AP Calculus AB Syllabus
Course Design and Philosophy
The primary objective of this course is to develop and enhance each student’s understanding of the concepts of calculus by providing experience with its methods and applications. Specifically, we will encourage students to comprehend the reasoning behind a theorem or a concept instead of just trying to remember the proper patterns for solving. In doing this, we hope to promote a higher level of abstract thinking while fostering an ability to analyze and evaluate new data based on this prior knowledge. Problems will be presented and solved via four precise modalities including: analytically, graphically, numerically, and verbally.
Teaching Strategies
Students will most likely begin the class by relying on their graphing calculators in order to determine the proper answers. As time progresses, however, it is important for them to understand that graphs and charts alone do not provide sufficient evidence to prove calculus-based ideas. Although it can be extremely valuable to dissect data both numerically and graphically, verification always requires an analytical approach. It is my hope and expectation that all students will rise to this level. In addition, this class will also stress the importance of communication skills. If a student is unable to literally or verbally explain the reasoning behind their solution, then it is stated that they do not really understand it.
In order to promote this verbal aspect of the class, students will be asked to do a variety of writing assignments through the Collin’s learning program which is our school’s current curriculum of study. Likewise, I will also be asking my students to orally present their solutions on the board, in front of the class, and to their peers for additional reinforcement of topics.
Daily class instruction will include time for students to work collaboratively on selected problems and homework assignments. In addition, I will host a study session or study group in order for students to gain mastery on concepts that may be more difficult or on ideas they continue to struggle with. After all, much of calculus depends upon the comprehension of previously learned items. Similarly, I plan on spiraling my teaching as a means of encouraging true understanding along with a positive retention factor. In other words, not only will I correlate ideas from varying chapters but I intend to put questions on my tests that include data from earlier chapters.
Major Text
Anton, Bivens, and Davis. Calculus – Early Transcendentals, Single Variable. 9th ed. Jefferson City: John Wiley and Sons, Inc., 2009.
Technology
Every student in this course will be provided with a graphing calculator, most likely a TI-83 Plus, if they do not have their own. In order to enhance the usage of these graphing utilities, appropriate class time will be spent training students on them with the help of Smartview software displayed on a Smartboard. Students can then become aware of the benefits of calculator usage yet must also determine when such usage is appropriate. This technology is provided as a tool to enhance learning not as a crutch to be dependent upon.
We will use the graphing calculators both to experiment and make discoveries about functions in calculus while also using them to support or confirm our conclusions. Such instances include:
· Graphing functions within arbitrary windows.
· Using the table feature to interpret results numerically.
· Solving equations or functions numerically by using the appropriate utility.
· Validating analytical solutions through graphing.
· Approximating values of derivatives and definite integrals through numerical methods.
Course Outline
Unit 1: Precalculus Review (2-3 weeks)
A. Functions
1. Definition of a function
2. Vertical line test
3. Domain and range
4. Piecewise functions
5. Absolute value functions
B. New functions from old
1. Composition of functions
2. Translations
3. Reflections
4. Stretches and compressions
5. Symmetry
6. Even and odd functions
C. Families of functions
1. Power functions
2. The family y = x-n
3. Polynomials
4. Rational functions
5. Algebraic functions
6. The families y = A sin Bx and y = A cos Bx
D. Inverse functions; Inverse trigonometric functions
1. Finding inverse functions
2. Existence of inverse functions
3. Graphs of inverse functions
4. Inverse trigonometric functions
5. Evaluating inverse trigonometric functions
6. Identities for inverse trigonometric functions
E. Exponential and logarithmic functions
1. Irrational exponents
2. The family of exponential functions
3. The natural exponential function
4. Logarithmic functions
5. Solving equations involving exponentials and logarithms
Unit 2: Limits and Continuity (3 weeks)
A. Limits
1. Tangent lines and limits
2. Areas and limits
3. One-sided limits
4. Two-sided limits
5. Infinite limits
6. Vertical asymptotes
B. Computing limits
1. Basic limits
2. Limits of polynomials and rational functions
3. Limits involving radicals
4. Limits of piecewise-defined functions
C. Limits at infinity; End behavior of a function
1. Limits at infinity and horizontal asymptotes
2. Limit laws for limits at infinity
3. End behavior models
4. Properties of limits
5. Visualizing limits
D. Continuity
1. Definition of continuity
2. Continuity on an interval
3. Properties of continuous functions
4. The intermediate-value theorem
E. Continuous functions
F. Discontinuous functions
Unit 3: The Derivative (3-4 weeks)
A. Tangent lines and rates of change
1. Tangent lines
2. Velocity
3. Slopes and rates of change
B. The derivative function
1. Definition of the derivative function
2. Instantaneous velocity
3. Differentiability
4. The relationship between differentiability and continuity
C. Introduction to techniques of differentiation
1. Derivative of a constant
