AP Calculus BC - Syllabus

AP Calculus BC - Syllabus

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Course Requirements

1) All students are expected to learn the material detailed in the Course Outline so that they are well-prepared for college-level mathematics and the AP Exam in May

a) Students are expected to participate in preparatory mock-exams as well as the actual AP exam

b) A variety of resources exist at the district and school level to insure that all students have the opportunity to take the exam

2) All students are expected to understand and explore the "Rule of 4" which provides for the representation of functions in various ways including graphically, numerically, analytically, and verbally

a) Students will be given a wide range of problems demonstrating their knowledge of how to work with each type of representation as well as the connections that exist between the different representations

b) Students will be expected to use TI-83/84 type calculators on a daily basis to explore the relationships between numeric data, formulas, and their corresponding graphs

3) All students are expected to be able to communicate mathematical concepts and describe solutions to problems both orally and in writing

a) Students will work extensively in small groups where they will be expected to collaborate with their peers as they develop their understanding of key mathematical concepts

b) Students will be expected to keep a journal of notes for key concepts taken in Cornell note form where students will be expected to write a summary using complete sentences of the math concept being explored

c) Students will be given homework assignments beginning early in the year where they will be expected to develop the necessary skills to successfully answer free- response questions from previous AP tests by explaining and justifying their answers with a written response

4) All students are expected to use a TI-83/84 type calculator on a daily basis for use in solving problems, experimenting with graphs and solutions, exploring the reasonableness of results, and verifying answers

a)Students will have access to school supplied TI-83/84 calculators for use in class and at home

b) Students are free to use their own graphing calculator provided that it meets the requirements for use on the actual AP Calculus exam

c) Students will be assessed in a manner similar to that of the AP exam with students being expected to answer questions which are calculator-active and questions which are calculator-inactive

d) Students are expected to be able to use the calculator to regularly perform the following:

i) Plot the graph of a function using an appropriate window, including the ability to zoom in on a point to demonstrate properties such as local linearity

ii) Find the zeros of functions

iii) Numerically calculate the derivative of a function

iv) Numerically calculate the value of a definite integral

5) It is strongly desired that all students will develop an appreciation of calculus as a truly amazing human accomplishment as it provides a cohesive mathematical approach to solving problems

Textbook

Larson, Hostetler and Edwards. Calculus of a Single Variable: Eighth Edition.

Boston: Houghton Mifflin Company, 2006.

Students will be given a copy of this textbook for use in the course. The Course Outline below references specific sections in the textbook. While this textbook is our primary source of material for this course, it will be supplemented as necessary and appropriate to enhance the experience of learning calculus.

Course Outline

Unit I: Functions, Graphs, and Limits (15 days)

A) Analysis of Graphs (2 Days)

i) Interplay between the geometric and analytic information (1.2, 1.3, 3.5)

ii)Use of calculus to predict and explain observed local and global behavior (1.2, 1.3, 3.5)

B)Limits of Functions (Including one-sided limits)(5 Days)

i) An intuitive understanding of the limiting process (1.2-1.4, 3.5)

ii) Calculating limits algebraically (1.2-1.4, 3.5)

iii) Estimating limits graphically and numerically (1.2-1.4, 3.5)

C) Continuity as a Property of Functions (3 Days)

i) An intuitive understanding of continuity using sufficiently close domain values (1.4)

ii) Understanding continuity in terms of limits (1.4, 1.5)

iii) Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem) (1.4, 3.1)

D) Asymptotic and Unbounded Behavior (2 Days)

i) Understanding asymptotes graphically (1.5, 3.5)

ii) Asymptotes and limits at infinity (3.5)

E) Parametric, Polar and Vector Functions (10.2, 10.4, 12.1*) (*Note: This material will be taken from the 9th edition of the textbook)(3 Days)

Unit II: Derivatives (45 days)

A) Concept of Derivative (5 Days)

i) Derivative presented graphically, numerically, and analytically (2.1)

ii) Relationship between differentiability and continuity (2.1)

iii) Derivative as the limit of the difference quotient (2.1)

iv) Derivative as an instantaneous rate of change (2.2 - 2.4)

B)Derivative at a Point (4 Days)

i) Slope of a curve at a point including vertical tangents (2.2)

ii) Instantaneous rate of change as the limit of average rate of change (2.2)

iii) Approximate rate of change from graphs and tables (2.3, 2.4)

iv) Tangent Line to a curve at a point and local linear approximation (2.2 - 2.5, 3.9)

C)Computation of Derivatives (8 Days)

i) Derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions (2.2, 2.3, 5.1, 5.4, 5.5, 5.6)

ii) Basic rules for the derivative of sums, products, and quotients (2.2, 2.3)

iii) Chain Rule and implicit differentiation (2.4, 2.5)

iv) Derivatives of parametric, polar, and vector functions (5.3-5.6, 10.3-10.4, 12.2-12.4*)(*Note: This material will be taken from the 9th edition of the textbook)

