Calculus II, Level 1 Course of Study

BOE Approved Spring 2006

Revised October 2011

Table of Contents

Members of Revision Committee

Statement of Purpose

Program of Studies Description

Core Content Standards

Textbook and Resource Materials

Course Objectives/Activities

Content Outline/Timeline

Members of Revision Committee

Statement of Purpose:

The purpose of all curriculum guides is to provide direction for instruction. They identify the written outcomes in a subject and /or grade as the basis for classroom activities and student assessment. In order to achieve maximum understanding, the objectives identified as learning outcomes must be written clearly and reflect the specific learning and behavior which are expected.

Objectives are written as major outcomes and stated to require critical thinking. Teachers should understand that they must make careful decisions about the specific sub skills and prior learning needed to reach these objectives. These professionals are encouraged to reflect with others teaching the same curriculum for this purpose and also to identify the most appropriate resources and methods of assessment. The assessments are directly aligned with the objectives. Therefore, the objectives in this guide are designed to provide direction to the teacher in order to facilitate instructional planning.

All teachers, parents and students should be informed of the expected outcomes (i.e. objectives) for the subject and/or grade level.

Program of Studies Description:

Expands the depth of study of some Calculus AB topics, prepares students to take the Advanced Placement Test in Calculus BC, and includes additional topics beyond the calculus BC curriculum. Students should receive college credit or advanced standing depending on their scores and the policy of the college. This course is designed for advanced mathematics students who have completed Calculus. This course may be offered as a BCC CAPS Course.

Common Core Curriculum Standards for Math

http://www.corestandards.org

Textbook and Resource Materials

Calculus with Analytic Geometry

Authors: Ron Larson, Robert P. Hostetler, and Bruce H. Edwards

Publisher: Houghton Mifflin Company

Copyright: 2006

1. Calculus Exploration (ISBN 1-55953-311-0

Foerster

Key Curriculum Press

2. Exploring Calculus with Geometric Sketchpad (ISBN 1-55953-535-0)

Clements, Pantizzi, Steketee

Key Curriculum Press

3. Connecting Mathematics with Science (ISBN 1-55953-538-5)

Lyublinskaya

Key Curriculum Press

4. Fast Track to a "5", Preparing for the AP Calculus AB and BC Examinations

Sharon Cade, Rhea Caldwell, Jeff Lucia

McDougal Littell

5. Teaching AP Calculus

Lin McMullin,

D&S Marketing Systems, Inc

6. Preparing for the Calculus AP Exam

Finney, Demana, Waits, Kennedy

Addison Wesley

7. Barron's How to Prepare for the AP Calculus Exam,

Hockett,

Bock

Course Objectives/Activities:

Objective 1:

Students will demonstrate knowledge of functions. They will be able to determine graphs in rectangular, polar, and parametric form, with or without a calculator, determine continuity, identify asymptotes and unbounded behavior. (Standards: F-IF:7a,b,c,d,e)

Activities to meet objective:

• Students will work in groups with cooperative learning to explore labs in Foerster's Resource as activators for review and new topics.
• Students will construct a personal handbook containing explanations of procedures for various commonly used functions on their graphing calculator.
• Students will complete worksheets on graphing functions of all types in all forms.

Assessment:

1) Students will demonstrate an ability to perform basic calculations on the graphing calculator.

2) Teacher made tests and quizzes.

3) AP, NCTM, and/or SAT II type problems.

Objective 2:

Students will demonstrate a familiarity with the definition of a limit. They will be able to approximate limits from tables or graphs, and evaluate limits by methods including L'Hopital's Rule. (Standard: F-IF:4)

Activities to meet objective:

• Students will create a graphic organizer showing procedures for evaluating limits.
• Students will use graphing calculators to approximate limits or show the non-existence of limits.
• Students will use cooperative learning groups to explore labs in Foerster's Resource book for further explorations on limits.

Assessment:

1) Students will do a self-assessment of their ability to take limits by using graphing calculators to verify limits obtained with algebraic techniques.

2) Students will define orally and by written explanation, procedures used for evaluating a limit.

3) Teacher made tests and quizzes.

