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Glen Ridge Public Schools –Mathematics Curriculum

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Course Title: CALCULUS

Subject: Mathematics

Grade Level: 9-12

Duration: FULL YEAR

Prerequisite: PRE-CALCULUS (Honors or College Prep) WITH A GRADE

OF B OR BETTER

Elective or Required: ELECTIVE

Mathematics Mission Statement

Since Mathematical and Computational thinking are an integral part of our lives and 21st Century learning, students must be actively involved in their mathematics education with problem solving being an essential part of the curriculum.  The mathematics and computer science curricula will emphasize thinking skills through a balance of computation, intuition, common sense, logic, analysis and technology. 

 

Students will be engaged and challenged in a developmentally appropriate, student-centered learning environment. Students will communicate mathematical ideas effectively and apply those ideas by using manipulatives, computational skills, mathematical models and technology in order to solve practical problems. 

 

To achieve these goals, students will be taught a standards-based curriculum that is aligned with the National Common Core Standards in Mathematics and the New Jersey Common Core Curriculum in Technology and 21st Century Life and Careers.

Course Description:

The Honors Calculus course is designed for those students with a solid foundation in algebra, geometry, and math analysis. Students in this class should possess an interest in studying advanced mathematical topics as well as a desire to spend time solving problems that are of a challenging nature. The course is meant to serve as an introduction to derivatives and their applications as well as integration and its applications. The course covers a majority of the topics also covered in Advanced Placement Calculus, but not as quickly or to the same depth. Therefore this course is equivalent to a one semester Calculus course at the college level.

The graphing calculator is used throughout the course to help students develop an intuitive feeling for concepts before they are approached through typical algebraic techniques. Although it is emphasized as a tool to illustrated ideas and topics, it is expected that students also become proficient in using the calculator to:

•        Find roots of an equation

•        Sketch functions in a specified window

•        Approximate the derivative at a point using numerical methods

•        Approximate the value of a definite integral using numerical methods.

The students are therefore required to supply their own graphing calculator and the teacher will very often use a graphing calculator view screen.

Author: Catherine McCarthy

Date Submitted: Summer 2012

TEXTBOOK: CALCULUS EARLY TRANSCENDENTALS SINGLE VARIABLE, Anton, Bivens, & Davis, Wiley, 2012

Calculus

I Prerequisites for Calculus & Functional Analysis

Approximate # Of Weeks: 3

Essential Questions:

• What are the essential concepts one must understand about functions before beginning the study of calculus?

• How can functions be studied numerically, graphically, through tables, and analytically?

• What is the significance of a domain and range to function behavior?

• How can one compare the graphs of piecewise, absolute value and composite functions?

• What is the relationship between the 6 trigonometric functions, their graphs, and the unit circle?

NJCCS: A-CED#1, A-REI #3, F-IF #1 & 2, F-IF#8a, F-BF#1c, F-IF#7B,

F-TF #.1, 2, &.5, 6, & 7

Upon completion of this unit students will be able to:

1. Define and develop the concept of a “function.”

2. Write domain and range for a function using interval notation.

3. Generate the graphs of equations and functions with a graphing calculator.

4. Set and apply appropriate viewing window when graphing functions on the calculator.

5. Use a graphing calculator to determine critical values of a function.

6. Analyze functions for even or odd symmetry.

7. Review characteristics of linear functions such as point-slope form of the equation and the relationships between parallel and perpendicular lines.

8. Find the composition of two functions.

9. Identify and analyze families of functions such as linear, power, polynomial, exponential, logarithmic, and trigonometric functions.

10. Recognize the characteristics of and graph special functions such as the absolute value functions and piecewise functions.

11. Apply the properties of transformations to translate the graphs of functions.

12. Define, describe, and use radian measure.

13. State and use the six basic trigonometric functions.

The following prerequisite topics should be taught in chapter 3 prior to teaching the derivatives of exponential and logarithmic functions

14. Define and describe exponential functions.

15. Compare exponential growth and decay functions.

16. Define a logarithmic function as the inverse of an exponential function.

17. Use properties of logarithms to solve equations.

18. Solve equations involving logarithmic and exponential expressions.

  

Interdisciplinary Standards ()

• Standard 9.1 21st Century Life & Career Skills

• Standard 8.1 Computer and Information Literacy

• Standard 8.2 Technology Education

• Standard 5.1 Science Practices

Activities:

• Students will read and study material presented in course textbook and then be challenged with questions about their reading through examples.

• Students will take notes on instructor’s lecture and participate in class discussions.

