Curriculum and Pacing Guide - Mathematics
Honors Calculus (120230) and AP Calculus AB (1202310)
(everyday block all year)
Text: McDougal Littell Calculus of a Single Variable
Revised February 2007
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
I.
AP Calculus AB Course Overview and Philosophy
In developing and using the topics of differential and integral calculus, students have the opportunity to solidify all their previous high school
math experience, since practically all the major ideas from the prerequisite math courses are necessary for understanding. Consequently, we have
high expectations which emphasize concepts reinforced with development and application rather than just procedure. Of course it is a goal that the
successful student of Advanced Placement Calculus validate their success with a more that adequate performance on the AP Exam. However,
beyond this, it is our desire that the successful student appreciates the high level of their mathematical success and at the same time come to
realize that this accomplishment is not an end, but a beginning point for many other opportunities in various fields of study and work.
II.
Course Outline for AP Calculus AB
The following is from the document, “Curriculum and Pacing Guide – Mathematics: Honors Calculus and AP Calculus AB (everyday block all
year).” This document is relatively dynamic in that is regularly reviewed and updated as it should be. Although it may have more detailed than
necessary, the columns in the pacing of each unit provide the general time line, sections of the text book, topics covered, major concepts, and
suggested approaches and activities.
Curriculum and Pacing Guide - Mathematics
Honors Calculus (120230) and AP Calculus AB (1202310)
(everyday block all year)
Revised February 2007
This pacing guide is intended for use with the everyday block and will go all year. Students who successfully complete the 1st term will
receive a credit for Honors Calculus. Students who successfully complete 1st and 2nd terms will receive one credit for Honors Calculus and
one credit for AP Calculus AB and also take the AP Calculus AB Exam. Students who take AP Calculus in this manner cannot begin AP
Calculus AB in the 2nd term. The intent is to cover the content of Calculus Honors and AP calculus AB without the intensity of completing
the course on an all year alternating block.
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
1. The beginning unit is a review of the concepts from algebra and pre-calculus concerning relations /functions will probably be necessary to
some extent. These include linear, polynomial, rational, trigonometric, the generic functions, and piecewise functions. Topics relative to these are;
slopes, domain and range, symmetry, intercepts, intersections, and the graphs of these relations/functions. The graphing calculator (TI-89) is used
to reinforce these concepts, referring to the solving, table, and graphing capabilities of the calculator.
Chapter P: Preparation for Calculus
Block
Section
Topic
Days
1
P.1
Graphs and models
Concepts
Symmetry, Intercepts,
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(T,G) Graphing Features of
calculator, solving equations
Introducing Families of Functions
Video Tutorial - Learn how to Adjust the
Graph Viewing Window
1
3
5
1
11
P.2
P.3*
Appendix
D*
Linear Models and
Rates of Change
Slope, point-slope form, slopeintercept form
Functions and Their
Graph
Elementary functions
Precalculus Review
Unit circle, solving trigonometric
functions, piece-wise functions
Test
Total
(V)
Move My Way--A CBR Analysis of Rates
of Change
X (orY) Marks the Spot
(T) Transformations of Functions
Domain and Range of Graphs
Analysis of a Bouncing Ball
Video Tutorial - Learn how to Evaluate
Composite Functions
Functions
Horizontal and Vertical Transformations TINavigator, Activity Centre and Learncheck
Transformation Graphing App
(V)
Back and Forth---Analysis of Spring Motion
The Trigonometric Functions and Their
Inverse Func
Special Ratios and the Unit Circle 1
Special Ratios and the Unit Circle 2
Special Ratios and the Unit Circle 3
Special Ratios and the Unit Circle 4
Graph It in Pieces: Piecewise-defined
Functions
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
2.
The concepts of a limit, as the independent variable approaches a particular value, are addressed numerically, analytically, and graphically.
This involves developing the ideas of the existence of a limit. The issues are, if the limit exist, what is it, and if it does not, why not? This
necessitates the understanding of “broken graph” (addressed in detail later with continuity), oscillating, and asymptotic behaviors. Examination of
limits, including one sided limits, is done using the various algebra techniques for the types of functions, the properties of limits and special
techniques for rational trig related functions. Continuity and the Intermediate Value theorem and their applications are also part of this unit.
Chapter 1: Limits and Their Properties
Block
Section
Topic
days
1
2
SUPP.
