AP Calculus AB Syllabus s1

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 Newton’s Method and Simpson’s 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 on) and know the values of the trigonometric functions of the numbers 0, π/6, π/4, π/3, π/2, and their multiples.

Course Content:

Prerequisites for Calculus:

Lines  Increments  Slope of a line  Parallel and perpendicular lines  Equations of lines

Functions 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±∞  Infinite limits as xA  End behavior models  “Seeing” limits as x±∞ Continuity  Continuity at a point  Continuous functions  Algebraic combinations  Composites  Intermediate Value Theorem for Continuous Functions

Rates of Change and Tangent Lines  Average rates 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 Newton’s Method  Linear approximation  Newton’s 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  Simpson’s 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  Newton’s 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

L’Hôpital’s Rule  Indeterminate form 0/0  Indeterminate forms ∞/∞, ∞•∞, ∞-∞  Indeterminate form 1∞, 00, ∞0

Review Basic skills and concepts Graphing calculators Solving problems by hand and by calculator Test skills, Multiple Choice v Free Repsonse Collaborative Free Response Writing

Grading Policy: Homework 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.

Students will also work in groups especially early in the learning process on some topics. You are asked to be helpful and explain to other students who might need help in understanding and listen to the approach others have taken in solving problems. Often, several approaches lead to the same answer and knowing multiple approaches helps your chance at success.

You will also be asked to volunteer or be called to the board to show answers from time to time. This is not to embarrass or pick on anyone, but should be viewed as another way to see others approaches at solving problems and for the instructor to see how you explain your thought processes in solving problems.