• Instructor will provide opportunity for both guided and independent practice.

• Homework assignments will be discussed to insure a good understanding of the prerequisites.

• Technology: the graphing calculator will be integrated into various exercises to help the student visualize the problems.

• Students will be asked to make conclusions after working through explorations scattered throughout the unit.

• Internet Activities: Interactive PreCalculus Review/Diagnostics

• Lab Activities: Hidden Behavior Lab, Investigating Transformations Lab both by Jim Rahn

• SmartBoard Powerpoint presentations

Enrichment Activities:

• Pendulum Lab by Jim Rahn

Methods of Assessments/Evaluation:

• Written quizzes

• Worksheets

• Responses to discussion questions

• Homework

• Classwork

• Verbal Assessment

• Think/Pair/Share

• Exit slips

•

Resources/Including Online Resources

• Teacher Webpage

•

• homepages/calculus_labs.htm

II LIMITS AND CONTINUITY

Approximate # Of Weeks: 3

Essential Questions:

• What is meant by a limit of a function?

• What are the graphical, numerical, and analytical ways to determine a limit of a function at a given value?

• How can a limit be used to determine the graphical behavior of a function?

• What is a continuous function?

• How does continuity depend on limits?

NJCCS: F-IF.#6, 7b, & 8a, F-BF #1c, & F-LE.#5

Upon completion of this unit students will be able to:

1. Define and describe a limit intuitively.

2. Find limits using a numerical, graphical, and graphing calculator approach (using tables and graphs)

3. Find limits using an analytical approach.

4. State and use properties of limits.

5. Distinguish between one-sided and two-sided limits

6. Compute finite limits as the domain values approach infinity (identify vertical and horizontal asymptotes).

7. State, describe, and apply the Sandwich Theorem..

8. Determine infinite limits as domain values approach a constant.

9. Determine end behavior models for functions.

10. Use visualization to determine limits as domain values approach pos/neg infinity.

11. Define and describe continuity at a point.

12. Identify intervals where a function is continuous.

13. Apply continuity criteria to algebraic functions.

14. Identify types of discontinuities.

15. Rewrite a removable discontinuity by extending or modifying the function.

16. State and use the Intermediate Value Theorem for Continuous Functions.

Interdisciplinary Standards ()

• Standard 9.1 21st Century Life & Career Skills

• Standard 8.1 Computer and Information Literacy

• Standard 8.2 Technology Education

• Standard 5.1 Science Practices

Activities

• Students will watch Tutorials for the Calculus Phobe: Limits

• Students will read and study material presented in course textbook and then be challenged with questions about their reading through examples.

• Students will take notes on instructor’s lecture and participate in class discussions.

• Instructor will provide opportunity for both guided and independent practice.

• Homework assignments will be discussed to insure a good understanding of the concepts presented.

• Technology: the graphing calculator will be integrated into various exercises to help the student visualize the problems.

• Students will be asked to make conclusions after working through explorations scattered throughout the unit.

• Online lectures: how to evaluate limits

• Internet activities: Interactive limit and continuity problems

• SmartBoard Powerpoint presentations

• Lab Activities: Investigating Limits Through Tables, Continuity Lab, Investigating Limits on the Calculator all by Jim Rahn

Enrichment Activities:

• Intermediate Value Theorem Lab by Jim Rahn

Methods of Assessments/Evaluation:

• Written quizzes

• Worksheets

• Responses to discussion questions

• Homework

• Classwork

• Verbal Assessment

• Think/Pair/Share

• Exit slips

Resources/Including Online Resources

• Teacher Webpage

• Handley Math Page Calculus Lesson Plans

•

• homepages/calculus_labs.htm

•

III DERIVATIVES

Approximate # Of Weeks: 8

Essential Questions:

• How can the limit of the average rate of change of a function lead to the instantaneous rate of change?

• How does one determine the slope of a function at a point?

• How does one find the derivative of a function?

• What are the conditions for differentiability?

• What are the various rules for differentiation and how are they applied?