4) AP, NCTM and/or SAT II type problems

Objective 3:

Students will demonstrate familiarity with the concept of a derivative and be able to evaluate the derivatives of algebraic, transcendental, parametric, polar and vector functions. They will be able to apply that knowledge to analyze curves given in function, parametric, polar, and vector form, solve optimization problems and related rate problems, solve differential equations with slope fields and Euler's method, and find zeros with Newton's method. (Standards: F-IF:6, N-VM:4a,b,c 5,a,b)

Activities to meet objective:

• Students will create a set of flash cards consisting of rules for differentiation, steps for solving optimization and related rate problems, and formulas for Euler's and Newton's Methods.
• Students will work in cooperative groups on labs from Foerster's book on the “Rubber Band Chain Rule Problem”, “Displacement and Acceleration from Velocity”, “Derivatives of Inverse Trig Functions”.
• Students will continue their handbook for finding derivatives at a point, graphing derivatives, finding zeros by Newton's Method and creating slope fields.
• Students will create a slope field by hand to provide a visual perspective for the solution of a given differential equation.

Assessment:

1) Students will give a written analysis of labs.

2) Students will students in groups will be given a list of problems to be differentiated. They will choose a formula

they think will work. Then the list will be passed to another group who will review and make corrections. The

list will then be passed to a third group who will differentiate using the formulas stated and a fourth group who

will check the work. The winners will be the groups who have correct lists of derivatives. This will assess

knowledge of appropriate use of derivative formulas.

3) Students will give a written explanation of the differences and similarities between slope fields and Euler's

Method as ways to solve differential equations.

4) Students will explain the solution of an application problem on how to differentiate a complex function.

5) Students will complete teacher made tests and quizzes.

6) AP, NCTM, and/or SAT II type problems.

Objective 4:

Students will be able to demonstrate an understanding of definite integrals as the limit on Riemann sums, the Fundamental Theorem, and properties including, additivity, and linearity. Students will be able to approximate definite integrals with trapezoidal rule, Simpson's rule, and rectangles, from a table of values. (Standard: G-MG: 1, 3 )

Activities to meet objective:

• Students will use Lab Resource Books by Foerster, Heyd and others to analyze various techniques for evaluating the definite integral.
• Students will continue their calculator handbook to include taking definite integral of a function and graphing the integral function.
• Students will complete the “Driving” group activity (Students will take a partner on a 20 minute trip in the car. At one minute intervals they will record the speed of the car. They will also record the distance traveled. They will come to class with their data. In class they will graph the data, and use the trapezoidal rule to approximate the distance traveled. They will then compare their distance to the actual distance as recorded by the car's odometer by giving the percent error.) to estimate the distance traveled from a table of velocity data.
• Students will complete the “Human Shadow” or the “land mass estimation” activity to approximate area.
• Students will use activities from Goldenberg and Greenwald to explore engineering and scientific applications of integration.

Assessment:

1) Students will give a written analysis of lab work.

2) Students will verify correct solutions from right, left and midpoint Riemann sums, trapezoidal and Simpson's rule approximations, and will use a graphing calculator to find exact area.

3) Students will prepare a written summary of science and engineering application problems.

4) Students will complete teacher made tests and quizzes.

5) AP, NCTM, and/or SAT II type problems.

Objective 5:

Students will demonstrate an understanding of the anti-derivative. They will be able to evaluate anti-derivatives of basic algebraic and transcendental functions and improper integrals using methods including: substitution, change of variables, parts, partial fractions, trigonometric substitutions, and tables of integrals. (Standard: G-MG:2)

Activities to meet objective:

• Students will continue creating flash cards on formulas for new methods of integration.
• Students will work together on sets of problems which can be done by more than one method.
• Students will develop a pattern for discerning the types of problems that can be done by tabular integration.
• Students will use a graphing calculator to determine whether improper integrals converge or diverge.
• Students will analyze which formulas from the table of integrals should be selected to integrate various functions.

Assessment:

1) Take home test on all types of integration.

2) Student groups will create problems which can be solved using more than one method.

3) Group made problems will be distributed to other groups and students will write out solutions using at least two

methods.

4) Teacher made tests and quizzes.

5) AP, NCTM, and/or SAT II type problems.

Objective 6:

Students will be able to apply the definite integral to solve separable differential equations as are found in exponential and logistic growth and decay problems. Students will use integrals as rates of change to give accumulated change in models for physical, social and economic situations. They will use definite integrals to find area of a region, volumes of solids with known cross sections, distance traveled and length of a plane curve when curves are given in function, polar, and parametric form. (Standards: F-IF:6 F-LE:1a,b 3,5)

Activities to meet objective:

• Students will work in groups to solve sets of real world application problems.
• Students will use graphing calculators to solve applied problems which require evaluation of definite integrals.
• Students will complete appropriate lab activities. For example, from Foerster's lab book, “Differential equations for compound interest”, “Differential equations for memory retention”.
• Students will continue to create flash cards for various applications of integration.
• Students will be given a rubric on how to assess a grade for an AP exam free response question and as a group the class will grade each other's papers.