1.2
Algebra Review
Finding Limits Graphically
and Numerically
Concepts
Factoring perfect cubes, complex
fractions, manipulating rational
expressions, rationalizing, absolute
value
Limit
Suggested Approaches
(T) Technology, (V) Verbal,
(G) Graphical, (N) Numerical
(V) verbal
Absolute Value and Piecewise
Functions
Solving Rational Expressions
Factoring trinomials and Expanding
binomial x binomial
(T) See www.calculushelp.com
(T) table features of calculator
(T) trace features of calculator
(V) p. 56/ 39-42, 9-52
Approaching Limits
TI: Limits, Numerically
Functions, Graphs, and Limits
TI: Find a Delta For Any Given
Epsilon
4
1.3
2
1.4
Evaluating Limits
Analytically
Continuity and One-Sided
Limits
Indeterminate form, Squeeze
Theorem
Continuity at a point, one-sided
limits, Intermediate Value Theorem
(V) verbal
The Derivative as a Limit
(T) See www.calculushelp.com
Is There a Limit to Which Side You
Can Take?
Graphical Consequences of
Continuity
3
1
1
14
1.5
Total
Infinite limits (may cover 3.5
after Section 1.5)
Review
Test
Infinite limit, vertical asymptote
(V) verbal
To Infinity and Beyond!
(V) How Do I? Worksheets
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
3.
The derivative is developed in this unit which involves the geometric interpretation of the tangent line at a point, leading to the limit definition
of the derivative of a function. The limit definition is used both to find derivatives at a point and to develop the basic derivative rules. The
relationship between continuity and differentiability is examined with attention given to continuous functions that have points where the derivative
does not exist. Basic derivative rules are used for first and second derivatives and also for implicit derivatives. Areas of application in this unit are;
finding equations of tangent lines (and normal lines) at a point, beginning motion problems, and related rates.
Chapter 2: Differentiation
Block Days
Section
3
2.1
Topic
Concepts
The derivative and the Tangent
line problem
Derivative, differentiable, normal
line
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical, (N)
Numerical
(V) p. 104/ 89-92
(V) p. 116/ 105
(T)
http://www.ima.umn.edu/~arnold/graphics.html
Seeing Is Believing
TI: Writing Equations for Tangent and Normal Lines
Graph of Functions and Their Derivatives
3
2
2.2
2.3
Differentiation rules and rates
of change.
Instantaneous rate of change,
velocity, speed
Product and Quotient rules
Higher order derivatives
(V) verbal
Average Rate of Change, Difference Quotients, and
Approximate Instantaneous Rate of Change
Derivatives (Trig Functions)
(V)
TI: Product Rule
TI: The Product and Quotient Rules
Local Linearity, Differentiability and Limits of
Difference Quotients
3
2.4
Chain Rule
Power rule
(T) Calculator: graphs of functions and tangent
lines
TI: Derivatives of Composite Functions-Chain Rule
TI: Logarithmic Differentiation
1
2
4
Test
2.5
2.6
Implicit Differentiation
Related rates
Explicit/implicit forms of
functions
(V) verbal
(V) verbal
Implicit Differentiation
Implicit Differentiation
TI: Implicit Differentiation on the TI-89
NUMB3RS - Season 3 - "Hardball" - Implicit Orbits
(V) verbal
Related Rates
Related Rates
TI: Visualizing Related Rates Problems
1
19
Test
Total
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
4. This unit continues the application of derivatives. Also three principle theorems are developed and used – the Extreme Value Theorem, Rolle’s
Theorem, and the Mean Value Theorem. First and second derivatives are used to determine for a given function the critical values, intervals of
increase and decrease, relative maxima and minima, points of inflection, and intervals concave up and concave down. These applications along
with examination of limits at infinity for horizontal asymptotes and the previous study of asymptotic behaviors (vertical and slant) leads to curve
sketching for the function. This application is done with and without graphing calculators. Included with this application is the examination of the
relationships of the graphs of a function, the graph of its 1st derivative, and the graph of its 2nd derivative. The very useful and important derivative
application of solving optimization problems are in this unit as well as linear approximations and differientials.
( Pacing on next page)
Honors Calculus (120230) and AP Calculus AB (1202310)
(everyday block all year)
Chapter 3: Applications of Differentiation
Block
Section
Topic
Days
2
3.1
Extrema on an
interval
2
3.2
Rolle’s Theorem
and Mean Value
Theorem
1
1
1
2
3.3
3.4
3.5
3.6
First Derivative
Test, increasing
and decreasing
functions
Concavity and the
Second Derivative
Test
Limits at infinity
Summary of curve
sketching
Concepts
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical, (N) Numerical
Critical numbers, absolute
and relative extrema
Functions and Their Extrema
TI: Absolute Extrema
(T) Demonstrate Theorems using graphing calculator
TI: Rolle's Theorem
Average Rate of Change, Difference Quotients, and Approximate Instantaneous Rate
of Change.
