Thursday, 3-15: Today everyone has had a chance to complete the polar introduction and the assignment is to work in the online text. The assignment in parentheses for the 27th should have already been done, so that day is a review and I will find more appropriate problems for that day.
Monday, 3-19: Today we answered questions about writing polar equations in Cartesian form and vice versa. Then students investigated the structures and effects of the parameters of special polar graphs. Although the assignment includes book work, I will be checking the worksheet only tomorrow and would like students to complete the book work after we compare conclusions from today.
Tuesday, 3-20: Today we summarized the conclusions from the special polar shapes worksheet Special Polar Graphs:
1. Limaçons: Basic equations: r = a + b×cos (theta) and r = a + b×sin (theta)
The difference between the cosine and the sine curves is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis.
When a > b, the graph has no loop because the radius "a" is larger than the amplitude of the trig function that alters it.
When a < b, the graph does have a loop because at some angles, a - b will produce a negative radius, so points will be plotted "behind" the quadrant that the theta values predict.
If b < 0, the graph will be reflected over the x axis for a sine curve or over the y axis for a cosine curve.
When a = 0, the graph is a circle. :-)
2. Cardioids: Basic equations: r = a + a×cos(theta) and r = a + a×sin(theta) These are supposed to look like hearts...
The difference between the cosine curve and the sine curve is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis. Are all these graphs similar??
3. Lemniscates: Basic equations: r^2 = a^2×cos (2theta) and r^2 = a^2×sin(2theta) These are all "infinity shaped" graphs.
The difference between the sine and cosine graphs is that cosine graphs are symmetric wrt the x axis and the sine graphs are symmetric wrt the Cartesian line y = x (which is the polar graph theta = pi/4.)
4. Roses: Basic equations: r = a×cos(b×theta) and r = a×sin(b×theta)
The difference between sine and cosine graphs is cosine curves always have a petal on the positive x axis. Sine curves have a petal where the value of theta is (pi/b).
When b is even there are 2×b petals.
When b is odd there are only b petals because the graph traces over itself.
The variable a affects the length of the petals.
5. Lines: Basic equation for A×x + B×y = C in polar is r = C/(A×cos(theta) + B×sin(theta))
The slope is -A/B. The x-intercept is x = C/A. The y-intercept is y = C/B.
Then we looked at the difference between where polar graphs cross and simultaneous solutions of polar equations. Their assignment is a worksheet that has about 6 intersection problems and many other review problems. Here are answers to that worksheet.
***Students have asked if the book work is really necessary and my answer must be "yes". Although I haven't checked it, page 720 (the other assignment from Monday night) has a great section about finding maximum radii (a topic we mentioned today but didn't delve into) and some good remarks about symmetry. Although I don't plan on having you use all their method to test for symmetry, I will ask you to discuss the symmetry of these graphs so you need to look over this section too.
Wednesday, 3-21: Today we looked at symmetry, maximum radii and project information. The assignment is still to look at the polar form of conic sections, but we didn't do that in class and the particulars of that are not on the quiz. Below is a derivation of the formula for the polar form of a conic and Friday we will look at that in more detail. The quiz is still tomorrow but the test will be Tuesday. Quiz topics include:
Points:
Convert polar to Cartesian
Convert Cartesian to polar
Plot Polar points
"Deal with negatives" both radii and angles
Write other names for polar points
Equations:
Convert polar to Cartesian
Convert Cartesian to polar
Sketch polar equations of limacons, cardioids, lemniskates, roses, circles, lines
Analyze polar equations (including symmetry and writing equations of graphs)
Find simultaneous solutions of polar equations.
Below is also the picture of the homework to be turned in Friday. Complete the 3 missing equations needed to complete this figure and include the window (complete with theta) values that were needed as well.
Monday, 3-19: Today we answered questions about writing polar equations in Cartesian form and vice versa. Then students investigated the structures and effects of the parameters of special polar graphs. Although the assignment includes book work, I will be checking the worksheet only tomorrow and would like students to complete the book work after we compare conclusions from today.
