AP CALCULUS AB & BC
COURSE SYLLABUS 2022-2023
St. George’s School
TEXT: Calculus: Graphical, Numeric, Algebraic
Finney, Demana, Waits, and Kennedy (4th edition)
CALCULATOR: TI 84 (or equivalent) – allowed in portions of class tests and AP 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 the material. An AP course in Calculus consists of a full high-school academic year
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 assumed that students are both well prepared and highly motivated.
Students who achieve a sufficiently high score on the AB AP exam (a 4 or 5) may earn college credit
for differential calculus (one semester). Students who achieve a sufficiently high score on the BC AP
exam (a 4 or 5) may earn college credit for both differential calculus and integral calculus (two
semesters). All students taking AB or BC are expected to write their designated AP Calculus exam in
May. Marks for this course are accepted by all Canadian and American Universities.
Based on past results, a majority of students complete this course in high standing, but it is evident that
a good one hour’s study for each class is essential if a student is to gain a creditable result. After the
AP exam in May, students are required to participate in and will be assessed on various deep dive
studies as well as concepts that are extensions of what they have already learned.
TOPICAL OUTLINE FOR AP CALCULUS (BC): This course closely follows the text; fondly
known as FDWK! Questions assigned from the text are given to the student before each class.
Both AP Calculus AB and BC are double credit courses at this time. Students will obtain credit for
Calculus 12 in addition to their AP course. As a result, students will receive two marks throughout the
year: one for AP Calculus (AB or BC) and another for Calculus 12.
i) Students in Calculus AB are likely to have a similar mark throughout the year as the two
curriculums are very similar. There are, however, a small number of topics which differ
between AP Calculus AB and Calculus 12. Thus, it is possible that a student’s Calculus 12
mark will not be the same. ii) Students in Calculus BC can also expect roughly the same
Calculus 12 mark throughout the first half of the year; however, it is possible the two marks will
diverge, somewhat. This is because there is a considerable amount of more advanced BC
material that will not be included as part of a student’s Calculus 12 mark. iii) In general, there
is no guarantee that a student’s Calculus 12 mark will be higher than their corresponding AP
mark.
UNIT I: PREREQUISITES FOR CALCULUS
[Due to time constraints, this section will only be explicitly covered in its entirety in Calculus AB.]
1.1 Lines [AB & BC]
Increments; Slope of a Line; Parallel and Perpendicular Lines; Equations and Lines; Applications
1.2 Functions and Graphs [AB & BC]
Functions; Domain and Range; Reviewing and Interpreting Graphs; Even Functions and Odd Functions
– Symmetry; Functions Defined in Pieces; Absolute Value Functions; Composite Functions
1.3 Exponential Functions [AB & BC]
Exponential Growth; Exponential Decay; Applications; The Number e
1.4 Parametric Equations [BC only]
Relations; Circles; Ellipses; Lines and Other Curves
1.5 Functions and Logarithms [AB & BC]
One-to-One Functions; Inverses; Finding Inverses; Logarithmic Functions; Properties of Logarithms;
Applications
1.6 Trigonometric Functions [AB & BC]
Radian Measure; Graphs of Trigonometric Functions; Periodicity; Even and Odd Trigonometric
Functions; Transformations of Trigonometric Graphs; Inverse Trigonometric Functions
UNIT II: LIMITS AND CONTINUITY AND THE DERIVATIVE
2.1 Rates of Change and Limits [AB & BC]
Average and Instantaneous Speed; Definition of Limit; Properties of Limits; One-Sided and Two-Sided
Limits; Sandwich Theorem
2.2 Limits Involving Infinity [AB & BC]
Finite Limits as x approaches infinity; Sandwich Theorem Revisited; Infinite Limits as x approaches a;
End Behaviour Models; “Seeing” Limits as x approaches infinity
2.3 Continuity [AB & BC]
Continuity at a Point; Continuous Functions; Algebraic Combinations; Composites; Intermediate Value
Theorem for Continuous Functions
2.4 Rates of Change and Tangent Lines [AB & BC]
Average Rate of Change; Tangent to a Curve; Slope of a Curve; Normal to a Curve; Speed Revisited
UNIT III: DERIVATIVES
3.1 Derivative of a Function [AB & BC]
Definition of a Derivative; Notation; Relationship Between Graphs of f and f’; Graphing Derivative from
Data; One-Sided Derivatives
3.2 Differentiability [AB & BC]
How f’(a) might Fail to Exist; Differentiability Implies Local Linearity; Derivatives on a Calculator;
