ࡱ > ^ bjbjT~T~ 4` 6 6 t, | | , t Ւ t s s s s s N N N $ I ~ & i N N & s s c c c s s c c c F D s փj " T 0 Ւ t i i D i D N h J c < < N N N & & N N N Ւ i N N N N N N N N N | : AP Calculus AB Syllabus
Course Objectives: In this course we will study everything in the Calculus AB guideline as it appears in the AP Calculus Course Description, as well as topics including Newtons Method and Simpsons Rule. The ultimate goal is to introduce students to the major concepts of a first-year college calculus course, with the overall objective of preparing the student for the Advanced Placement exam in May. Throughout the course, instructional practices to enhance understanding, skills, and use of technology are emphasized to incorporate multiple representations.
Textbook: Demana, Waits, Foley, & Kennedy, Calculus: Graphical, Numerical, Algebraic, Prentice Hall 2003
Requirements: School policy requires students enrolled in AP classes to take the AP exam in May.
Prerequisites: Algebra I, Algebra II, Geometry, and Pre-Calculus
The College Board AP Calculus Course Description booklet recommends: Before studying calculus, all students should complete four years of secondary mathematics designed for college-bound students; courses in which they study algebra, geometry, trigonometry, analytic geometry, and elementary functions. These functions include those that are linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise defined. In particular, before studying calculus, students must be familiar with the properties of functions, the algebra of functions (domain and range, add and even, periodic, symmetry, zeros, intercepts, and so o n ) a n d k n o w t h e v a l u e s o f t h e t r i g o n o m e t r i c f u n c t i o n s o f t h e n u m b e r s 0 , / 6 , / 4 , / 3 , / 2 , a n d t h e i r m u l t i p l e s .
C o u r s e C o n t e n t :
P r e r e q u i s i t e s f o r C a l c u l u s :
L i n e s
I n c r e m e n t s
S l o p e o f a l i n e
P a r a l l e l a n d p e r p e n d i c u l a r l i n e s
E q u a t i o n s o f l i n e s
F u n c tions and Graphs
Functions
Domains and ranges
Viewing and interpreting graphs
Even and odd functions
Symmetry
Functions defined in pieces
Absolute value functions
Composite functions
Exponential Functions
Exponential growth
Exponential decay
Applications
The number e
Parametric Equations
Relations
Circles
Ellipses
Lines and other curves
Functions and Logarithms
One-to-one functions
Inverses
Finding inverses
Logarithmic functions
Properties of logarithms
Applications
Trigonometric Functions
Radian measure
Graphs of trigonometric functions
Periodicity
Even and odd trigonometric functions
Transformations of trigonometric graphs
Applications
Inverse trigonometric functions
Limits and Continuity
Rates of Change and Limits
Average and instantaneous speed
Definition of limit
Properties of limits
One-sided and two-sided limits
Sandwich theorm
Limits Involving Infinity
Finite limits as x( "
I n f i n i t e l i m i t s a s x ( A
E n d b e h a v i o r m o d e l s
S e e i n g l i m i t s a s x ( "
C o n t i n u i t y
C o n t i n u i t y a t a p o i n t
C o n t i n u o u s f u n c t i o n s
A l g e b r a i c c o m b i n a t i o n s
C o m p o s i t e s
I n t e r m e d i a t e V a l u e T h e o r e m f o r C o n t i n u o u s F u n c t i o n s
R a t e s o f C h a n g e a n d T a n g e n t L i n e s
A v e r a g e r a t es of change
Tangent to a curve
Slope of a curve
Normal to a curve
Speed revisited
Derivatives
Derivative of a Function
Definition of derivative
Notation
Relationships between the graphs of f and f
Graphing the derivative from data
One-sided derivatives
Differentiability
How f(a) might fail to exist
Differentiability implies local linearity
Derivatives on a calculator
Differentiability implies continuity
Intermediate Value Theorem for Derivatives
Rules for Differentiation
Positive integer powers
Multiples, sums, and differences
Products and quotients
Negative integer powers of x
Second and higher order derivatives
Velocity and Other Rates of Change
Instantaneous rates of change
Motion along a line (velocity, acceleration, free-fall, speed)
Sensitivity to change
Derivatives in economics
Derivatives of Trigonometric Functions
Derivative of the sine function
Derivative of the cosine function
Simple harmonic motion
Jerk
Derivatives of other basic trigonometric functions
Chain Rule
Derivative of a composite function
Outside-Inside Rule
Repeated use of Chain Rule
Slopes of parametrized curves
Power Chain Rule
Implicit Differentiation
Implicitly defined functions
Lenses, tangents, and normal lines
Derivatives of higher order
Rational powers of differentiable functions
Derivatives of Inverse Trigonometric Functions
Derivatives of inverse functions
Derivatives of the arcsine
Derivatives of arctangent
Derivatives of arc secant
Derivatives other three
Derivatives of Exponential and Logarithmic Functions
Derivatives of ex
Derivative of ax
Derivative of ln x
Derivative of logax
Power Rule for arbitrary real powers
Applications of Derivatives
Extreme Values of Functions
Absolute (global) extreme values
Local (relative) extreme values
Finding extreme values
Mean Value Theorem
Mean Value Theorem
Physical interpretation
Increasing and decreasing functions
Other consequences
Connecting f and f
First derivative test for local extrema
Concavity
Points of inflection
Second derivative test for local extrema
Learning about functions from derivatives
Modeling and Optimization
Examples from business and industry
Examples from mathematics
Examples from economics
Modeling discrete phenomena with differentiable functions
