTEGRALS

9.2 L’Hopital’s Rule  

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

9.3 Relative Rates of Growth   *

Comparing Rates of Growth of Functions Transitivity of Growing Rates

9.4 Improper Integrals   *

Infinite Limits of Integration; The Integral; Integrands With Infinite Discontinuities; Tests For Convergence and Divergence.

 

TERM 3 about here: 14 classes before the AP exam.

 

UNIT X: POLYNOMIAL APPROXIMATIONS AND SERIES

10.1 Power Series   *

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

Identifying Series,

10.2 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 Maclaurin Series

10.3 Taylors Theorem   *

Taylor Polynomials; The Remainder and Taylors Theorem.

10.4 Radius of Convergence   *

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

10.5 Testing Convergence at End Points   *

Integral Test; Harmonic Series and ‘p’ Series; Comparison Tests (Direct & Limit); Alternating Series and Alternating Series Error Bound; Absolute and Conditional Convergence; Intervals of Convergence.

  

UNIT XI: PARAMETRIC, VECTOR, AND POLAR FUNCTIONS

11.1 Parametric Functions   *

Parametric Curves in the Plane; Slope and Concavity; Arc Length; Cycloids

11.2 Vectors in the Plane

Two-Dimensional Vectors; Vector Operations; Modeling Planar Motion; Velocity,    Acceleration, and Speed; Displacement and Distance Traveled

11.3 Polar Functions

Polar Coordinates; Polar Curves; Slopes of Polar Curves; Areas Enclosed by Polar Curves; A Small Polar Gallery

 

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:

10%  Limits [L]

40%  Derivatives [D]

40%  Integrals [I]

10%  Power Series and Parametric, Vector, & Polar Functions [P]

 

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

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 learned 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”.