Differentiability Implies Continuity; Intermediate Value Theorem for Derivatives
3.3 Rules for Differentiation [AB & BC]
Positive Integer Powers, Multiples, Sums, and Differences; Products and Quotients; Negative Integer
Powers of x; Second and Higher Order Derivatives
3.4 Velocity and Other Rates of Change [AB & BC]
Instantaneous Rates of Change; Motion on a Line; Sensitivity to Change; Derivatives in Economics
(optional)
3.5 Derivatives of Trigonometric Functions [AB & BC]
Derivative of the Sine Function; Derivative of the Cosine Function; Simple Harmonic Motion; Jerk;
Derivation of Derivatives to Other Basic Trigonometric Functions
UNIT IV: MORE DERIVATIVES
4.1 The Chain Rule [AB & BC]
Derivative of a Composite Function; “Outside-Inside” Rule; Repeated Use of the Chain Rule; Slopes of
Parametrized Curves; Power Chain Rule
4.2 Implicit Differentiation [AB & BC]
Implicitly Defined Functions; Lenses, Tangents, and Normal Lines; Derivatives of Higher Order;
Rational Powers of Differentiable Functions
4.3 Derivatives of Inverse Trigonometric Functions [AB & BC]
Derivatives of Inverse Functions; Derivative of the Arcsine; Derivative of the Arctangent; Derivatives of
the Other Three
4.4 Derivatives of Exponential and Logarithmic Functions [AB & BC]
Derivative of ex; Derivative of ax; Derivative of ln x; Derivative of loga x; Power Rule for Arbitrary Real
Powers
UNIT V: APPLICATIONS OF DERIVATIVES
5.1 Extreme Values of Functions [AB & BC]
Absolute (Global) Extreme Values; Local (Relative) Extreme Values; Finding Extreme Values
5.2 Mean Value Theorem [AB & BC]
Mean Value Theorem; Physical Interpretation; Increasing and Decreasing Functions; Other
Consequences
5.3 First and Second Derivatives [AB & BC]
First Derivative Test for Local Extrema; Concavity; Points of Inflection; Second Derivative Test for
Local Extrema; Learning about Functions from Derivatives
5.4 Modeling and Optimization [AB & BC]
Examples from Mathematics; Examples from Business and Industry; Examples from Economics
(optional); Modelling Discrete Phenomena with Differentiable Functions (optional)
5.5 Linearization [AB & BC]
Linear Approximation; Differentials; Estimating Change with Differentials; Absolute, Relative, and
Percentage Change; Sensitivity to Change; Newton’s Method [Calculus 12 only]
5.6 Related Rates [AB & BC]
Related Rate Equations; Solution Strategy; Simulating Related Motion
UNIT VI: THE DEFINITE INTEGRAL
6.1 Estimating with Finite Sums [AB & BC]
Distance Traveled, Rectangular Approximation Method (RAM); Volume of a Sphere; Cardiac Output
6.2 Definite Integrals [AB & BC]
Riemann Sums; Terminology and Notation of Integration; Definite Integral and Area; Constant
Functions; Integrals on a Calculator; Discontinuous Integrable Functions
6.3 Definite Integrals and Antiderivatives [AB & BC]
Properties of Definite Integrals; Average Value of a Function; Mean Value Theorem for Definite
Integrals; Connecting Differential and Integral Calculus
6.4 Fundamental Theorem of Calculus [AB & BC]
Fundamental Theorem, Part I; Graphing the Function; Fundamental Theorem, Part II; Area Connection;
Analyzing Antiderivatives Graphically
6.5 Trapezoidal Rule [AB & BC]
Trapezoidal Approximation; Other Algorithms; Error Analysis
UNIT VII: DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING
7.1 Antiderivatives [AB & BC]
Differential Equations; Slope Fields; Euler’s Method [BC only]
7.2 Integration by Substitution [AB & BC]
Indefinite Integrals; Leibniz Notation and Antiderivatives; Substitution in Indefinite Integrals;
Substitution in Definite Integrals
7.3 Integration by Parts [BC only]
Product Rule in Integral Form; Solving for the Unknown Integrand; Tabular Integration; Inverse
Trigonometric and Logarithmic Functions
7.4 Exponential Growth and Decay [AB & BC]
Separable Differential Equations; Law of Exponential Change; Continuously Compounded Interest;
Radioactivity; Modelling Growth and Other Bases; Newton’s Law of Cooling
7.5 Partial Fractions and Logistic Growth [BC only]
How Populations Grow; Partial Fractions; The Logistic Differential Equation; Logistic Growth Models
UNIT VIII: APPLICATIONS OF DEFINITE INTEGRALS
8.1 Integral as Net Change [AB & BC]
Linear Motion Revisited; General Strategy; Consumption Over Time; Net Change from Data; Work
8.2 Areas in The Plane [AB & BC]
Area Between Curves; Area Enclosed by Intersecting Curves; Boundaries with Changing Functions;
Integrating with Respect to y; Saving Time with Geometry Formulas
8.3 Volumes [AB & BC]
Volume As an Integral; Square Cross Sections; Circular Cross Sections; Cylindrical Shells (Optional);
