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Polar Equations!!!!!! What are they good for?

Polar Equations!!!!!! What are they good for? Anything that orbits or involves circular motion (helixes!). There are some graphs in math that are best described in polar…take for example a rose curve….have fun sketching the curve in rectangular form with implicit piecewise functions!!!.

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Polar Equations!!!!!! What are they good for?

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  1. Polar Equations!!!!!! What are they good for? Anything that orbits or involves circular motion (helixes!). There are some graphs in math that are best described in polar…take for example a rose curve….have fun sketching the curve in rectangular form with implicit piecewise functions!!!

  2. Let’s go way back to the way back!!! WHAT THE HECK DOES THIS LOOK LIKE?

  3. Plot the following polar coordinate on your polar graph. *AP will either give you a sketch of the polar curve being analyzed or simply give you words…so your ability to see what you are finding the area or derivative of is crucial! I.O.W. – You best be great at graphing!!!

  4. Multiple representations of one point Cyclical nature of polar coords. a point can be represented in infinite ways…unlike rectangular coords. Example

  5. Recognition of identical location w/ diff. coords. will become important as we sweep out area in polar curves.

  6. Review of Coordinate Conversion y Can be useful for convincing ourselves of a polar curve’s shape and necessary for theorems and calculations involving calc. concepts r y x x polar axis (x – axis) Rect polar Polar Rect

  7. Convert each of the following rect. coords. to polar. Be mindful of the quadrant!!!

  8. Graphing Basic Polar Equations Through Conversion YOU SHALL BECOME PROFICIENT IN THE WISE WAYS OF: “FOLLOWING THE RADIAL ARM” AND “IMAGINING SWEEPING OUT THE CURVE”

  9. Graph r = 3 a.) Convert to rect. and graph. b.) Plot in GC in polar coords!!! c.) How does this make sense by “following the radial arm?” (thinking in terms of polar coordinates)

  10. Graph a.) Convert to rect. and graph. b.) How does this make sense by “following the radial arm?” (thinking in terms of polar coordinates)

  11. WHY THE FOCUS ON CONCEPTUAL AND ABILITY TO QUICKLY SKETCH POLAR GRAPHS???? You will be applying the concepts of i-gration & differentiation to polar graphs and you must have a great handle on polar curves in order to understand the conceptual and analytical components of the calculus of them!! I.O.W. – IMPORTANT TO KNOW OF WHAT YOU ARE TAKING THE AREA, ARC LENGTH, DERIVATIVE, ETC.

  12. One more basic graph… a.) Graph by first converting to rect. Form. b.) Does the graph make sense polarly? c.) Graph in your GC…make your theta – step = .01 Why is the top half graphed first? Any other obs?

  13. Good idea of what’s going on by sketching from . “FOLLOW THE RADIAL ARM” DON’T BE AFRAID TO GENERALIZE…YOUR CURVE DOES NOT HAVE TO BE PERFECT…BUT CLOSE TO THE ACTUAL . Convert the polar equation to rectangular form.

  14. What might you be able to generalize from the last two graphs that will aid in your polar curve sketching?

  15. Properties of Polar Curves Cosine functions are symmetric to the polar axis (x-axis) Sine functions are symmetric to the line perp. to the polar axis (y-axis) Polar axis

  16. Continue to use symmetry as you plot curves!!! a.) Ask yourself : “Self. What do I know about this curve before I plot it? b.) Plot on GC c.) Plot by hand

  17. You just sketched a rose curve!!! Sometimes called a petal curve!!! A couple of facts for you:

  18. Petal Curves If n is a positive even integer, the curve has 2n petals.

  19. If n is a positive odd integer, the curve has n petals.

  20. 1.) As you graph, look for zeros, maxes, and a couple in b/n…generalize…do not graph every point you can think of!! – I actually list all my info out first. 2.) Use your rules of symmetry and function recognition.

  21. GOAL Know your polar functions well enough that you have a clear idea of what a curve will look like before you graph it. FUNCTION RECOGNITION VISUAL!!! Study the curves and corresponding functions and their conditions that make them unique on p. 735!

  22. What causes the inner loop?

  23. Limacon Also called a cardiod

  24. 1.) Get a picture in your head first based on what you now know: a. Sym. about what axis? b. What should curve generally look like? c. What direction will curve face? Why? d. Inner loop? 2.) Then plot using the mental image and “convenient points”

  25. HW Add – On (Do without polar graph handout) For each curve listed below: 1. List all info you know about the graph. 2. Plot convenient points and sketch graph of curve using “mental image” and “following the radial arm” 3. Confirm your graph by plotting in GC

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