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Course Description Class Structure Grading/Assignments
Textbook:
Larson, Ron, et. al. Calculus of a Single Variable, Seventh Edition,second printing. Boston: Houghton-Mifflin Company, 2002.
Classroom Supplies
The only outside supplies needed for this course are pencil and paper (preferably in a spiral notebook for convenience) and a good supply of coffee for the late hours. We issue students a TI-83 Plus graphing calculator for their use during the course. We ask only that they return them with a fresh pack of batteries. However, for students intending to pursue technical fields in college, I strongly suggest that they eventually purchase their own TI-83 Plus Silver edition or TI-84 Silver edition and download the programs I've written into it (note: the Silver editions have much faster and larger RAM). Cost varies from about $90 to $130. Students with access to a computer may want to invest in one of the Calculus tutorial programs such as Math Advantage 2004, or use the internet to access some of the AP online exams and resource materials.
AP Calculus AB / IB Mathematics SL is an introduction to the Calculus. After a brief review of coordinate and analytical geometry and an introduction to the notion of the limit of a function, the BTW Calculus I class builds the theoretical foundation for the derivative, its properties, and its application to work-a-day world problem solving with particular attention to slope of tangent lines and curve sketching, related rates, and maxima and minima. Later, in like manner, we establish the theoretical basis for the anti-derivative and its properties. We spend much time on the mechanics of computing derivatives and anti-derivatives of various functions until they become second nature. We also devote considerable time to interpretation of graphs of the function, its 1st and 2nd derivatives, and its integral. We spend the bulk of the remaining time applying principles of calculus to real-world problem solving, including area under a curve, area and volume of solids of revolution and projection, length of a curve, related rates, and work. Because not all students received underclassmen instruction in the subjects, my curriculum also includes an investigation of vectors, series and sequences with mathematical induction, common trigonometric functions and their identities, inverse trigonometric functions, exponential, logarithmic, logistical, and other transcendental functions. And if there is time, I may also include some work on statistics. I include these additional topics because IB Mathematics SL covers these topics in addition to the calculus.
Grade
Level:
While AP students preparing for Calculus II or Higher Level mathematics and IB students taking IB Mathematics at the Standard Level comprise the majority of the students enrolled in the class, the course is open to any student who has successfully completed Pre-Calculus, IBSL Advanced Math Studies, or Algebra III with a "B" or better and recommendation from the previous teacher. However, receiving weighted credit for the class is contingent upon the student’s taking the AP or IB exams in May. Failure to take the test will result in an unweighted grade only.
Lesson Structure:
This course is a
college-preparatory class, and as such, I conduct it in a style similar to
college lecture/discussion but with cooperative learning structures.
I teach with the TI-83 Plus and Capri graphic software as discovery tools to
reinforce the abstract concepts. Typically, the lesson begins with a call for
questions from previous assignment, proceeds into the presentation/discovery
of new material, and ends with a relevant assignment (usually selections from a
problem set from the textbook or a worksheet that doubles as a study guide or a
portfolio writing assignment), begun in class and finished at home. However, we
may spend an entire class discussing questions if students demonstrate a need
for more background for a specific topic.
In accordance with the IB Program, a Portfolio Project is worth 20% of the course grade and consists of two of four, teacher-made assignments that differ in type: Type I—a discovery / investigation assignment; and Type II—an applications / modeling assignment. (Each student submits his best one of each type for the portfolio even though he may complete more than that.) WARNING: The British concept of a portfolio is quite different from that of us Yanks. We educators in America typically think of a portfolio as a representative sample of student work; however, the IB Curriculum Officials in Cardiff consider the portfolio to be a collection of mathematical essays on a specific assignment. They place primary emphasis on good mathematical writing—complete with introduction, main body development with appropriate mathematical rigor, and a conclusion/evaluation of the assignment.
Each test counts twice, but I allow one make-up test per six-week grading period with no questions asked. However, students must schedule the make-up test within one week of receiving the original and schedule it for a non-class time period. Tests are scored against performance standards, not norms. Students will have one to three testing opportunities per chapter, depending upon the volume of material covered. Since most students are preparing for the IB Mathematics SL Exam or the AP Calculus AB exam, I typically do not give final exams. The course is extremely cumulative in nature anyway, and students must have mastered previous material in order to perform later tasks. Staying current with assignments is imperative for success.
The student daily score counts once for the grading period and consists of a ratio of problems completed to problems assigned. Students who do extra work may achieve a daily score in excess of 100%, but not to exceed 200%.
