Notes from April 30 - Introduction to Limits

Today, we were introduced to limits. At the moment, it's a calculator-heavy concept, but we eventually will learn to do evaluate them without the calculator.

The actual definition of a limit will not be pleasing right now, but for now, we have an "intuitive" definition:

lim f (x) = L

x -> c

where for x values "close" to c, the y values are "close" to L.

April 12 - Proof by Induction

Proof by induction is a tricky little concept and it will take some time to figure it out. It really emphasizes algebra skills (of all things...) and the really disconcerting thing is that it is sometimes hard to recognize you're finished; it may not feel as though you've really done anything.

But.

It is a powerful tool in a mathematician's arsenal and if you continue to take math past calculus, you can most certainly expect to see it again.

Here are my notes from today:

These are the notes from today. We learned to evaluate the sumation of geometric sequences. That equation is on the first page. We also looked at examples of doing this, and I made a note that for the example with the 6th power, there were for complex roots that we DO NOT need to know.

March 31 - Arithmetic Sequences

We defined an arithmetic sequence today in class as a sequence in which there is a constant difference between terms. I gave my version of the formula for the nth term in an arithmetic sequence; note that it differs slightly from the formula in the book.

Thanks to Jaime for today's notes. :)

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.