Other Cross Sections
8.4 Lengths of Curves [BC only]
A Sine Wave; Length of Smooth Curve; Vertical Tangents; Corners, and Cusps
8.5 Applications from Science and Statistics (optional) [AB & BC]
Work Revisited; Fluid Force and Fluid Pressure; Normal Probabilities
UNIT IX: L’HOPITAL’S RULE, IMPROPER INTEGRALS AND POLAR FUNCTIONS
9.1 Sequences [BC only]
Defining a Sequence; Arithmetic and Geometric Sequences; Graphing a Sequence; Limit of a Sequence
9.2 L’Hopital’s Rule [AB & BC]
Indeterminate Forms: 0/0, ∞/∞; Indeterminate Forms (optional): ∞ ⋅ 0 , ∞ − ∞ , 1∞ , 00, ∞ 0
9.3 Relative Rates of Growth (optional) [BC only]
Comparing Rates of Growth; Using L’Hopital’s Rule to Compare Growth Rates, Sequential versus
Binary Search
9.4 Improper Integrals [BC only]
Infinite Limits of Integration; Integrands with Infinite Discontinuities; Test for Convergence and
Divergence; Applications
UNIT X: POLYNOMIAL APPROXIMATIONS AND SERIES [BC only]
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; Beauty Bare; Maclaurin and Taylor Series;
Combining Taylor Series; Table of Maclaurin Series
10.3 Taylors Theorem
Taylor Polynomials; The Remainder; Bounding the Remainder; Euler’s Formula
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; Absolute and Conditional Convergence; Intervals of Convergence; A Word of Caution
UNIT XI: PARAMETRIC, VECTOR, AND POLAR FUNCTIONS [BC only]
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
HOMEWORK may be checked for quality and completion. Students are encouraged to collaborate on
homework problems and rely on one another as a resource. Any additional questions may be brought
to the teacher’s attention.
UNIT TESTS may be similar to College Board questions, but will strongly reflect assigned text
questions. An assessment may follow the format used for the final College Board exam and thus 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 to arrange a tutorial session is acceptable
– asking for a lesson to be repeated because of a missed class is not.
PHILOSOPHY: The course is aimed at developing a student’s understanding of the concepts of
calculus and offering experience with its methods and applications. The concepts are presented in
many different forms.
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
• ‘Summing’ each cross-sectional area
• 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 from their Physics course, but with little
understanding. For some students, this will also be their first experience with mathematical modelling.
The concept of “integral as net change” is often cited when dealing with such topics as pollution or
consumption of limited reserves.
This course provides the students with the skills to answer questions analytically, but also makes many
of those questions more 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 student to gain
some grasp of how empirical mathematics can be. Problem solving is an important life skill. Given
this course requires knowledge of varying topics to effectively solve one problem emphasizes just how
interrelated mathematics can be.
Mathematical communication is often emphasized. Class discussions will often begin with the
question, “Can you tell me why, or how?”, “The calculator will do the rest for us once we know what
to enter.”
ASSESSMENT: There will be additional material covered after the AP exam is completed. Students
are expected to fully participate and will be required to complete various tasks/assignments for course
credit.
The following mark breakdowns reflect approximate weightings of various course components; they
reflect the estimated amount of time spent on each major topic. Actual weights for grading purposes
will be determined based on the number of points accolated on a given assessment. [This is to help
ensure that a student’s course mark remains aligned with assessment outcomes without over-inflating
the value of any particular outcome.]
It is at the teacher’s professional discretion to adjust any weightings which may impact
a student’s grade. It should be understood that such an occurrence is not open to
negotiation – the teacher’s word is final.
AP Calculus BC:
AP Calculus AB:
5%
12%
18%
Limits and Continuity
Differentiation
Applications of Differentiation
10%
16%
24%
Limits and Continuity
Differentiation
Applications of Differentiation
18%
16%
Integration
Applications of Integration
24%
22%
Integration
Applications of Integration
16%
10%
Infinite Sequences and Series
Parametric Equations, Polar Coordinates,
and Vector-Valued Functions
5%
Additional Topics
___________
Total: 100%
4%
Additional Topics
___________
Total: 100%