Categorical Listing of Topics Discussed
| Topic Grouping | Specific Concepts (Note: "PK" = Presumed Knowledge) |
| Linear
Functions PK |
Slope, x-intercept (zeros of the function), y-intercept, slopes of parallel and perpendicular (normal) lines, solving systems of linear equations to determine intersection points, four forms of linear equations (general/ standard form, slope-intercept form, point-slope form, intercept form), plotting points, distance, midpoint, geometric applications of linear functions, graphing linear functions, angle of inclination, applications and mathematical modeling |
|
Quadratic functions PK |
Complex numbers and imaginary solutions, factoring techniques and zeros, quadratic formula, translations (horizontal shift, vertical shift), dilation (stretch), reflection, rotation, graphing, standard form, graphing form, focus, line of directrix, applications and mathematical modeling |
|
Absolute Value PK |
Definition, translations, dilation, reflection, graphing, inequalities with absolute value, practical meaning, applications and mathematical modeling |
|
Polynomials PK |
Definition, degree, odd-even, roots of a polynomial, synthetic division, properties of polynomials, rational functions, applications and mathematical modeling |
|
Functions PK |
Definition, distinguish function and relation, domain, range, mapping notation, function notation, x-y notation, operations on functions, composition of functions, inverse of a function, translations, dilation, reflection, rotation, applications and mathematical modeling |
|
Trigonometry PK |
Six basic trigonometric functions (including their graphs and inverses), radian measure of angles, right-triangle trigonometry, trigonometry in the Cartesian plane, solving trigonometric expressions, trigonometric identities and proofs, applications of trigonometry |
|
Parametric Equations PK |
Parametric equations of x and y as functions of time, parametric equations of x and y as functions of angle θ of rotation about the origin, applications to vectors and matrices |
|
Transcendental Functions PK |
Inverses of trigonometric functions, the natural logarithm, properties and operations on logarithms, solving logarithmic equations, applications of logarithms, logarithms in other bases, exponential functions, the Euler constant, properties and operations on exponential expressions, solving exponential expressions, practical applications of exponents |
| Limits | Cauchy's definition of a limit, properties of limits, continuity and discontinuity, left- and right-hand limits, limits and infinity, limits in the definition of the derivative, limits of indeterminate forms |
| Derivatives | Difference quotient, derivative as limit of difference quotient (i.e. definition of a derivative), derivative as slope of a tangent to a curve at a given point, slope of secant as average rate of change, derivative as instantaneous rate of change, Leibniz differential notation, power rule, chain rule, product rule, quotient rule, linear approximation to a curve, derivatives of trigonometric functions, derivatives of trigonometric inverses, Newton-Raphson Method of approximating roots, function differentiability, graph interpretation and curve sketching, absolute and local maxima and minima on a closed interval, concavity, inflection points, asymptotes and symmetry, mean value theorem, derivative of the natural logarithm, derivative of exponential expressions, derivatives and technology |
| Integration |
Indefinite integral as anti-derivative, discovering the constant of
integration, substitution method in integration, Fundamental Theorem of
Calculus, indefinite integrals of trigonometric functions, interpreting
graphs, integration with related rates, Riemann Sums,
definite integrals and the limits of integration, properties of definite
integrals, definite integrals as area under the curve, evaluating definite
integrals, substitution in definite integrals and critical values,
approximating irregular definite integrals using the trapezium rule and
Simpson’s rule, integrals as net change in position of and distance traveled
by a moving body, integrals as area between two curves, volumes of solids by
slicing, washer and shell methods of calculating the volume of solids of
revolution about an axis, integrals as length of plane curves, surface area
of a solid of revolution, average value of a function, moments and center of
mass, integration in calculating work, Integration by parts, integral tables (maybe--time permitting) |
| Miscellaneous topics | Mathematical induction and proof, arithmetic and geometric series and sequences, vectors, matrices, conic sections, probability and statistics with regression equations applied to curve sketching, discrete math topics |
Tentative Time Table (varies according to student progress)
The following table is an approximate weekly schedule of topics and concepts
covered in my class in order of coverage and the corresponding objectives,
tasks, and skills to be mastered for that topic. However, since the school
year is often subject to short-notice changes, I live by the 11th
Beatitude: "Blessed are the flexible, for they shall not get bent out of
shape."
