Math Analysis CP, 2010



Math Analysis CP, 2013 Due Date 12/17/2013

Polar Project 100 points

The purpose of the Polar Project is to familiarize students with polar coordinates and polar equations. You will do graphing calculator investigations, solve simultaneous polar equations, create your own polar design, and research historical polar curves. You will complete this project in groups of 3 or 4 students. This project is worth 100 points and is due on Tuesday, December 17 at the beginning of the period. Ten points will be deducted for each day late.

The polar project consists of the ten assignments listed below. Each assignment is worth 10 points. You will be graded on completeness, accuracy, neatness, and format. The assignments marked with a * must be completed by each individual student. The remaining assignments are to be completed as a group. Assignments 1, 6, and 7 must be typed. You will turn in:

a) A ½" rigid (non-flexible) binder with all assignments in the order listed

b) A title page (This may be put inside a "view binder", but is not required)

c) A table of contents matching the order of the assignments

d) Each individual assignment * clearly labeled with student’s name

e) DO NOT put any pages into sheet protectors

Assignments for Polar Project

1. * Math Analysis Letter – Write a letter to a new student describing this course. Be specific with your comments and suggestions. This should be about 1 full page and it must be typed.

• What kinds of things have you learned?

• Is it similar to previous math courses?

• Is it easier or harder than other math classes?

• What advice would you give to a student at the beginning of the course?

2. * Textbook assignments – All graphs must be on drawn on polar graph paper (Found on Q).

i. P. 558 (1-3, 5 – 13, 15)

ii. P. 565 (1, 2, 11 – 19) Include a table of values (from the calculator) for each graph.

iii. P. 571 (4 – 12)

3. Graphing Investigation I – Print and complete the Changing Values of n Worksheet from Q.

4. Graphing Investigation II – Print and complete the Polar Curves Investigation from Q.

5. Systems of Polar Equations – Solve the following…Groups of three students must complete any six systems; groups of four students must do all eight.

• Solve each system algebraically and show your work.

• Write your answer as an ordered pair.

• Sketch a graph of each system.

• Plot and label the points of intersection on your graph.

|i. [pic] |ii. [pic] |iii. [pic] |iv. [pic] |

|v. [pic] |vi. [pic] |vii. [pic] |viii. [pic] |

6. Spiral of Archimedes – Write a brief (two or three paragraph) report on the Spiral of Archimedes.

• Include historical facts about Archimedes, the spiral, and why a polar curve is named after him.

• Sketch a graph of his curve (by hand) on polar graph paper and include its general equation.

• You must use at least two resources for your research and reference them using MLA format.

• This report must be typed and printed.

7. Other Curves – Choose two polar curves (three curves if your group has four members) from the list below.

• Write a brief report (2-3 paragraphs) on each curve.

• Sketch a graph of each curve (by hand) on graph paper and include its general equation.

• You must use at least two resources for your research and reference them using MLA format.

• This report must be typed and printed.

8. * Original Design – Create an original design using at least two polar equations on your graphing calculator.

• You must print the graph itself, the equations used, and your window settings.

• You may color or embellish your design after you have printed it.

9. * Polar Multiple Choice WS – You must print and complete Worksheet from Q.

• Your answers must be written on the printed worksheet in the space provided.

• You must show your work on the back or separate sheet.

• The worksheet will be graded for accuracy.

10. Can You Study Guide and Evaluation – Write a "Can You" Study Guide for this unit.

• List the topics a student should be able to do after completing the polar project.

• Include a question and an answer for each topic.

• Be sure each member of your group keeps a copy.

• Groups with four members must write a brief evaluation of the project, describing what you liked and disliked about the process. This should be at least half of a page.

List of Polar Curves

|Astroid |Bicorn |Cartesian Oval |Cassinian Ovals |

|Catenary |Cayley's Sextic |Cissoid of Diocles |Cochleoid |

|Conchoid |Conchoid of de Sluze |Cycloid |Devil's Curve |

|Double Folium |Durer's Shell Curves |Eight Curve |Epicycloid |

|Epitrochoid |Equiangular Spiral |Fermat's Spiral |Folium of Descartes |

|Freeth's Nephroid |Hyperbolic Spiral |Hypocycloid |Hypotrochoid |

|Kampyle of Eudoxus |Kappa Curve |Lame Curves |Lissajous Curves |

|Lituus |Neile's Parabola |Nephroid |Pear-shaped Quartic |

|Pearls |Plateau Curves |Pursuit Curve |Quadratix of Hippias |

|Rhodonea Curves |Right Strophoid |Serpentine |Sinusoidal Spirals |

|Talbot's Curve |Tractrix |Tricuspoid |Trident of Newton |

|Trifolium |Trisectrix of Maxlaurin |Tschirnhaus' Cubic |Watt's Curve |

Note: We will be in the library on 12/11, 12/12 and 12/13. You should plan on using this time to research your Polar Curves and print your original Polar Design.