NJCCS: A-CED #1, A-REI #3, F-IF # 1, 2, 7b, 7c, 8a, F-BF #1c, F-LE #5

Upon completion of this unit students will be able to:

1. Define, describe, and compute the average rate of change of a function.

2. Distinguish between average rate and instantaneous rate of change given the curve and a point.

3. Describe the concept of slope of a curve at a point.

4. Define and describe a normal line to a curve and determine its equation given the curve and a point.

5. Define and describe the derivative and use its notation.

6. Approximate derivatives numerically using the calculator or graphically.

7. Analyze the relationship between the graphs of f and f’.

8. Describe the 4 ways a derivative might fail to exist.

9. Describe how differentiability implies continuity.

10. State and use the Intermediate Value Theorem for Derivatives.

11. Differentiate positive integer powers, multiples, sums and differences of functions.

12. Differentiate products and quotients of functions.

13. Differentiate negative integer powers of x.

14. Use rules for differentiating trigonometric functions.

15. Differentiate a composite function.

16. Apply the “Outside-Inside” Rule

17. Apply the chain rule repeatedly when necessary.

18. Apply the power chain rule.

19. Define and describe an implicitly defined function.

20. Find derivatives using the process of implicit differentiation.

21. Solve problems involving tangents and normal lines.

22. Solve problems involving higher order derivatives.

23. Solve problems involving rational powers of differentiable functions.

24. Calculate the derivatives of exponential and logarithmic functions

Interdisciplinary Standards ()

• Standard 9.1 21st Century Life & Career Skills

• Standard 8.1 Computer and Information Literacy

• Standard 8.2 Technology Education

• Standard 5.1 Science Practices

Activities :

• Students will watch Tutorials for the Calculus Phobe: Derivatives

• Students will read and study material presented in course textbook and then be challenged with questions about their reading through examples.

• Students will take notes on instructor’s lecture and participate in class discussions.

• Instructor will provide opportunity for both guided and independent practice.

• Homework assignments will be discussed to insure a good understanding of the

concepts presented.

• Students will work on several exercises throughout the unit cooperatively to enhance their understanding of the process of differentiation.

• Technology: the graphing calculator will be integrated into various exercises to help the student visualize the problems.

• Students will be asked to make conclusions after working through explorations scattered throughout the unit.

• Lab Activities: Zooming In to Estimate Slope of a Function Lab, Discovering Derivative Relationships Lab both by Jim Rahn

• SmartBoard Powerpoint presentations

• Derivative Review Puzzle

• Internet Activities: Interactive derivative graphing activity; Definition of the derivative activity

•

Enrichment Activities:

• Exploration Lab on Derivatives by Jim Rahn

Methods of Assessments/Evaluation:

• Written quizzes

• Worksheets

• Responses to discussion questions

• Homework

• Classwork

• Verbal Assessment

• Think/Pair/Share

• Exit slips

Resources/Including Online Resources

• Teacher Webpage

• homepages/calculus_labs.htm

•

•

• Handley Math Page Calculus Lesson Plans

IV APPLICATIONS OF THE DERIVATIVE

Approximate # Of Weeks: 6

Essential Questions:

• How does implicit differentiation aid in solving related rates problems?

• How can derivatives be used to draw conclusions about extreme values of a function and the general shape of a function’s graph?

• How can optimizations problems be modeled algebraically using differentiation?

• How can a linearization model be used to find the zeroes of a function?

NJCCS: A-CED #1, A-REI #3, F-IF # 1, 2, 7b, 7c, 8a, F-BF #1c, F-LE #5

Upon completion of this unit students will be able to:

1. Write a strategy for solving related rates problems

2. Model a physical problem using related rates.

3. Solve related rates problems.

4. Distinguish between absolute and relative extreme values

5. Find extreme values by applying derivatives

6. State and use the Mean Value Theorem

7. Determine local extrema by applying the first derivative test

8. Determine concavity of a function by applying the second derivative test.

9. Identify all points of inflection by applying the second derivative test.

10. Apply the second derivative test for local extrema

11. Find all absolute extrema and points of inflection of the graph of f given only f’ and f”

12. Discuss a general strategy for max-min problems

13. Solve max-min problems from mathematics, business, industry and economics

14. Model discrete phenomena with differentiable functions.

15. Apply a linearization process to approximate functional values

16. Describe both verbally and in writing the steps of Newton’s Method.

17. Use the calculator to apply Newton’s Method of finding roots.

18. Estimate change with differentials.

19. Define and describe the indeterminate form 0/0, ∞/∞, ∞ * 0, ∞ - ∞

20. Use L’Hopital’s Rule to find limits involving indeterminate forms.

Interdisciplinary Standards ()

• Standard 9.1 21st Century Life & Career Skills

• Standard 8.1 Computer and Information Literacy

• Standard 8.2 Technology Education

• Standard 5.1 Science Practices

Activities:

• Students will read and study material presented in course textbook and then be challenged with questions about their reading through examples.

• Students will take notes on instructor’s lecture and participate in class discussions.

• Instructor will provide opportunity for both guided and independent practice.

