*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)

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**