| Time Period |
Item / Chapter . Section |
| Week
1
Introductions, Class/Team Building, Intro to Learning Structures, Memory techniques |
Pair
Match into Interview Introductions Identify Individual Bird Type and Color Spectrum, Create Learning Colonies & HIVE, Develop HIVE agreements, Demonstrate thinking/study skills, Execute collaborative learning structures |
| Weeks 1-2 | Linear, Quadratic, Polynomial, Rational, and Exponential Functions Review: Worksheets and Sections P.1-P.3 |
| Week 3 |
Applications with Data Modeling of Functions
& their graphs with regression equations: Section P.4 Test on Chapter P |
| Weeks 4 |
A Preview of the Calculus with an
Introduction to Limits: Section 1.1 Portfolio Project on Exploring Perfect Squares or Portfolio Project on Generating Pythagorean Triples |
| Week 5 |
Cauchy's Formal Definition of a limit at a
point. Finding Limits Graphically/Numerically: Section 1.2 Evaluating Limits Analytically: Section 1.3 |
| Week 6 |
Continuity and One-sided Limits:
Section 1.4 Infinite Limits (not an oxymoron!): Section 1.5 Test on Chapter 1 The Difference quotient and its Limit: Section 2.1 |
| Week 7 |
Intuitive Definition of the Derivative as the
slope of the line tangent to a curve: Section 2.1 Basic Differentiation Rules: Section 2.2 Rates of Change Portfolio Project on Sudden Impact: Investigating Kinematics |
| Week 8 |
Product Rule, Quotient Rule:
Section 2.3 Higher Order Derivatives of Functions The Chain Rule with Composite Functions: Section 2.4 |
| Week 9 |
Implicit Differentiation of
Functions: Section 2.5 Related Rates Application Problems: Section 2.6 Test over Non-Trig Derivatives with Related Rates |
| Week 10 |
Review of Trigonometry with outside
worksheets Derivatives of Trigonometric Functions: Sections 2.2-2.3 |
| Week 11 |
Higher order Trig Functions with Chain Rule:
Section 2.4 Implicit Differentiation with Trig-Functions: Section 2.5 More Related Rates Application Problems: Section 2.6 |
| Weeks 12 |
Portfolio Project
on Related Rates Test over Trig Functions, Derivatives, & Related Rates Applications of Derivatives: Extrema on Interval--Section 3.1 |
| Week 14 |
Rolle's Theorem and Mean Value Theorem:
Section 3.2 First Derivative Test and Curve Direction: Section 3.3 Tests for Concavity and Inflection Points: Section 3.4 |
| Week 15 |
Interpreting Infinite Limits and Limits
approaching Infinity Summary of Curve Sketching Techniques: Sections 3.5-3.6 Portfolio Project on Curve Sketching Indeterminate Forms and L'Hopital's Rule: Section 7.7 |
| Week 16 |
Test over Curve
Sketching
Optimization Problems (i.e. Maxima and Minima):
Section 3.7 |
| Week 17 |
Newton's Method of
approximating zeros: Section 3.8 Differentials and Implicit Differentiation: Section 3.9 Marginal Values and Business Applications: Section 3.10 Test over Applications of the Derivative |
| Week 18 |
Intro to Integration Indefinite Integration as Anti-derivative: Section 4.1 Portfolio Project on Sequences and Series or Portfolio Project on Elizabeth's Champaign Fountain Definite Integral as Riemann Sum and Area under a curve: Sections 4.2-4.3 |
| Weeks 19 |
The Fundamental Theorem of Calculus:
Section 4.4 Integration using u-variable substitutions: Section 4.5 |
| Weeks 20 |
Numerical Integration Techniques (Trapezoid
Rule and Simpson's Rule): Section 4.6 Test over Integration Review of Logarithms and Exponents from worksheets |
| Week 21 |
The Reciprocal Function as a slope field of
tangents to the curve of the Natural Logarithm: Section 5.1 The Natural Logarithm Defined as an Integral: Section 5.2 Inverse Functions and Composition: Section 5.3 The Euler Constant as the base e |
| Weeks 22 |
Exponential Functions--Derivatives &
Integrals: Section 5.4 Bases other than e : Section 5.5 Solving Exponential and Logarithmic Equations with emphasis on Growth and Decay: Section 5.6 Solving Differential Equations using the Separation of Variables technique: Section 5.7 |
| Weeks 23 |
Test over
Exponential and Logarithmic Functions Intro to Inverse Trigonometric Functions Derivatives of Inverse Trigonometric Functions: Section 5.8 |
| Week 24 |
Integrals of Inverse Trigonometric Functions:
Section 5.9 Using a and u-variable substitutions with Trig. Functions Test over Inverse Trigonometric Functions Introduction to Applications of the Integral |
| Week 25 |
Area of a Region between two Curves:
Section 6.1 Volume of a Rotational Solid (Disc Method): Section 6.2 Volume of a Rotational Solid (Shell Method): Section 6.3 Volume of a Solid using Cross-Sectional Areas: Section 6.3 |
| Week 26 |
The Integral applied to Arc Length and
Distance along a curve: Section 6.4 The Integral applied to Surface Area of Rotational Solid: Section 6.4 Work as an Integral of Force and Distance: Section 6.5 Other Physics applications of the Integral: Section 6.6-6.7 |
| Week 27 |
Integration by Tables and other techniques:
Section 7.6 Indeterminate forms and L'Hopital's Rule: Section 7.7 Test over Applications of the Integral |
| Weeks 28-36 |
Review for AP Calculus AB Exam and IB Mathematics SL Exam |