AP Calculus BC - Syllabus
Teacher:
School:
District:
Course Requirements
1) All students are expected to learn the material detailed in the Course Outline so that they
are well-prepared for college-level mathematics and the AP Exam in May
a) Students are expected to participate in preparatory mock-exams as well as the
actual AP exam
b) A variety of resources exist at the district and school level to insure that all
students have the opportunity to take the exam
2) All students are expected to understand and explore the "Rule of 4" which provides for
the representation of functions in various ways including graphically, numerically,
analytically, and verbally
a) Students will be given a wide range of problems demonstrating their knowledge
of how to work with each type of representation as well as the connections that
exist between the different representations
b) Students will be expected to use TI-83/84 type calculators on a daily basis to
explore the relationships between numeric data, formulas, and their corresponding
graphs
3) All students are expected to be able to communicate mathematical concepts and
describe solutions to problems both orally and in writing
a) Students will work extensively in small groups where they will be expected to
collaborate with their peers as they develop their understanding of key
mathematical concepts
b) Students will be expected to keep a journal of notes for key concepts taken in
Cornell note form where students will be expected to write a summary using
complete sentences of the math concept being explored
c) Students will be given homework assignments beginning early in the year where
they will be expected to develop the necessary skills to successfully answer freeresponse questions from previous AP tests by explaining and justifying their
answers with a written response
4) All students are expected to use a TI-83/84 type calculator on a daily basis for use in
solving problems, experimenting with graphs and solutions, exploring the reasonableness
of results, and verifying answers
a)Students will have access to school supplied TI-83/84 calculators for use in class
and at home
b) Students are free to use their own graphing calculator provided that it meets the
requirements for use on the actual AP Calculus exam
c) Students will be assessed in a manner similar to that of the AP exam with
students being expected to answer questions which are calculator-active and
questions which are calculator-inactive
d) Students are expected to be able to use the calculator to regularly perform the
following:
i) Plot the graph of a function using an appropriate window, including the
ability to zoom in on a point to demonstrate properties such as local linearity
ii) Find the zeros of functions
iii) Numerically calculate the derivative of a function
iv) Numerically calculate the value of a definite integral
5) It is strongly desired that all students will develop an appreciation of calculus as a truly
amazing human accomplishment as it provides a cohesive mathematical approach to
solving problems
Textbook
Larson, Hostetler and Edwards. Calculus of a Single Variable: Eighth Edition.
Boston: Houghton Mifflin Company, 2006.
Students will be given a copy of this textbook for use in the course. The Course Outline
below references specific sections in the textbook. While this textbook is our primary
source of material for this course, it will be supplemented as necessary and appropriate to
enhance the experience of learning calculus.
Course Outline
Unit I: Functions, Graphs, and Limits (15 days)
A) Analysis of Graphs (2 Days)
i) Interplay between the geometric and analytic information (1.2, 1.3, 3.5)
ii)Use of calculus to predict and explain observed local and global behavior
(1.2, 1.3, 3.5)
B)Limits of Functions (Including one-sided limits)(5 Days)
i) An intuitive understanding of the limiting process (1.2-1.4, 3.5)
ii) Calculating limits algebraically (1.2-1.4, 3.5)
iii) Estimating limits graphically and numerically (1.2-1.4, 3.5)
C) Continuity as a Property of Functions (3 Days)
i) An intuitive understanding of continuity using sufficiently close domain
values (1.4)
ii) Understanding continuity in terms of limits (1.4, 1.5)
iii) Geometric understanding of graphs of continuous functions (Intermediate
Value Theorem and Extreme Value Theorem) (1.4, 3.1)
D) Asymptotic and Unbounded Behavior (2 Days)
i) Understanding asymptotes graphically (1.5, 3.5)
ii) Asymptotes and limits at infinity (3.5)
E) Parametric, Polar and Vector Functions (10.2, 10.4, 12.1*) (*Note: This material
will be taken from the 9th edition of the textbook)(3 Days)
Unit II: Derivatives (45 days)
A) Concept of Derivative (5 Days)
i) Derivative presented graphically, numerically, and analytically (2.1)
ii) Relationship between differentiability and continuity (2.1)
iii) Derivative as the limit of the difference quotient (2.1)
iv) Derivative as an instantaneous rate of change (2.2 - 2.4)
B)Derivative at a Point (4 Days)
i) Slope of a curve at a point including vertical tangents (2.2)
ii) Instantaneous rate of change as the limit of average rate of change (2.2)
iii) Approximate rate of change from graphs and tables (2.3, 2.4)
iv) Tangent Line to a curve at a point and local linear approximation (2.2 2.5, 3.9)
C)Computation of Derivatives (8 Days)
i) Derivatives of basic functions, including power, exponential, logarithmic,
trigonometric, and inverse trigonometric functions (2.2, 2.3, 5.1, 5.4, 5.5, 5.6)
ii) Basic rules for the derivative of sums, products, and quotients (2.2, 2.3)
iii) Chain Rule and implicit differentiation (2.4, 2.5)
iv) Derivatives of parametric, polar, and vector functions (5.3-5.6, 10.3-10.4,
