| Credit- Degree applicable | | Effective Quarter: Fall 2020 | I. Catalog Information
| MATH 43 | Precalculus III: Advanced Topics | 5 Unit(s) |
| | (See general education pages for the requirement this course meets.) (Not open to students with credit in MATH 43H.)
Prerequisite: MATH 41 (or MATH 41H) and MATH 42 (or MATH 42H) (both with a grade of C or better); or a satisfactory score on Calculus Readiness Test within the last calendar year.
Advisory: EWRT 211 and READ 211, or ESL 272 and 273. Lec Hrs: 60.00
Out of Class Hrs: 120.00
Total Student Learning Hrs: 180.00 Hyperbolic functions, parametric equations, systems of equations and inequalities, vectors, lines and planes, sequences and series, polar coordinates, mathematical induction, and the binomial theorem. |
| Student Learning Outcome Statements (SLO)
| | | Analyze, investigate, and evaluate linear systems, vectors, and matrices related to two or three dimensional geometric objects. |
| | | Graph and analyze regions/curves represented by inequalities or trigonometric, polar, and parametric equations, including conic sections. |
| | | Analyze, develop, and evaluate formulas for sequences and series; Justify those formulas by mathematical induction. |
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II. Course Objectives | A. | Graph and analyze curves in polar coordinates |
| B. | Graph and analyze parametric equations. |
| C. | Explore matrices, matrix reduction and determinants in the context of solving systems of linear equations |
| D. | Solve systems of inequalities and systems of non-linear equations |
| E. | Perform operations with 3D vectors |
| F. | Explore equations of lines and planes in 3-space. |
| G. | Develop and use sequences and series |
| H. | Write proofs using mathematical induction |
| I. | Use the binomial theorem to calculate binomial coefficients and to expand binomial expressions |
| J. | Examine the logic of conditional and bi-conditional statements as they appear in mathematical statements |
| K. | Examine Hyperbolic functions, their graphs and verify and use common hyperbolic identities, and solve equations containing hyperbolic expressions |
III. Essential Student Materials | | Graphing calculator or computer software |
IV. Essential College Facilities V. Expanded Description: Content and Form | A. | Graph and analyze curves in polar coordinates |
| 1. | Study polar equations and convert between rectangular and polar forms
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| 2. | Graph polar equations and explore polar graph symmetries |
| 3. | Explore conic sections in polar form |
| 4. | Investigate application problems using polar coordinates such as, but not limited to |
| c. | Historical development and use of polar coordinates
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| B. | Graph and analyze parametric equations. |
| 1. | Find parametric forms of plane curves, including conic sections. |
| 2. | Convert between equations in parametric form and rectangular form. |
| 3. | Investigate application problems using parametric equations such as, but not limited to |
| c. | Historical development and use of parametric equations
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| C. | Explore matrices, matrix reduction and determinants in the context of solving systems of linear equations |
| 1. | Use Gaussian elimination to solve linear systems
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| a. | Systems with unique solutions
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| b. | Systems with infinite solutions
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| c. | Parameterization of solutions
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| d. | Identify inconsistent systems
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| 2. | Define and calculate the determinant of a matrix
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| 3. | Use Cramer's Rule to solve linear systems (optional)
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| 4. | Use determinants to analyze systems of equations
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| 5. | Study the algebra of matrices |
| 6. | Find the inverse of a matrix (optional) |
| 7. | Use matrix inversion to solve linear systems (optional) |
| 8. | Investigate application problems using matrices such as, but not limited to |
| a. | Business and economics such as production, inventory, revenue and profit |
| b. | Network analysis such as traffic flow |
| c. | Historical development and use of systems of equations and matrices in solving problems in the sciences
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| D. | Solve systems of inequalities and systems of non-linear equations
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| 1. | Solve systems of inequalities and represent solution regions graphically |
| 2. | Solve non-linear systems (optional)
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| E. | Perform operations with 3D vectors |
| 1. | Define the rectangular coordinate system in 3D using Right Hand Rule |
| b. | Identify the 8 octants and the position of ordered triples in 3D
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| c. | Plot lines and planes in 3D
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| 2. | Develop the distance formula in 3D and use it to
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| a. | Find the distance between ordered triples
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| b. | The midpoint between ordered triples
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| c. | Write the equation of a sphere
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| d. | Compute the magnitude of 3D vectors
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| 3. | Study the dot and cross products of vectors in 3D
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| c. | Study the properties of the dot and the cross products |
| 4. | Investigate application problems using vectors such as, but not limited to |
| a. | Static equilibrium problems |
| c. | Area and volume problems |
| F. | Explore equations of lines and planes in 3-space. |
| 1. | Write the equation of a line in 3D using a point and a parallel vector |
| 2. | Use scalar components of the vector equation of a line to write the equation of a line |
| 3. | Write the equation of a plane using a point and a normal vector
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| 4. | Use vector projection to find |
| a. | The distance from a point to a plane |
| b. | An angle between two planes |
| c. | Find the distance from a point to a plane.