Resources posted on Q (You should print these as soon as possible):

- Detailed grading rubric

- Notes

- Polar graph paper

- Graphing Investigation I

- Graphing Investigation II

- Polar Review Worksheet

POLAR PROJECT RUBRIC

|Assignment |Points Possible |Student 1 |Student 2 |Student 3 |Student 4 |

| | | | | | |

|0. Binder | | | | | |

|Title Page |5 | | | | |

|Table of Contents/Pages Labeled | | | | | |

|1. Letter to future student | | | | | |

|Reflective |5* | | | | |

|Typed, double spaced | | | | | |

|2. Textbook Assignments | | | | | |

|Complete |10* | | | | |

|Accurate | | | | | |

|3. Graphing Investigation I | | | | | |

|Neat and accurate graphs on polar grid |10 | | | | |

|Changes to “n” clearly described | | | | | |

|4. Graphing Investigation II | | | | | |

|Complete | | | | | |

|Neat and Accurate Graphs |10 | | | | |

|Questions correctly answered | | | | | |

|5. Systems of Equations | | | | | |

|Complete (6 or 8) | | | | | |

|Algebraic solutions accurate |10 | | | | |

|Graphs on polar grid, neat | | | | | |

|POI clearly labeled | | | | | |

|6. Spiral of Archimedes Report | | | | | |

|Typed, double spaced | | | | | |

|Historical facts provided |10 | | | | |

|Graph and equation provided | | | | | |

|2 references, MLA format | | | | | |

|7. Other Polar Curves Reports | | | | | |

|Typed, double spaced | | | | | |

|Historical facts provided |10 | | | | |

|Graph and equation provided | | | | | |

|2 references, MLA format | | | | | |

|8. Original Design | | | | | |

|At least 2 equations | | | | | |

|Graph, Eqn, and Window Printed |10* | | | | |

|Title | | | | | |

|9. Review Worksheet | | | | | |

|Complete |10* | | | | |

|Accurate | | | | | |

|10. Study Guide/Group Evaluation | | | | | |

|Complete |10 | | | | |

|Typed, double spaced | | | | | |

|TOTAL | | | | | |

| |100 | | | | |

WS- Graphing Investigation I

Changing Values of n

Graph the following equations on the polar grid provided using different colored pens or pencils.

Graph [pic] in blue.

Graph [pic] in green.

Graph [pic] in red.

Graph [pic] in yellow.

Verbally describe the changes to each curve as n changes:

Graph the following equations on the polar grid provided using different colored pens or pencils.

Graph [pic] in blue.

Graph [pic] in green.

Graph [pic] in red.

Graph [pic] in yellow.

Verbally describe the changes to each curve as n changes:

Graph the following equations on the polar grid provided using different colored pens or pencils.

Graph [pic] in blue.

Graph [pic] in green.

Graph [pic] in red.

Verbally describe the changes to each curve as n changes:

Graph the following equations on the polar grid provided using different colored pens or pencils.

Graph [pic] in blue.

Graph [pic] in green.

Graph [pic] in red.

Verbally describe the changes to each curve as n changes:

WS- Graphing Investigation II

Polar Curves Investigation

In this investigation, you will look at different equations and their graphs. Many of the pictures will look neat! Later, you will explore a design using your own equations. It will be helpful to have a grasp of the terms on your calculator.

Switch your calculator mode to POL (Polar Mode). Pressing Y= now takes you to a menu containing [pic]. Further, your calculator will no longer display an "X" when you press the [pic] button. Instead, it will display [pic]. So you will input an angle measurement and will get an output that is the length of a point to the origin. Thus, you are graphing the ordered pairs [pic].

Finally, your window will be extremely important. In general, your window settings should be as follows. Although you may use radians, it is recommended that you use degrees.

|[pic] Min: 0 or [pic] |X Min: -6 |Y Min: -4 |

|[pic] Max: 360 or [pic] |X Max: 6 |Y Max: 4 |

|[pic] Step: 2 or 0.03 Radians |X Scl: 1 |Y Scl: 1 |

For different problems, however, you may wish to change the max/min on the two axes to better see the graph.

The Rose

General Form: [pic] or [pic]

Graph: [pic] Graph: [pic] Graph: [pic]

Dist. Between Circles: 0.25 Dist. Between Circles: 0.5 Dist. Between Circles: 1

Complete the table for [pic] Complete the table for [pic]

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

So, what is the general shape of the Rose graph? _______________________________________________________________

In general, what does the a in [pic] do? ______

Graph: [pic] (Use the table below) Graph: [pic] Graph: [pic]

Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1

So in general, what does the b in [pic] do?

How does the graph change when the b value is even vs. odd?

