May 5 - Shortcuts to the Derivative

If the main concept yesterday (Tuesday) was the definition of the derivative, today's main idea was how to best calculate a derivative. It is important that you always keep in mind that the derivative is defined to be the limit of the slope between points, but we will rarely (never) use the limit definition to find the derivative of a function.

For now it's all about the Power Rule. Examples are included of how to apply the Power Rule even when it isn't immediately obvious.

Thanks again to Rachel for providing her notes for today.

May 4 - Slope of a curve

The main concept for today was the definition of the derivative. Imagine two points on a curve - call them (x, f(x)) and (c, f(c)). Calculate the slope between those points (change in y over change in x) and then take the limit of that expression as x -> c (which implies that the two points "move" ever closer together). The value of the limit can be interpreted as the slope at that point (c, f(c)).

Thanks to Rachel for her notes for today.

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.)