• Homework assignments will be discussed to insure a good understanding of the

concepts presented.

• Students will work on several exercises throughout the unit cooperatively to enhance their understanding of the applications of derivatives.

• Technology: the graphing calculator will be integrated into various exercises to help the student visualize the problems.

• Students will be asked to make conclusions after working through explorations scattered throughout the unit.

• Class Activity: Derivative Matching Cards

• Lab Activities: Maximization Lab, Understanding the Relationship between a Function and its Derivative Lab, Curve Sketching Lab, Newton’s Method all by Jim Rahn

• Internet Activities: Interactive Derivative Puzzles and Applets; Maxima & Minima Examples; Related Rates and Min/Max word problems

• SmartBoard Powerpoint presentations

Enrichment Activities:

• Lab Activity: Motion along a line Lab by Jim Rahn

Methods of Assessments/Evaluation:

• Written quizzes

• Worksheets

• Responses to discussion questions

• Homework

• Classwork

• Verbal Assessment

• Think/Pair/Share

• Exit slips

Resources/Including Online Resources

• Teacher Webpage

• homepages/calculus_labs.htm

• Handley Math Page Calculus Lesson Plans

•

•

•

•

•

V THE DEFINITE INTEGRAL

Approximate # Of Weeks: 5

Essential Questions:

• How does instantaneous change accumulate over an interval to produce a function?

• How is an integral a representation of a summation?

• How can the definite integral be applied to finding area under a curve?

• How does the limit of a Riemann Sum relate to integration?

• How does the Fundamental Theorem of Calculus connect integral and differential calculus?

NJCCS: A-CED #1, A-REI #3, F-IF # 1, 2, 7b, 7c, 8a, F-BF #1c, F-LE #5

Upon completion of this unit students will be able to:

1. Approximate the area under the graph of a nonnegative continuous function by using rectangle approximation methods.

2. Interpret the area under a graph as a net accumulation of a rate of change.

3. Express the area under a curve as a definite integral and as a limit of Riemann Sums.

4. Define and describe a definite integral.

5. Evaluate a definite integral using area formulas.

6. State and use the Rules for Definite Integrals

7. Integrate numerically using the fnInt command of the TI-83 Plus graphing calculator.

8. Apply rules for definite integrals and find the average value of a function over a closed interval.

9. Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus.

10. Apply the Fundamental Theorem of Calculus.

11. Use the definite integral to find the average value of a function on a closed interval.

Interdisciplinary Standards ()

• Standard 9.1 21st Century Life & Career Skills

• Standard 8.1 Computer and Information Literacy

• Standard 8.2 Technology Education

• Standard 5.1 Science Practices

Activities:

• Students will read and study material presented in course textbook and then be challenged with questions about their reading through examples.

• Students will take notes on instructor’s lecture and participate in class discussions.

• Instructor will provide opportunity for both guided and independent practice.

• Homework assignments will be discussed to insure a good understanding of the

concepts presented

• Students will work on several exercises throughout the unit cooperatively to enhance their understanding of the definite integral.

• Technology: the graphing calculator will be integrated into various exercises to help the student visualize the problems.

• Students will be asked to make conclusions after working through explorations scattered throughout the unit.

• SmartBoard Powerpoint presentations

• Lab Activities: Investigating Riemann Sums Lab by Jim Rahn

• Internet Activities: Riemann Sum Applet; Visuals of Riemann Sums

Enrichment Activities:

• Lab Activity: Understanding the 2nd Fundamental Theorem of Calculus Lab by Jim Rahn

Methods of Assessments/Evaluation:

• Written quizzes

• Worksheets

• Responses to discussion questions

• Homework

• Classwork

• Verbal Assessment

• Think/Pair/Share

• Exit slips

Resources/Including Online Resources

• Teacher Webpage

• homepages/calculus_labs.htm

• Handley Math Page Calculus Lesson Plans

•

•

•

VI DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING

Approximate # Of Weeks: 4

Essential Questions:

• How can differential equations be used to predict the behavior of a function?

• What is the relationship between slope fields and differential equations?

• How can one reverse the chain rule in integration?

• What are some alternative techniques of integration?