Tuesday, 3-20: Today we summarized the conclusions from the special polar shapes worksheet Special Polar Graphs:
1. Limaçons: Basic equations: r = a + b×cos (theta) and r = a + b×sin (theta)
The difference between the cosine and the sine curves is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis.
When a > b, the graph has no loop because the radius "a" is larger than the amplitude of the trig function that alters it.
When a < b, the graph does have a loop because at some angles, a - b will produce a negative radius, so points will be plotted "behind" the quadrant that the theta values predict.
If b < 0, the graph will be reflected over the x axis for a sine curve or over the y axis for a cosine curve.
When a = 0, the graph is a circle. :-)
2. Cardioids: Basic equations: r = a + a×cos(theta) and r = a + a×sin(theta) These are supposed to look like hearts...
The difference between the cosine curve and the sine curve is that the cosine curves are symmetric wrt the x axis and the sine curves are symmetric wrt the y axis. Are all these graphs similar??
3. Lemniscates: Basic equations: r^2 = a^2×cos (2theta) and r^2 = a^2×sin(2theta) These are all "infinity shaped" graphs.
The difference between the sine and cosine graphs is that cosine graphs are symmetric wrt the x axis and the sine graphs are symmetric wrt the Cartesian line y = x (which is the polar graph theta = pi/4.)
4. Roses: Basic equations: r = a×cos(b×theta) and r = a×sin(b×theta)
The difference between sine and cosine graphs is cosine curves always have a petal on the positive x axis. Sine curves have a petal where the value of theta is (pi/b).
When b is even there are 2×b petals.
When b is odd there are only b petals because the graph traces over itself.
The variable a affects the length of the petals.
5. Lines: Basic equation for A×x + B×y = C in polar is r = C/(A×cos(theta) + B×sin(theta))
The slope is -A/B. The x-intercept is x = C/A. The y-intercept is y = C/B.
Then we looked at the difference between where polar graphs cross and simultaneous solutions of polar equations. Their assignment is a worksheet that has about 6 intersection problems and many other review problems. Here are answers to that worksheet.
***Students have asked if the book work is really necessary and my answer must be "yes". Although I haven't checked it, page 720 (the other assignment from Monday night) has a great section about finding maximum radii (a topic we mentioned today but didn't delve into) and some good remarks about symmetry. Although I don't plan on having you use all their method to test for symmetry, I will ask you to discuss the symmetry of these graphs so you need to look over this section too.
Wednesday, 3-21: Today we looked at symmetry, maximum radii and project information. The assignment is still to look at the polar form of conic sections, but we didn't do that in class and the particulars of that are not on the quiz. Below is a derivation of the formula for the polar form of a conic and Friday we will look at that in more detail. The quiz is still tomorrow but the test will be Tuesday. Quiz topics include:
Points:
Convert polar to Cartesian
Convert Cartesian to polar
Plot Polar points
"Deal with negatives" both radii and angles
Write other names for polar points
Equations:
Convert polar to Cartesian
Convert Cartesian to polar
Sketch polar equations of limacons, cardioids, lemniskates, roses, circles, lines
Analyze polar equations (including symmetry and writing equations of graphs)
Find simultaneous solutions of polar equations.
Below is also the picture of the homework to be turned in Friday. Complete the 3 missing equations needed to complete this figure and include the window (complete with theta) values that were needed as well.
Wednesday, 3 - 21 (Continued): Here are some sample quiz problems that I used in class last year with their solutions. Once more, the quiz is does not include question #6 which is to identify a conic, but I could ask you to write that equation in Cartesian standard form or I could ask you for the simultaneous solutions of that equation and the equation r = 3 (or r = sin(theta)).
Tuesday, 3 - 2 : Students took the polar quiz today and need to look at conics tonight!! The assignment is to READ from the online text Page 722 - 725 and work the problems from the Wednesday night's assignment. Also have the shamrock equations HW ready to turn in tomorrow.
Tuesday, 3 - 2 : Students took the polar quiz today and need to look at conics tonight!! The assignment is to READ from the online text Page 722 - 725 and work the problems from the Wednesday night's assignment. Also have the shamrock equations HW ready to turn in tomorrow.