Strictly monotonic
(V) verbal (G) Graphical
Graph of Functions and Their Derivatives
Graphing Relationships
Projectile Motion
Point of inflection,
concave down, concave up
Horizontal asymptote
Slant asymptote,
polynomial long division
(V) verbal (G) Graphical
Graphing Relationships
Using Traffic Data to Reinforce Inflection Points in Calculus
(V) verbal (G)Graphical (T) Technology
(T) www.unitedstreaming.com : “applications of derivatives”
(G) “Getting at the concept” questions
Relating Graphs of a Function To Derivatives
2
1
5
3.6
SUPP.
3.7
Examine relationships
between graphs of f, f’, &
f”
Test
Optimization
Problems
Primary/secondary
equations
THE BIG DERIVATIVE PUZZLE:
http://www.univie.ac.at/future.media/moe/tests/diff1/ablerkennen.html
(V) verbal
(V) verbal
TI: Analyzing Optimization Problems
TI: Optimization
2
3.9
1
1
21
Differentials
differential forms,
propagated error, tangent
line approximations
(V) verbal (G) Graphical
Review
Test
Total
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
5. Integrals are begun in this unit through the introduction of the antiderivative (indefinite integral) which answers the question: What is a
function whose derivative is given? This leads immediately to the basic integration rules. These are practiced. The notions of a family of curves
and a general solution to a differential equation are introduced followed by the idea of a particular solution for an initial condition is developed and
applied. This allows for more applications of motion problems.
The definite integral is developed by first examining estimates of the areas of plane regions as sums of rectangles constructed by using a
partitioning of an interval and the right, left, midpoint or any point of the partition. The definition of a definite integral can then be given as a limit
to an infinite Riemann Sum, the exact area of the plane region. The Fundamental Theorem of Calculus is developed along with the Mean Value
Theorem for Integrals leading to the Average Value of a function on an interval. The Second Fundamental Theorem is also given. The main
applications here are areas of simple plane regions, accumulation problems, and average-value-of-a-function problems. More integration techniques
are developed through pattern recognition, and substitution. This unit also includes estimation of plane regions by using trapezoidal approximation.
Chapter 4: Integration
Block Days
3
Section
Topic
Concepts
4.1
Antiderivative, Indefinite
Integration
initial condition, general
solution, differential
equation
Upper sum, lower sum, area
approximation
1
4.2
Area
3
4.3
Riemann Sums and Definite
Integral
4
4.4
Fundamental Theorem of
Calculus
Definite integral defined as
limit of Riemann Sum
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(V) p.252/ 89-94
Computing Indefinite Integrals
Integration - Advanced Placement
(T, N) Graphing calculator: “sum(seq”
commands to find approximate areas
using rectangles
(V) verbal
Accumulation Functions
Approximating Integrals with Riemann Sums
Mean value Theorem,
(V) verbal
Average value of a function, Riemann Sums and the Fundamental Theorem
of Calculus
Second Fundamental
Fundamental Theorem of Calculus
Theorem of calculus, Integral
as accumulation function
5
1
4.5
4.6
1
1
19
Integration by Substitution
Numerical Integration
General power rule. usubstitution
(Trapezoidal rule only)
(V) verbal
Exploring derivatives using lists
(V) verbal (G) Graphical (T)
Technology
Numerical Integration
Listen and Line Up!
AP Calculus Study Cards
Review
Test
Total
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
Total instructional days in the pacing : 84
First term days: 90
This is the end of the 1st term. Students who successfully complete this term will be given a credit for Honors Calculus. It is intended that
student continue to end of the year for the completion of AP calculus AB.
6.
This is a substantial unit which addresses the calculus of transcendental functions – primarily logarithmic and exponential as well as the
calculus of inverse functions. The integral definition of the natural logarithm function is used. Various derivative techniques are developed for
natural logarithm functions and integral techniques for rational form functions, including trigonometric. Inverse functions are addressed with
particular attention given to the natural exponential function as the inverse of the natural logarithm function. Solving logarithmic and exponential
equations is reviewed. The derivative and integral of the natural exponential function are given. Logarithmic and exponential functions of any base
are given along with their corresponding derivatives and integrals. This then gives the foundation for more application of differential equations and
the examination of slope fields. Particular attention is given to differential equations in the form “ y’=ky” ( exponential growth and decay).