NJCCS: A-CED #1, A-REI #3, F-IF # 1, 2, 7b, 7c, 8a, F-BF #1c, F-LE #5

Upon completion of this unit students will be able to:

1. Solve a first order differential equation by finding an antiderivative.

2. Find a particular solution of an initial-value problem

3. Define, describe, and sketch a slope-field.

4. Distinguish between a definite and indefinite integral.

5. Define and describe what is meant by a family of all antiderivatives of a function.

6. Use Leibniz notation to represent indefinite integrals

7. Stat and use properties of indefinite integrals.

8. Evaluate indefinite integrals by applying appropriate basic integration formulas.

9. Evaluate indefinite and definite integrals by using substitution

Interdisciplinary Standards ()

• Standard 9.1 21st Century Life & Career Skills

• Standard 8.1 Computer and Information Literacy

• Standard 8.2 Technology Education

• Standard 5.1 Science Practices

Activities:

• Students will read and study material presented in course textbook and then be challenged with questions about their reading through examples.

• Students will take notes on instructor’s lecture and participate in class discussions.

• Instructor will provide opportunity for both guided and independent practice.

• Homework assignments will be discussed to insure a good understanding of the

concepts presented

• Students will work on several exercises throughout the unit cooperatively to enhance their understanding of the definite integral.

• Technology: the graphing calculator will be integrated into various exercises to help the student visualize the problems.

• Students will be asked to make conclusions after working through explorations scattered throughout the unit.

• SmartBoard Powerpoint presentations

• Internet Activity: Slope-Field Applet

Methods of Assessments/Evaluation:

• Written quizzes

• Worksheets

• Responses to discussion questions

• Homework

• Classwork

• Verbal Assessment

• Think/Pair/Share

• Exit slips

Resources/Including Online Resources

• Teacher Webpage

•

SHOULD THERE BE MANY STUDENTS IN THE CLASS WHO ARE TAKING AP CLASSES, INCLUDE THE FOLLOWING UNIT DURING THE TWO WEEKS OF AP TESTING!

VII REVIEW FOR COLLEGE PLACEMENT TESTS

Approximate # of Weeks: 2

Essential Questions:

• What do I need to know when taking my college placement test?

Upon completion of this unit the student will be able to:

1. Review those algebra, trigonometry and pre-calculus concepts necessary to successful complete a college placement test.

Activities:

• Students will work on their own college placement test practice questions as well as those which can be found online from other colleges such as Montclair University, Trinity College etc.

Methods of Assessment:

• Students will be quizzed on the concepts presented in the practice placement tests completed.

VIII APPLICATIONS OF INTEGRALS

Approximate # Of Weeks: 3

Essential Questions:

• How can an integral be used to solve problems?

• How is an integral related to the net change made over time?

• How can integrals be used to estimate and calculate the area enclosed by a curve and an axis or by two or more curves?

NJCCS: A-CED #1, A-REI #3, F-IF # 1, 2, 7b, 7c, 8a, F-BF #1c, F-LE #5

Upon completion of this unit students will be able to:

1. Distinguish between displacement and total distance traveled.

2. Solve problems involving linear motion by applying definite integrals.

3. Find the area between curves by applying definite integrals.

4. Find the area between intersecting curves by applying definite integrals.

5. Find the area of a region whose boundary is defined by more than one function by applying definite integrals.

6. Integrate with respect to y to simplify certain area problems.

7. Combine the use of basic geometric formulas with definite integrals to simplify certain area problems.

Interdisciplinary Standards ()

• Standard 9.1 21st Century Life & Career Skills

• Standard 8.1 Computer and Information Literacy

• Standard 8.2 Technology Education

• Standard 5.1 Science Practices

Activities:

• Students will read and study material presented in course textbook and then be challenged with questions about their reading through examples.

• Students will take notes on instructor’s lecture and participate in class discussions.

• Instructor will provide opportunity for both guided and independent practice.

• Homework assignments will be discussed to insure a good understanding of the

concepts presented

• Students will work on several exercises throughout the unit cooperatively to enhance their understanding of the definite integral.

• Technology: the graphing calculator will be integrated into various exercises to help the student visualize the problems.

• Students will be asked to make conclusions after working through explorations scattered throughout the unit.

• SmartBoard Powerpoint presentations

• Internet Activities: Finding Areas Enclosed by Two Curves

Methods of Assessments/Evaluation:

• Written quizzes

• Worksheets

• Responses to discussion questions

• Homework

• Classwork

• Verbal Assessment

• Think/Pair/Share

• Exit slips

Resources/Including Online Resources

• Teacher Webpage

•

Primary Textbook:

Anton, Bivens, & Davis, Calculus Early Transcendentals Single Variable. John Wiley, Inc 2012

Supplemental Text:

Finney, Demana, Waits, Kennedy, Calculus Graphical, Numerical, Algebraic, Pearson Education Inc., 2012

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