Assessment:

1) Students will provide a written explanation of the best procedure for finding the volume of a solid of revolution.

2) Students will compile a written presentation on the details of the solution of the extended group project.

3) Graded take home assignments.

4) Teacher made tests and quizzes.

5) AP, NCTM, and/or SAT II type problems.

Objective 7:

Students will demonstrate an understanding of the concept of a series as a sequence of partial sums and use technology to explore convergence and divergence. Furthermore, they will demonstrate an understanding of a series of constants with the ability to apply appropriate convergence tests. (Standard: F-LE: 2)

Activities to meet objective:

• Students will create a graphic organizer to test for convergence of a given series.
• Students will use a graphing calculator to explore convergence and/or divergence.
• Students will continue to create flash cards for various convergence theorems.
• Students will do appropriate labs. For example, from Foerster's lab book, “Power series for familiar functions”, “A power series for a definite integral”, Introduction to the ratio technique”.

Assessment:

1) Give a written explanation of the differences among convergence tests.

2) Students will complete take home activity sheets or explorations.

3) Drill and practice sheets to hone discrimination skills.

4) Teacher made tests and quizzes.

5) AP, NCTM, and/or SAT II type problems.

Objective 8:

Students will demonstrate an understanding of Taylor series. They will be able to perform a Taylor polynomial approximation with graphic demonstration of convergence, construct a general Taylor and Maclaurin series for a given function at a given value of x, manipulate Taylor series by integration and differentiation term by term to form a new series from known series, discover the interval and radius of convergence for functions defined as power series, and evaluate the Lagrange error bound for Taylor polynomials. (Standard: F-BF:2)

Activities to meet objective:

• Students will expand their set of flash cards to include formulas for Maclaurin series for common functions.
• Students will use a graphing calculator to compare an elementary transcendental function with successive Taylor polynomial approximations.
• Students will expand their calculator handbook to include sum/sequence function.
• Students will use calculator to explore the interval and radius of convergence.
• Students will work in groups to explore appropriate lab activities – Foerster's book for example “Improper Integrals to test for convergence”, Introduction to error analysis”.

Assessment:

1) Students will use graphing to self assess the correctness of calculated Taylor series.

2) Students will evaluate transcendental functions at non-traditional values using simple and constructed Taylor series.

3) Students will complete take home activity sheets or explorations.

4) Teacher made tests and quizzes

5) AP, NCTM, and/or SAT II type problems.

Objective 9:

Students will demonstrate an ability to take partial derivatives of functions of two, three, and more variables and higher order partial derivatives. (Standards: F-IF:6 F-BF:1a)

Activities to meet the objective

• Students will work in groups to review partial derivatives and discover higher order partial derivatives.
• Students will extend their sets of flash cards to include notation, and order for the procedure.
• Students will do the project on “Moire Fringes” detailed on page 915 in Larson, Hostetler.

Assessments

1) Students will hand in completed review and explore new topics.

2) Teacher made tests and quizzes.

3) Students will do a web assignment on the history of the development of partial derivatives as well as discussing some of their practical uses in physics.

4) Students will develop a quiz of ten questions that would test their knowledge in the ability to take partial

derivatives of functions of two, three and more variables and higher order partial derivatives.

5) AP, NCTM, and/or SAT II type problems.

Objective 10:

Students will demonstrate the ability to take double and triple integrals and apply double integrals to the solution of volume problems. (Standards: F-IF:6 F-BF:1a)

Activities to meet objective

• Students will work in groups to review and explore new topics.
• Students will extend their sets of flash cards to include uses for double integrals, properties of double integrals, and steps in the process.
• Students will work in groups to complete the explorations on page 992 and 995 of the Larson book.
• Students will develop a quiz of ten questions that would test their knowledge in the ability to take double and triple integrals and apply double integrals to the solution of volume problems.

Assessments

1) Students will hand in completed group work.

2) Students will complete teacher made tests and quizzes.

3) Students will do a web assignment on the history of the development of double and triple integrals, as well as

discussing some of their practical uses in physics.

4) AP, NCTM, and/or SAT II type problems.