D) Derivative as a Function (8 Days)

i) Curve sketching and corresponding characteristics between and by finding critical values (2.3, 2.4, 3.1)

ii) Mean Value Theorem and its graphical and analytic implications (3.2)

iii) Relationship between the increasing and decreasing behavior of and (3.3)

iv) Equations involving derivatives (3.7)

E) Second Derivative (2 Days)

i) Corresponding characteristics of , , and (3.4)

ii) Relationship between sign of second derivative and concavity (3.4)

iii) Points of inflection as a location where concavity changes with many examples and counterexamples (no sign change) (3.4)

F) Applications of Derivatives (18 Days)

i) Analysis of curves, including the notions of monotocity and concavity (3.6)

ii) Optimization, absolute and relative extrema (3.7)

iii) Modeling rates of change including related rates problems (2.6)

iv) Use of implicit differentiation to find the derivative of an inverse (2.5)

v) The derivative as a rate of change in applied context including velocity, speed, and acceleration (3.2)

vi) Geometric interpretation of differential equations via slope fields (6.1)

vii) Analysis of planar curves in parametric, polar, and vector form including velocity and acceleration and arc length (5.3-5.6, 10.3-10.4, 12.2- 12.4*)(*Note: This material will be taken from the 9th edition of the textbook)

viii) Numerical solution of differential equations using Euler's Method (6.1)

ix) L'Hospital's Rule, including its use in determining limits and convergence of improper integrals and series (8.7)

Unit III: Integrals (40 days)

A) Numerical Approximations to Definite Integrals (4 Days)

i) Use of Riemann sums (left, right, midpoint, and trapezoidal) (4.2, 4.3)

B)Interpretations and Properties of Definite Integrals (4 Days)

i) Definite integral as a limit of Riemann sums (4.3)

ii) Properties of definite integrals (4.3)

iii) Definite integral of the rate of change of a quantity over an interval abf'x=fb-f(a) (4.4)

C) Techniques of Anti-differentiation (8 Days)

i) Anti-derivatives following directly from derivatives of basic functions (4.4)

ii) Anti-derivatives by substitution of variables including change of limits for definite integrals, parts, and simple partial fractions(4.5, 8.1-8.6)

iii) Improper integrals (8.8)

D) Fundamental Theorem of Calculus (4 Days)

i) Use of the Fundamental Theorem to evaluate definite integrals

ii) Use of the Fundamental Theorem to represent a particular anti-derivative and the analytical and graphical analysis of functions so defined (4.5)

E) Applications of Anti-differentiation (4 Days)

i) Finding specific anti-derivatives using initial conditions, including applications to motion along a line (4.1, 4.4)

ii) Solving separable differential equations and using them in modeling with exponential growth and the equation (6.1 - 6.3)

iii) Solving logistic differential equations and using them in modeling (6.3)

F) Applications of Integrals (16 Days)

i) Physical, biological, and/or economic situations (6.2, 6.3)

ii) Using the integral of a rate of change, to give accumulated change, area of a region, volume of a solid with known cross-sections, the average value of a function, and the distance traveled by a particle along a line (6.2, 6.3, 7.1, 7.2)

Unit IV: Polynomial Approximations and Series (35 days)

A) Concept of series (10 Days)

i) Definition of series and convergence (9.1-9.2)

B) Series of Constants (20 Days)

i) Motivating examples including decimal expansion (9.2)

ii) Geometric series with applications (9.2)

iii) The harmonic series (9.3)

iv) Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing convergence of

p-series (9.3)

v) Comparing series to test for convergence and divergence (9.4)

vi) Alternating series with error bound (9.5)

vii) The ratio test for convergence and divergence (9.6)

C) Taylor Series (15 Days)

i) Taylor polynomial approximation with graphical demonstration (9.7, 9.10)

ii) Maclaurin series and the general Taylor series centered at x=a (9.7, 9.10)

iii) Maclaurin series for the functions ex,sinx,cosx, 11-x(9.7, 9.10)

iv) Formal manipulation of Taylor series including substitution, differentiation, anti-differentiation, and the formation of new series from known series (9.7, 9.10)

v) Functions defined by power series (9.8)

vi) Radius and interval of convergence of power series (9.8)

vii) Lagrange error bound for Taylor polynomials (9.7)

Unit V: Comprehensive Review (10+ days)

A) Mock Exam

B) Comprehensive Review of Limits, Derivatives, Integrals, and Polynomial Approximations and Series using previously released tests

Grading

This class will utilize the approved grading scale for the district for all AP courses. A minimum of 6 daily grades will be given each grading period that constitutes 40% of a student's grade. Daily grades include daily assignments, homework, quizzes, written summaries and free-response questions. A minimum of 3 test grades will be given each grading period that constitutes 60% of a student's grade. Test grades include major tests given at appropriate points during the Course Outline and any major projects given.

Sample of Additional Materials

·  AP Calculus Multiple Choice Questions, 1998 - present

·  AP Calculus Free Response Questions, 1998 - present

·  AP Central Curriculum Modules and Special Focus Materials

·  Additional materials collected during Summer AP Institute

·  Computer generated images for visual representations of rotated solids