Linearization and Newtons Method
Linear approximation
Newtons Method
Differentials
Estimating change with differentials
Absolute, relative, and percentage change
Sensitivity to change
Related Rates
Related rate equations
Solution strategy
Simulating related motion
The Definite Integral
Estimating Finite Sums
Distance traveled
Rectangular Approximation Method (RAM)
Volume of a sphere
Cardiac output
Definite Integrals
Riemann Sums
Terminology and notation of integration
Definite integral and area
Constant functions
Integrals on a calculator
Discontinuous integrable functions
Definite Integrals and Antiderivatives
Properties of definite integrals
Average value of a function
Mean Value Theorem for Definite Integrals
Connecting differential and integral calculus
Fundamental Theorem of Calculus
Fundamental Theorem, Part 1
Graphing the function integral ((f(t)), (t), (a), (x))
Fundamental Theorem, Part 2
Area connection
More applications
Trapezoidal Rule
Trapezoidal approximations
Other algorithms
Simpsons Rule
Differential Equations and Mathematical Modeling
Antiderivatives and Slope Fields
Solving initial value problems
Antiderivatives and indefinite integrals
Properties of indefinite integrals
Applications
Integration by Substitution
Power Rule in integral form
Trigonometric integrands
Substitution in indefinite integrals
Substitution in definite integrals
Separable differential equations
Exponential Growth and Decay
Law of Exponential Change
Continuously compounded interest
Radioactivity
Newtons Law of Cooling
Resistance proportional to velocity
Population Growth
Exponential model
Logistic growth model
Logistic regression
Applications of Definite Integrals
Integral as Net Change
Linear motion revisited
General strategy
Consumption over time
Net change from data
Work
Areas in the Plane
Area between curves
Area enclosed by intersecting curves
Boundaries with changing functions
Integrating with respect to y
Saving time with geometry formulas
Volumes
Volume as an integral
Square cross sections
Circular cross sections
Cylindrical shells
Other cross sections
LHpitals Rule
Indeterminate form 0/0
Indeterminate forms "/ ", "" ", "- "
I n d e t e r m i n a t e f o r m 1 ", 0 0 , "0
R e v i e w
B a s i c s k i l l s a n d c o n c e p t s
G r a p h i n g c a l c u l a t o r s
S o l v i n g p r o b l e m s b y h a n d a n d b y c a l c u l a t o r
T e s t s k i l l s , M u l t i p l e C h o i c e v F r e e R e p s o n s e
C o l l a b o r a t i v e F r e e R e s p o n s e W r i t i n g
G r a d i n g P o l i c y :
H o m e w o rk will be assigned and collected on a daily basis. Full homework credit will only be given to work that is complete, shows all necessary steps, and is turned in on time. Late homework must still be completed, but will receive lower point values. Homework problems may be graded at random to check for completion/comprehension of problems. Not completing homework 3 times in one 9-week period will result in parent conference and/or dismissal from the program.
Small 1-2 question quizzes will be given nearly daily in class. These will be given the day after the homework problems have been gone over in class.
Larger 5-10 question quizzes will be given about once per week over 2-4 sections.
Exams will be given at the end of each chapter. Standardized test questions will be included on each test. Also, tests will be divided into calculator/non-calculator parts as well as have multiple choice and written free response style questions to follow the format of the AP exam. These questions will not only ask you to solve problems, but they will ask you to explain your understanding of the question and answer in paragraph format.
Approximate grade schedule:
60% Examinations
20% Quizzes
20% Homework
Technology:
All students enrolled in this class must have a TI-84 or higher calculator of their own. All in class examples and instructions using graphing calculators will be based on the TI-84. Help will be provided for using graphing calculators in class, but will primarily be limited to new items. If you need help using basic graphing calculator skills you may want to set up a time after school to get extra help. The following is a list of items that students must be proficient on the graphing calculator:
Graphing functions, finding zeros, changing window sizes, finding points of discontinuity, finding limits, solving derivatives using nDER, finding maxima and minima, solving integrals using nINT, finding regression equations, finding carrying capacity, finding slope fields, as well as basic arithmetic and algebraic skills.
Students will be responsible for using the calculators in and out of class as well as group settings. The use of the calculator may be at times the only way to solve problems (graphical approach), but the calculator is often used as a tool to verify and support your analytical answers. You must be prepared to use a graphing calculator or not use one on most problems.
Students will also have to complete two lab type explorations using data collected from our schools weather station, including finding rates, finding totals from rates, and basic function skills.
Class Work:
Students are advised to devote time every day for homework and review. Since this is a college level course, they will be expected to spend the necessary time to practice and learn the material. Class attendance is extremely important.
Students will be responsible for all of their own work. Working with others on homework is great for help in understanding, but everyone must have their own answers.
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