Course of Study For Calculus
Mathematics Department
March 2007
Winegar
PURPOSE AND PHILOSOPHY
Calculus is the study of limits, derivatives and integrals through the Fundamental Theorem of Calculus and some applications of the definite integral. This study corresponds to slightly less than the first semester course taught at many colleges and universities. The Calculus course is designed as a full year, challenging course on limits, derivatives and integrals which will focus on real world methodology and application and lead the successful student to a working knowledge of single variable calculus. Students who elect to may receive college credit through the CAP program at BCC. This course requires that students will have successfully completed Precalculus (C or better) and are capable of self motivation and direction. All students will have and use a graphing calculator on a daily basis. While most students have their own, the school issues graphing calculators for those students who do not. Given the successful completion of this course, students will be prepared to undertake a first year college level Calculus course.
This course will cover:
Ø Limits and continuity (2 weeks) including rates of change and limits, limits involving infinity, continuity, rates of change and tangent lines
Ø Derivatives ( 5 weeks) including differentiability, rules, velocity and rates of change, the chain rule, implicit differentiation, L’Hopital’s rule and the derivatives of trigonometric, inverse trigonometric, exponential and logarithmic functions
Ø Applications of Derivatives (5 weeks) including extreme values, mean values, graph theory, modeling and optimization and related rates
Ø The Definite Integral (5 weeks) including finite sums, antidifferentiation, the Fundamental Theorem of Calculus and the Trapezoidal and Simpson rules
Ø Differential Equations and Mathematical Modeling (5 weeks) including the relationship between antiderivatives and slopes fields, integration techniques such as substitution, integration by parts and tabular integration, exponential growth and decay and population models
Ø Applications of the Definite Integral (4 weeks) including net change, areas in the plane, calculation of volume by cross section, disc and washer, and arc length.
The course emphasizes a multi-representational approach to calculus with concepts, results and problems being expressed in multiple ways: geometrically, analytically, verbally and numerically. The connections among these representations are emphasized.
Broad concepts and widely applicable methods are emphasized. The focus of the course is neither manipulation nor memorization of an extensive taxonomy of functions, curves, theorems or problem types, thus, although facility with manipulation and computational competence are important outcomes of the course they are not the core. The course is a cohesive whole not a collection of unrelated topics.
STUDENT OUTCOMES
Major Goals
At the conclusion of this course:
Ø Students will be able to analyze and describe functions given in a variety of representations: graphical, tabular, algebraic or verbal. Students will be able to understand the connections among these representations.
Ø Students will understand the derivative of a function as a rate of change and be able to apply the derivative to solve a variety of problems.
Ø Students will understand the definite integral of a function as a limit of Riemann sums or the net accumulation of a rate of change and be able to use integrals to solve a variety of problems.
Ø Students will understand the inverse relationship between differentiation and integration.
Ø Students will be able to communicate mathematics orally and in well written sentences. Student must be able to explain solutions to problems as well as solve them.
Ø Student will be able to model a written description of a physical situation with a function, a differential equation or an integral.
Ø Students will use technology (most often TI graphing calculators with downloaded software) to help solve problems, experiment, decide upon appropriate models, interpret results and verify conclusions.
Ø Students will develop an appreciation of calculus as a coherent body of knowledge and human accomplishment.
Evaluation
To evaluate student progress, both formal and informal assessment will be used. Homework, quizzes and tests will compose the formal assessment while group work, projects. notebooks, and technological explorations will comprise the informal assessment. Formal assessment will multiple choice and open ended questions. Students will be provided ample opportunity to practice across a variety of learning styles.
LEARNING ACTIVITIES
In order to develop mastery of stated objectives, the following may be used:
Ø Extra practice problems
Ø Reinforcement exercises
Ø Transparencies
Ø Teacher prepared review and practice packets – topic specific
Ø On-line applets for motion and development problems
Ø Power Point presentations that illustrate complex topics and applications
Ø Maintenance of a notebook and organizational material
Ø Computer assisted instruction
Ø On-line quizzes and exercises for self evaluation and self assessment
Ø Group explorations and chapter projects including presentations
Ø Graphing calculator explorations (TI Explorations)
ASSESSMENT
To evaluate student progress, homework will be checked by teacher and by peers, written quizzes and chapter tests will be given, projects will be presented, notebooks will be evaluated and a final examination in June will be administered.
ENRICHMENT
There are extensive enrichment readings and instructional materials on line for students who wish to pursue the study of a related concept in greater depth.
SOFTWARE
Software for the TI-83, 84 and 89 graphing calculators
TEXTBOOK
Calculus, Finney, Demana, Waits, Kennedy; Pearson Prentice Hall; 2004.
Available in the classroom:
Calculus, Larson, Hostetler, Edwards, 1998.
Calculus. Anton, 1998
Student Solution Manual
Graphing Calculator Manuals
Core Curriculum Content Standards for Mathematics for New
Standard 4.1 All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.
Standard 4.2 All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.
Standard 4.3 All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions and algebraic concepts and processes.
Standard 4.4 All students will develop an understanding of the concepts and techniques of data analysis, probability and discrete mathematics and will use them to model situations, solve problems, and analyze and draw appropriate inferences from data.
Standard 4.5 All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology t6o solve problems and communicate mathematical ideas.