This will be short due to pressing time constraints!

Today in 121, we looked at the ways limits can screw up. These include the presence of jumps in the graph, vertical asymptotes, and too much oscillation. We could also have a limit not exist if the left-hand and right-hand limits both exist at our point but don't agree. Then we began talking (as briefly as possible) about the formal definition of a limit.

Tomorrow we will finish our look at the formal definition of limit and prove the existence of a couple of easy limits. Then we'll start talking about "limit shortcuts and rules." You'll have earned it by then.

We spent today talking more about limits. We showed how to make tables of function values as we approach a particular x-value of interest. We noted that for the polynomial, we could have just substituted the x-value of interest and gotten the right limit value. It turns out that this approach always works for limits involving polynomial functions.

We can often try the same technique on limits for rational functions, as long as we stay away from "bad" x's. If we approach an x-value that makes the denominator zero, our only hope is to factor the numerator and denominator of our rational function and hope for some cancellation. If we cannot both factor and cancel, the limit will either not exist or be infinite.

Thursday, we'll talk about the ways limits can screw up and the formal definition of limit. We'll also prove the existence of a couple of easy limits. See you then.

Today was the end of the "review period." We answered some questions on the homework from Section 1.6 and then began talking about the difference between algebraic functions and transcendental functions. Two types of transcendental functions are the exponential and the logarithmic functions. These are discussed in Section 1.7. We reviewed the relevant properties of each.

We then briefly recalled the trigonometric functions. We recalled the SOHCAHTOA version of sine, cosine, and tangent, and then expanded to the unit circle. Using the circle, we were able to show the periodic properties of sine and cosine.

On Wednesday, we move into Chapter 2, talking about limits of functions.

Today we kept the lecture short and sweet due to the quiz. We reviewed the idea of inverse functions, functions that "reverse" the action of a given function. We recalled that the functions with an inverse were exactly the one-to-one functions. Just as a graph is the graph of a function if it passed the vertical line test, a function is one-to-one if and only if its graph passes the horizontal line test. (Those are the tests that say a graph is the graph of a function (a one-to-one function) if no vertical (horizontal) line touches the graph in more than one point.)

We also recalled how to find a formula for the inverse of a one-to-one function. (It can't always be done due to the algebra difficulty.) We also recalled how to graph an inverse, even if we don't have a formula in hand.

Then we took the quiz. Friday, we'll talk about exponentials, logarithms, and maybe even trig functions.

Today in Calculus I, we talked about how to make new functions from old ones. We did this in two ways. Graphically, we recalled all the ways we can transform a graph through shifting, stretching, and squashing. We then talked about how adding, subtracting, multiplying, dividing, or composing two functions works and how to find the domains of such combined functions.

Thursday, we'll talk about inverse functions, and recall exponential and logarithmic functions.

Monday in Calculus I we went over the syllabus, homework schedule, grading policies, and expectations for the semester.

We then talked about a list of "Things I Think You Know," which included the topics from Sections 1.1 through 1.5 of our book. If you haven't looked those sections over to check for knowledge gaps please do so before the end of add/drop.

The goal for Wednesday will be to go over the material in Section 1.6 on algebra of functions, and possibly start Section 1.7 on exponentials and logarithms.

See you then!