Sequences, continued

Today, we learned another way to write sequences. In addition to the closed form we learned in class yesterday, we also learned the recursive definition for sequences. The recursive definition allows you to find the next number in the sequence one by one. It consists of two equations: the initial value for a (a₁= some constant) and then the equation for a_n as a function of a_n-1 (a_n= a_n-1 with some operation.)


Also, we learned how to find the sum of a sequence of terms. In the examples in the notes, the "n=" term underneath the Σ tells us the starting value for n. The term to the right of the Σ tells us the operation to preform for each sequential n value. The number above the Σ is a constant, which tells us the final value of n to include.
(In the 1st example, the number above Σ is 5. Since the operation given is n², the final number we include should be 5² = 25.)

Notes 3/29

Today in class we learned about sequences. A sequence is an ordered set of mathematical objects, like numbers. We have seen sequences before, since they are like pattern recognition where you must find the next set of numbers. For instance, the next three numbers in the ordered set 3, 6, 9, 12 are 15, 18, 21. There are different types of sequences, like linear, quadratic, exponential, and factorial. There are examples of each on the posted notes.

Today in class we learned about using DeMoivre's Theorem with roots. We learned that the root affects how many answers you should look for (for instance, square root has 2 answers, and cube root has 3). We also learned that in order to find the other answers after using DeMoivre's Theorem, you have to divide 360 (or 2 pi for radians) by the number of answers that you need and add that number to the theta you had until you have the number of answers you need. After that, the answers will start to repeat.

Notes 3/22

Today we both reviewed material from last week and covered a bit of new material. The stuff we reviewed was converting complex numbers to trig form (which can be done by drawing a triangle to find the r and theta). Next we learned about multiplying and dividing complex numbers in trig form: we use certain formulas (which are found boxed at the bottom of the first notes page and on the second page) and plug in the respective Rs and thetas. We found that when multiplying, the Rs multiply and the thetas add, and when dividing, the Rs divide and the thetas subtract. We also covered complex numbers to a power where we saw that Demoirvre's Theorem could be used.

March 4 Notes: Cardioids and Limaçons

Today in class we learned about the polar graphs of Cardioids and Limaçons.

Polar Graphing of Circles and Roses

We covered graphing of two main different shapes today using polar equations. First, we covered the main equation for graphing a circle, using sin and cos, depending on its orientation. Second, is the main equation for graphing a rose and what information can be taken from the polar equation about the rose. At the top is just a diagram and equation used for the last problem on the yesterday's homework.

In class today, we explored various methods used in converting polar equations to rectangular equations and rectangular equations to polar equations. Our previous notion about the importance of mastering trigonometry identities was reaffirmed, as solutions for problems often involved these identities. Rather than being presented as a new topic all together, the notes served as an additive to yesterday's notes, using similar/the same techniques to serve slightly different purposes.

Polar Graphing (3/1)

Today in class, we learned about polar graphing. The points are written as (r (the directed distance from the origin, θ (the angle drawn in standard position)). An example would be (4, 60°). We also learned how to convert from polar to rectangular by using the formula x=rcosθ and y=rsinθ. For example, you would insert the formula into (4, 180°) to become (4cos180°, 4sin180°) which would solve to (-4, 0). Finally we learned how to convert from rectangular to polar. You would use the formula, x^2 + y^2 = r^2 and y/x = tanθ, and there may be more than one solution.