Calculus I Grade Level 12

Transcription

1 Calculus I Calculus I introduces students to the following concepts, functions and more: calculating x-values and corresponding values continuous functions asymptotes graphing tangents secants, and cosecants sign graphs antiderivatives with negative exponents 1

2 TEACHERS' GUIDE COURSE DESCRIPTION: The Anywhere Learning System (ALS) Calculus I is part of the comprehensive, completely integrated math courses for grades The Mathematics courses develop strategies to solve real-world problems and are designed to move the student beyond the level of basic knowledge into critical thinking and learning activities. COURSE FEATURES: Mathematics is a collection of one- and two- semester long courses. All lessons in these titles contain a study guide, a practice test, and mastery test. Some courses are enriched by National Library of Virtual Manipulatives (NLVM) applets or Encyclopædia Britannica Online School Edition (EB) workspaces that contain learning materials. Learning materials may contain articles, games, images, maps, and/or videos. COURSE OBJECTIVES: The content in these titles is designed to meet and exceed the requirements of the National Council of Teachers in Mathematics. The standards for school mathematics describe a comprehensive set of goals for mathematics instruction. 2

3 TEACHERS' GUIDE COURSE CONTENT Course Name Number of Lessons Length of Course in Semesters Grade Levels Calculus I

4 SCOPE & SEQUENCE Lesson Name Lesson Description Activities 1 Limits Calculating x-values and corresponding values; approaching function values, limits, and notation 2 Continuous Functions Definition of continuous function; continuous graphs of polynomial functions, sine, and cosine; evaluating the limits of continuous function 3 Discontinuous Functions 1 Examining various types of discontinuities: holes, asymptotes, jumps, and their graphs 4 Discontinuous Functions 2 Approaching negative and positive infinities 5 Discontinuous Functions 3 One-sided limits 6 Special Trig Functions 7 Limits at Infinity Trigonometric limits of sine and cosine functions; graphing tangents, cotangents, secants, and cosecants Polynomials as they approach infinity, negative infinity, and infinity squared; definition of infinity squared; examples of how changing the argument of the function changes the limit 8 Limit Unit Review Review of limit lessons 9 Derivatives 10 Derivative Shortcuts 1 Derivatives and determining the slope of a tangent at a given point, using the derivative as a velocity, the derivative as a function; Leibniz notation Using the mathematical definition of derivatives; the Power Rule and coefficients of sums and differences Essay: Problem Solving 11 Derivative Shortcuts 2 Negative exponents; derivatives of sine and cosine; derivatives at specific points 4

5 SCOPE & SEQUENCE Lesson Name Lesson Description Activities 12 Some Derivative Rules Functions that are products; the Product Rule; rational functions and the Quotient Rule; the derivative as a reciprocal of sine 13 The Chain Rule Derivatives of composite functions; definition of the chain rule; extending the chain rule 14 Higher Derivatives Acceleration as a derivative of velocity; notation and use of higher derivatives 15 Implicit Differentiation Examples of finding the derivative implicitly without solving for y 16 Derivative Unit Review Review of derivatives 17 Values 1 Determining maximum and minimum values of given functions on closed intervals 18 Values 2 Using zero-slope to determine maximum and minimum values, critical points, and relative extrema 19 Tests 1 The first derivative tests; increasing and decreasing slopes; finding relative extrema 20 Tests 2 Second derivative tests; finding relative extrema 21 The Second Derivative Concavity and inflection points of graphs; definition and determination of inflection points; sign graphs 22 Application Unit Review 1 Review of maximum and minimum values and tests 23 Applications of Extrema Determining the need to find maximum and minimum values in real life situations 5

6 SCOPE & SEQUENCE Lesson Name Lesson Description Activities 24 Related Rates 1 Problems with derivatives that are related; problems involving related rates and spheres 25 Related Rates 2 Using related rates to determine the volume of cones; using the Pythagorean relationship in related rate problems Graphing Using Extremes 1 Graphing Using Extremes 2 Application Unit Review 2 Understanding the nature of graphing; determining graphing data Asymptotes as related to graphs Review of related rates and graphing 29 Antiderivatives Determining the original function from the derivative; definition of antiderivatives; proving antiderivatives; antiderivatives with negative exponents. 30 Comprehensive Exam Review of all material presented in Calculus I 6