( Pacing on next page.)
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
Chapter 5: Logarithmic, Exponential, and other Transcendental Functions
Block days
Section
Topic
3
2
2
1
1
3
5.1
5.2
5.3
5.4
2
5.5
5
5.6
2
SUPP.
Concepts
Natural Logarithmic Function:
Differentiation
Definition of e
Natural Logarithmic Function:
Integration
Inverse Function
Log rule for integration
Review
Test
Exponential Functions:
Differentiation and Integration
Bases other than e and
application
Differential Equation: Growth
and decay
Suggested Approaches
(T) Technology, (V) Verbal, (G)
Graphical, (N) Numerical
(V) verbal
Investigating the Derivatives of Some
Common Functions
What Is the Number "e"?
(V) verbal (G) Graphical
TI: Logarithmic Differentiation
Derivative of inverse functions
(V) verbal
Derivative Inverse
(V) verbal
(V) verbal
(V) verbal
Using Derive
(V) verbal
Exponential Law of growth and
decay, separation of variable,
Newton’s law of cooling
Slope fields
(T, G) CBL Lab (Cooling)
Exploring the Exponential Function
Newton's Law Of Cooling and The
Calculus Behind It
Applications of Integrals
(G) www.apcentral.com :
Teacher Resource: Slope fields
Using Slope Fields
Introduction to Slope Fields
Differential Equations and Slope Fields Activity 7
3
1
1
3
5.7
5.8
Differential Equation:
Review
Test
Inverse Trigonometric
Functions: Derivatives
Separation of variables, initial
condition, particular solutions
Range restrictions of inverse trig
functions
(V) p. 379/ 111-114
TI: Introduction to Solving Differential
Equations
(V) verbal (G) Graphical
The Trigonometric Functions and Their
Inverse Func
3
1
1
34
5.9
Inverse Trigonometric
Functions: Integration
(V)
The Area Function
Space Exploration AP: Next Generation
Spacecraft - Orion
Planting & Harvesting - An Application
of the Definite Integral
Integral Calculus
Review
Test
Total
Honors Calculus (120230) and AP Calculus AB (1202310) (everyday block all year)
7. This unit involves the integral applications of finding areas and volumes. The definite Integral is used to find the areas of regions between
curves using all types of functions. These are areas on an interval, areas between curves including curves with more than two intersections, also
incorporating change of axis. The next application is volume. These are volumes of rotation using the disk, and washer method incorporating the
change of axis. Volumes of three dimensional shapes with known cross sections completes this unit.
Chapter 6: Applications of Integration
Block Days
Section
3
6
6.1
6.2
Topic
Concepts
Area of a Region between two
curves
Area of an accumulator
Volume of Solids of Revolution
Disk & Washer Methods
Suggested Approaches
(T) Technology, (V) Verbal, (G) Graphical,
(N) Numerical
(V) verbal
Area Under the Curve
TI: Area Between the Curves—The Game
(V) verbal
The Region Between Two Curves
4
6.2
Solids of Known Cross-sections
2
6.3
Volume: Shell method
Optional
6.4
Optional
Optional
6.5
6.6
Optional
6.7
Arc length and surfaces of
revolution
Work
Moments, Centers of Mass, and
Centroids
Fluid pressure and fluid force
1
1
Review
Test
Integrating area to find
volume
Horizontal/Vertical axis of
revolution
Smooth curve,
continuously differentiable
(V) verbal
(V) verbal
(V) verbal
(V) verbal
(V) verbal
Exploring Hooke's Law
(V) verbal
Boyle's Law
17
Total
Total Instructional in the pacing: 51
Second term Days:
90
The remaining time in the 2nd term should be devoted to an over all review of the subject matter with the goal of preparing for the Advanced
Placement AB exam. Depending on the scheduled dates for the exam this leaves about 25 to 30 days. Teachers should pick topics of interest
from calculus such as moments and centers of mass and fluid pressure from chapter six for students to do after the AP exam.
III.
Major Text.
Larson, Ron, Robert P. Hostettler, and Bruce H. Edwards. Calculus with Analytic Geometry. 7th ad. Boston , New York: Houghton Mifflin, 2002
IV.
Supplementary Materials
College Board AP Calculus Released Exams.
Getting Started with the Calculus Tools App