12.2-12.4*)(*Note: This material will be taken from the 9th edition of the
textbook)
D) Derivative as a Function (8 Days)
i) Curve sketching and corresponding characteristics between f and f  by
finding critical values (2.3, 2.4, 3.1)
ii) Mean Value Theorem and its graphical and analytic implications (3.2)
iii) Relationship between the increasing and decreasing behavior of f and
f  (3.3)
iv) Equations involving derivatives (3.7)
E) Second Derivative (2 Days)
i) Corresponding characteristics of f , f  , and f  (3.4)
ii) Relationship between sign of second derivative and concavity (3.4)
iii) Points of inflection as a location where concavity changes with many
examples and counterexamples (no sign change) (3.4)
F) Applications of Derivatives (18 Days)
i) Analysis of curves, including the notions of monotocity and concavity (3.6)
ii) Optimization, absolute and relative extrema (3.7)
iii) Modeling rates of change including related rates problems (2.6)
iv) Use of implicit differentiation to find the derivative of an inverse (2.5)
v) The derivative as a rate of change in applied context including velocity,
speed, and acceleration (3.2)
vi) Geometric interpretation of differential equations via slope fields (6.1)
vii) Analysis of planar curves in parametric, polar, and vector form including
velocity and acceleration and arc length (5.3-5.6, 10.3-10.4, 12.212.4*)(*Note: This material will be taken from the 9th edition of the
textbook)
viii) Numerical solution of differential equations using Euler's Method (6.1)
ix) L'Hospital's Rule, including its use in determining limits and convergence
of improper integrals and series (8.7)
Unit III: Integrals (40 days)
A) Numerical Approximations to Definite Integrals (4 Days)
i) Use of Riemann sums (left, right, midpoint, and trapezoidal) (4.2, 4.3)
B)Interpretations and Properties of Definite Integrals (4 Days)
i) Definite integral as a limit of Riemann sums (4.3)
ii) Properties of definite integrals (4.3)
iii) Definite integral of the rate of change of a quantity over an interval
𝑏
∫𝑎 𝑓 ′ (𝑥) = 𝑓(𝑏) − 𝑓(𝑎) (4.4)
C) Techniques of Anti-differentiation (8 Days)
i) Anti-derivatives following directly from derivatives of basic functions (4.4)
ii) Anti-derivatives by substitution of variables including change of limits for
definite integrals, parts, and simple partial fractions(4.5, 8.1-8.6)
iii) Improper integrals (8.8)
D) Fundamental Theorem of Calculus (4 Days)
i) Use of the Fundamental Theorem to evaluate definite integrals
ii) Use of the Fundamental Theorem to represent a particular anti-derivative
and the analytical and graphical analysis of functions so defined (4.5)
E) Applications of Anti-differentiation (4 Days)
i) Finding specific anti-derivatives using initial conditions, including
applications to motion along a line (4.1, 4.4)
ii) Solving separable differential equations and using them in modeling with
exponential growth and the equation y   ky (6.1 - 6.3)
iii) Solving logistic differential equations and using them in modeling (6.3)
F) Applications of Integrals (16 Days)
i) Physical, biological, and/or economic situations (6.2, 6.3)
ii) Using the integral of a rate of change, to give accumulated change, area of a
region, volume of a solid with known cross-sections, the average value of a
function, and the distance traveled by a particle along a line (6.2, 6.3, 7.1, 7.2)
Unit IV: Polynomial Approximations and Series (35 days)
A) Concept of series (10 Days)
i) Definition of series and convergence (9.1-9.2)
B) Series of Constants (20 Days)
i) Motivating examples including decimal expansion (9.2)
ii) Geometric series with applications (9.2)
iii) The harmonic series (9.3)
iv) Terms of series as areas of rectangles and their relationship to improper
integrals, including the integral test and its use in testing convergence of
p-series (9.3)
v) Comparing series to test for convergence and divergence (9.4)
vi) Alternating series with error bound (9.5)
vii) The ratio test for convergence and divergence (9.6)
C) Taylor Series (15 Days)
i) Taylor polynomial approximation with graphical demonstration (9.7, 9.10)
ii) Maclaurin series and the general Taylor series centered at x=a (9.7, 9.10)
1
iii) Maclaurin series for the functions 𝑒 𝑥 , sin 𝑥, cos 𝑥, 1−𝑥(9.7, 9.10)
iv) Formal manipulation of Taylor series including substitution,
differentiation, anti-differentiation, and the formation of new series from
known series (9.7, 9.10)
v) Functions defined by power series (9.8)
vi) Radius and interval of convergence of power series (9.8)
vii) Lagrange error bound for Taylor polynomials (9.7)
Unit V: Comprehensive Review (10+ days)
A) Mock Exam
B) Comprehensive Review of Limits, Derivatives, Integrals, and Polynomial
Approximations and Series using previously released tests
Grading
This class will utilize the approved grading scale for the district for all AP courses. A
minimum of 6 daily grades will be given each grading period that constitutes 40% of a
student's grade. Daily grades include daily assignments, homework, quizzes, written
summaries and free-response questions. A minimum of 3 test grades will be given each
grading period that constitutes 60% of a student's grade. Test grades include major tests
given at appropriate points during the Course Outline and any major projects given.
Sample of Additional Materials
 AP Calculus Multiple Choice Questions, 1998 - present
 AP Calculus Free Response Questions, 1998 - present
 AP Central Curriculum Modules and Special Focus Materials
 Additional materials collected during Summer AP Institute
 Computer generated images for visual representations of rotated solids