2. Derivative of power functions
3. Derivative of a constant times a function
4. Derivatives of sums and differences
5. Higher derivatives
D. The product and quotient rules of derivatives
E. Derivatives of trigonometric functions
F. The chain rule
Unit 4: Topics in Differentiation (3-4 weeks)
A. Implicit differentiation
1. Functions defined explicitly and implicitly
2. Implicit differentiation
B. Derivatives of logarithmic functions
C. Derivatives of exponential and inverse trigonometric functions
D. Related Rates
E. Local linear approximation; Differentials
F. L’Hopital’s rule; Indeterminate forms
Unit 5: The Derivative in Graphing and Applications (4-5 weeks)
A. Analysis of functions
1. Increasing and decreasing functions
2. Concavity
3. Inflection points
4. Relative maxima and minima
5. First derivative test
6. Second derivative test
7. Geometric implications of multiplicity
8. Analysis of polynomials
9. Graphing rational functions
10. Graphs with vertical tangents and cusps
11. Graphing using calculus and technology together
B. Absolute maxima and minima
1. The extreme value theorem
2. Absolute extrema on infinite intervals
3. Absolute extrema on open intervals
4. Absolute extrema of functions with one relative extremum
C. Applied maximum and minimum problems
1. Problems involving finite closed intervals
2. Problems involving intervals that are not both finite and closed
3. An application to economics
4. Marginal analysis
D. Rectilinear motion
1. Velocity and speed
2. Acceleration
3. Analyzing the position versus time curve
E. Newton’s method
F. Rolle’s theorem
G. Mean-value theorem
Unit 6: Integration (7 weeks)
A. An overview of the area problem
1. The area problem
2. The rectangle method for finding areas
3. The antiderivative method for finding areas
B. The indefinite integral
1. Antiderivatives
2. The indefinite integral
3. Properties of the indefinite integral
4. Integral curves
5. Integration from the viewpoint of differential equations
6. Slope fields
C. Integration by u-substitution
D. The definition of area as a limit; Sigma notation
1. Sigma notation
2. Summation formulas
3. A definition of area
4. Numerical approximations of area
5. Net signed area
E. The definite integral
1. Riemann sums and the definite integral
2. Properties of the definite integral
3. Discontinuities and integrability
F. The fundamental theorem of calculus
1. The fundamental theorem of calculus – Part 1
2. The fundamental theorem of calculus – Part 2
G. Rectilinear motion revisited using integration
1. Finding position and velocity by integration
2. Computing displacement and distance by integration
3. Analyzing the velocity versus time curve
4. Constant acceleration
5. Free-fall model
H. Average value of a function and its applications
I. Evaluating definite integrals by substitution
J. Logarithmic and other functions defined by integrals
Unit 7: Applications of the Definite Integral in Geometry, Science, and Engineering (3 weeks)
A. Area between two curves
B. Volumes by slicing
1. Volumes by disks and washers perpendicular to the x-axis
2. Volumes by disks and washers perpendicular to the y-axis
3. Other axes of revolution
C. Volumes by cylindrical shells
D. Length of a plane curve
E. Area of a surface of revolution
F. Work
1. Work done by a constant force applied in the direction of motion
2. Work done by a variable force applied in the direction of motion
3. Calculating work from basic principles
4. The work-energy relationship
AP Review (3 weeks) – AP EXAM
At this point, students will view items from past exams, in particular, and practice testing strategies for both the multiple-choice and the free-response sections. They need to learn and be aware of the types of questions in which calculators will or will not be allowed. Students will be asked to work on these tests both individually and in teams. In regards to the latter, students will be expected to collaborate with their peers in order to formulate their team’s solutions. Finally, we will attempt to establish a rubric whereas the students are able to realize the importance of complete answers.
Unit 8: Principles of Integral Evaluation (2 weeks) – POST EXAM
A. Integration by parts
1. The product rule and integration by parts
2. Repeated integration by parts
3. Integration by parts for definite integrals
B. Integrating trigonometric functions
1. Integrating powers of sine and cosine
2. Integrating products of sine and cosine
3. Integrating powers of tangent and secant
4. Integrating products of tangents and secants
C. Trigonometric substitutions
D. Numerical integration; Simpson’s rule
1. Trapezoidal approximation
2. Comparison of the midpoint and trapezoidal approximations
3. Simpson’s rule
4. Error bounds
Unit 9: Mathematical Modeling with Differential Equations (2 weeks)
A. Modeling with differential equations
1. Solutions of differential equations
2. Initial-value problems
3. Population growth
4. Newton’s law of cooling
5. Vibrations of springs
B. Separation of variables
1. First-order separable equations
2. Exponential growth and decay models
3. Doubling time and half-life
4. Radioactive decay
5. Carbon dating
C. Slope fields; Euler’s method
D. First-order differential equations and applications
1. First-order linear equations
2. Mixing problems
3. A model of free-fall motion retarded by air resistance
Activities/Projects
Students will be asked to get into teams in order to develop a video in regards to one of the main concepts discussed this year in calculus. Their goal is to correctly define this idea and properly illustrate to their peers when and how to use this model in solving specific problems. Moreover, students may be asked to find websites and links that provide additional learning resources and function as tools in order to help exemplify and prove many of our newly learned calculus concepts.