Graph: [pic] Graph: [pic] Graph: [pic] (Use the table)

Dist. Between Circles: 0.25 Dist. Between Circles: 0.5 Dist. Between Circles: 1

How do the cosine Rose graphs differ from the sine graphs? ____________

Complete the table for [pic] Complete the table for [pic]

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

The Leminscate

General Form: [pic] or [pic] NOTE: You must take the square root of both sides since it is [pic].

Graph: [pic] (Use the table) Graph: [pic] Graph: [pic]

Dist. Between Circles: 0.25 Dist. Between Circles: 1 Dist. Between Circles: 1

What is the general shape of the Leminscate? _________________________________________________________________

How do the sine Leminscates differ from the cosine graphs? ______________________________________________________

_______________________________________________________________________________________________________

Now let’s see what happens when a is imaginary

Graph: [pic] (Use the table) Graph: [pic] Graph: [pic]

Dist. Between Circles: 0.25 Dist. Between Circles: 1 Dist. Between Circles: 1

So what does the a do in [pic] or [pic]? Be sure to discuss both real and imaginary values. _____________________________________________________________________________

Complete the table for [pic] Complete the table for [pic]

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

The Limacon

General Form: [pic] or [pic]

Graph: [pic] Graph: [pic] Graph: [pic]

Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1

Graph: [pic] Graph: [pic] Graph: [pic]

Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1

So, what is the basic shape (s) of the Limacon graph? ___________________________________________________________

_______________________________________________________________________________________________________

How do the sine and cosine graphs differ? ____________________________________________________________________

_______________________________________________________________________________________________________

In general, what do a and b in [pic] or [pic] do? ______

What is/are the requirements to have a loop in the middle versus just an indent?

Starting with [pic] can you find a value for a that does not make a loop or an indentation? ______________________

_______________________________________________________________________________________________________

Create a cosine equation where there is a loop in the middle:

Create a sine equation where there is only an indentation:

The Cardioid (A Special Limacon)

General Form: [pic] or [pic]

Graph: [pic] (Use the table) Graph: [pic] Graph: [pic] (Use the table)

Dist. Between Circles: 1 Dist. Between Circles: 1 Dist. Between Circles: 1

Why do you suppose we call this graph a Cardoid?

____________

In general, what does the value of a in [pic] or [pic] do? _______________________________________

_______________________________________________________________________________________________________

What, if anything, changes when the "+" is changed to a "-"?

Complete the table for [pic] Complete the table for [pic]

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

|[pic] |r |[pic] |r |

|0° | |210° | |

|30° | |225° | |

|45° | |240° | |

|60° | |270° | |

|90° | |300° | |

|120° | |315° | |

|135° | |330° | |

|150° | |360° | |

|180° | | | |

Worksheet #01- Polar Project Review

Write the letter of the correct answer on the blank. YOU MUST INCLUDE YOUR WORK!!!

1. Which is a value for [pic] if the points [pic] and [pic] are equivalent?

|[pic] |[pic] |[pic] |[pic] |

2. What is the graph of [pic]?

|a point |a vertical line |a horizontal line | |

3. Which is the polar equation for a circle with center at the pole and radius [pic]?

|[pic] |[pic] |[pic] |[pic] |

4. Which of the following is an equation of a spiral of Archimedes?

|[pic] |[pic] |[pic] |[pic] |

5. How many petals does the rose given by [pic] have?

|2 |8 |16 |4 |

6. What is the name of the classical curve represented by [pic] ?

|rose |cardiod |leminscate |limacon |

7. What is the solution of this system of equations?

[pic]

|[pic] |[pic] |[pic] & [pic] |[pic] & [pic] |

8. What are the polar coordinates of [pic]?

|[pic] |[pic] |[pic] |[pic] |

9. What are the rectangular coordinates of [pic]?

|[pic] |[pic] |[pic] |[pic] |

10. What is the polar form of [pic]?

|[pic] |[pic] |[pic] |[pic] |

11. Which of the following is equivalent to [pic]?

|[pic] |[pic] |[pic] |[pic] |

12. What is the simplest form of [pic]?

|[pic] |[pic] |[pic] |[pic] |

13. What is the polar form of [pic]?

|[pic] |[pic] |

|[pic] |[pic] |

14. What is the rectangular form of [pic]?

|[pic] |[pic] |[pic] |[pic] |

15. What is the product of [pic] and [pic]?

|[pic] |[pic] |

|[pic] |[pic] |

16. What is [pic]

|[pic] |[pic] |

|[pic] |[pic] |

17. What is [pic]?

|[pic] |[pic] |[pic] |[pic] |

18. What is [pic]?

|[pic] |[pic] |[pic] |[pic] |

19. What is [pic]?

|[pic] |[pic] |[pic] |[pic] |

20. Which of the following is a root of the equation [pic]?

|[pic] |[pic] |[pic] |[pic] |

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