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| G. | Develop and use sequences and series
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| 1. | Examine sequences and series of real numbers |
| 2. | Define arithmetic sequences and series and find an expression for their general terms |
| 3. | Find the sum of finite arithmetic and geometric series |
| 4. | Find the sum of infinite convergent geometric series |
| 5. | Investigate application problems using arithmetic and geometric sequence and series such as, but not limited to |
| a. | Falling objects and projectiles
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| b. | Simple and compound interest, depreciation and annuities |
| c. | Historical development and use of sequence and series in solving problems in the sciences |
| H. | Write proofs using mathematical induction |
| 1. | Recognize and apply the Principle of Mathematical Induction
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| 2. | Verify conjectures about mathematical patterns using Mathematical
Induction. |
| I. | Use the binomial theorem to calculate binomial coefficients and to expand binomial expressions
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| 1. | Calculate binomial coefficients |
| 2. | Expand binomial expressions using the binomial theorem |
| J. | Examine the logic of conditional and bi-conditional statements as they appear in mathematical statements
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| 1. | Explore the relationships between a conditional statement and its converse, inverse, and contrapositive |
| 2. | Explore the use of conditional and bi-conditional statements in mathematical statements, definitions, and theorems.
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| K. | Examine Hyperbolic functions, their graphs and verify and use common hyperbolic identities, and solve equations containing hyperbolic expressions
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| 1. | Define the six hyperbolic functions |
| a. | Find the domain of the six hyperbolic functions |
| b. | Sketch the graphs of hyperbolic sine, cosine and tangent functions |
| 2. | Establish common hyperbolic identities. |
| 3. | Establish formulae and graphs for the inverses of the hyperbolic sine, cosine and tangent functions |
| 4. | Solve equations containing hyperbolic and inverse hyperbolic expressions. |
| 5. | Solve application problems such as, but not limited to: |
| b. | The Brachistochrone problem |
VI. Assignments | A. | Required readings from text
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| B. | Problem solving exercises, some including technology |
| C. | A selection of homework/quizzes, group projects, exploratory worksheets |
| D. | Optional project synthesizing various concepts and skills from the course content |
VII. Methods of Instruction | | Lecture and visual aids
Discussion of assigned reading
Discussion and problem-solving performed in class
In-class exploration of internet sites
Quiz and examination review performed in class
Homework and extended projects
Guest speakers
Collaborative learning and small group exercises
Collaborative projects
Problem solving and exploration activities using applications software
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VIII. Methods of Evaluating Objectives | A. | Periodic quizzes and/or assignments from sources related to the topics listed in the curriculum are evaluated for completion and accuracy in order to assess student’s comprehension and ability to communicate orally and in writing of course content and provide timely feedback to students on their progress |
| B. | Projects (optional)
Projects may be used to enhance the student's understanding of topics studied in the course in group or individual formats where communicating their understanding orally through classroom presentation or in writing. The evaluation to be based on completion and comprehension of course content and the students shall receive timely feed back on their progress. |
| C. | At least three one-hour exams without projects, or at least two one-hour exams with projects are required. In these evaluations the student is expected to provide complete and accurate solutions to problems that include both theory and applications by integrating methods and techniques studied in the course. The student shall receive timely feed back on their progress |
| D. | One two-hour comprehensive final examination in which the student is expected to display comprehension of course content and be able to choose methods and techniques appropriate to the various types of problems that cover course content. The student shall have access to the final exam for review with the instructor for a period determined by college and departmental rules.
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IX. Texts and Supporting References | A. | Examples of Primary Texts and References |
| 1. | *Larson , Hostettler, Precalculus with Limits. Boston: Houghton Mifflin, 3rd edition 2014 |
| 2. | Connally, Hughes-Hallett, Gleason, et al. Functions Modeling Change, 5th Edition. New York: Wiley, 2011 |
| B. | Examples of Supporting Texts and References |
| 1. | Aufmann, Barker, Nation. Precalculus with Limits. Boston: Houghton-Mifflin, 2013 |
| 2. | Blitzer, Robert, Precalculus, 5th Edition, Prentice Hall, 2013 |
| 3. | Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics, Princeton, NJ: Princeton Univ. Press, 2010 |
| 4. | Kline, Morris, Mathematical Thought from Ancient to Modern Times, Vol. 1-3, 1972, New York and Oxford, Oxford University Press |
| 5. | Maor, Eli, Trigonometric Delights, Princeton, NJ, 1998 Princeton University Press |
| 6. | Maor, Eli, e - The Story of a Number, Princeton, NJ, 1994 Princeton University Press |
| 7. | The MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of St. Andrews, Fife, Scotland, http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html, http://www-groups.dcs.st-and.ac.uk/~history |
| 8. | Rouche, John J. The Mathematics of Measurements. Springer, The Athlone Press, London |
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