Objective 11:

Students will demonstrate the use of integrating factors to solve first order non-separable differential equations. (Standards: F-IF:6 F-BF:1a)

Activities to meet objective

• Students will work in groups to review and explore new topics.
• Students will work in groups to experiment with Logistic Models by using an activity worksheet on “Simulating the Spread of Disease” from Lyblinskaya's Connecting Mathematics with Science.
• Students will do written projects describing in their own words - how to recognize and solve differential equations by separation of variables, stating the test for determining whether an equation is homogeneous, and explaining the relationship between two families of curves that are mutually orthogonal.
• Students will complete the project on “Weight Loss” on page 440 in the Larson book.

Assessments

1) Students will hand in completed group work.

2) Students will complete teacher made tests and quizzes.

3) Students will hand in group experiments on Logistic Models.

4) AP, NCTM, and/or SAT II type problems.

5) Students will develop an appreciation of calculus as a coherent body of knowledge as a human accomplishment by doing a web assignment on the history of the development of calculus and its relation to some developments in physics and other scientific disciplines.

6) Students will develop a quiz of ten questions that would test their knowledge of integrating factors to solve first-order non-separable differential equations.

Content Outline/Timeline

I. Functions, Graphs and Limits (~ 10 days)

1. Asymptotes and unbounded behavior

2. Continuity of Functions

3. Analysis of graphs with graphing calculators

4. Delta-Epsilon definition of a limit

5. Calculating limits using Algebra

6. Estimating limits from graphs or tables of data

7. Evaluating limits using L'Hopital's rule

II. Computation and Applications of the Derivative of a Function (~ 35 days)

1. Basic rules for sums, products, quotients, chain rule & implicit differentiation

2. Derivatives of power, exponential, logarithmic, trigonometric & inverse

Trigonometric functions

3. Derivatives of parametric, polar & vector functions

4. Analysis of curves using first and second derivative tests

5. Analysis of planar curves given in parametric form, polar form and vector

form, including velocity and acceleration vectors

6. Optimization (Max/Min) & related rate problems

7. Slope fields & differential equations

8. Euler's method for solving differential equations

9. Newton's method

III. Integrals (~ 30 days)

1. Computation of Reimann Sums using left, right & midpoint sums

2. Reimann sums- Trapezoidal

3. Approximating the definite integral from a table of values

4. Limit of Reimann Sums

5. Fundamental Theorem of Calculus

6. Graphical analysis of functions using the Fundamental Theorem

7. Basic properties such as Additivity and Linearity

8. Anti-derivatives of basic functions, i.e. Trig, e, ln, algebraic functions

9. Anti-derivatives using substitution and change of variable

10. Anti-derivatives using parts, partial fractions, trig. substitution and

tables of integrals

11. Evaluation of improper integrals

End of Semester One…approximately 4 days of review for Midterm Exam

IV. Applications of Integrals (~ 20 days)

1. Solving separable differential equations and using them in modeling and

exponential growth and decay problems

2. Solving logistic differential equations

3. Using the integral as a rate of change to give accumulated change as a model

for physical, social or economic situations

4. Finding the area of a region, including polar curves

5. Finding the volume of a solid with known cross section

6. Finding distance traveled by a particle and the length of a curve including

parametric form

V. Polynomial Approximation and Series (~ 35 days)

1. Sequences of partial sums

2. Convergence of the limit of partial sums

3. Using technology to explore convergence or divergence

4. Geometric series with applications

5. Harmonic series

6. Alternating series with error bound

7. Series and the relationship to improper integrals

8. Convergence of p-series

9. Ratio test for convergence and divergence

10. Comparison of series to test for convergence or divergence

11. Taylor Polynomial Approximation with graphical demonstration of convergence

12. The general Taylor series centered at x = a

13. Maclauren series for the functions: e ^x , sin x , cos x,

 

14. Formal manipulation of Taylor series including differentiation, anti-differentiation and the formation

of new series from known series

15. Function defined by power series and radius of convergence

16. Lagrange error bound for Taylor polynomials

VI. Review for BC Exam (~ 20 days)

VII. Topics to be covered after the BC Exam (~ 20 days)

1. Functions of two variables

2. Functions of three or more variables

3. Higher order partial derivatives

4. Double integrals and volume

5. Triple integrals

6. Separation of variables in first order differential equations

7. First order non separable differentiable equations

End of Semester Two…approximately 4 days of review for Final Exam