August 31 - The derivative continued

Today introduced two new concepts - one is fairly minor but the other is more significant.

The first concept is notational. Students know that if I'm referring to a function f(x), then its derivative is written f ' (x). New for today is the idea that if I'm referring to a function y, then its derivative is written dy/dx. I do not care for the notation y', though note that your book will use it on occasion. I think it's sloppy and doesn't impart as much information as is needed.

The second concept is the derivative of sin x. We worked out the derivative using the definition as well as a trig ID, some algebra and our known trig limits. Once sin x has been differentiated, the others follow quite nicely. For now, you only need to know:

f(x) = sin x, then f ' (x) = cos x

and

f(x) = cos x, then f ' (x) = –sin x

The proof of the derivative is shown in my notes below.

August 30 - The derivative

So while today should be fairly momentous as we begin talking about the derivative, it is in fact a review of concepts previously seen at the end of last semester in pre-calculus.

The most important concern is that students remember that the derivative is defined as the limit of the slope between two points. That's it. It's the limit of the difference quotient as the delta x approaches zero.

We began today to study the methods for finding the derivative of a given function. I will continually refer to these methods as "shortcuts" since they do not involve limits at all. I want you to do your best though to always keep in mind the fact that limits are in play however.

The definitions of the derivative, the initial shortcut rules and a couple of examples are included in my notes below.

August 26 - Unbounded behavior

It's very exciting I think in math class to get to use the infinity sign ∞.

Infinity is a cool, confusing, complicated idea. In many ways, capturing what is meant by both the infinite (very large) and infinitesimal (very small) is what calculus is all about.

Today we talked about using ∞ and -∞ to describe when limits do not exist. Often you can but sometimes you can't. Any time direct substitution leads to 1/0, infinity is in play. The hardest part for students is determining whether the expression will be positive or negative.

Notes from today are below.

August 25 - One Sided Limits, Continuity

After a second day of practicing evaluating limits, today focused on two smaller (but still important) ideas.

The first idea is that of the one-sided limit. The limit normally asks you to consider the behavior of a function for x values that are a little less than and a little greater than some value c. Sometimes this results in a limit not existing because of different left/right behavior.

Evaluating a one-sided limit is just like evaluating a normal limit. However, in the instance where behavior is different on the left and right sides of x = c, the one-sided limit simply asks for the particular side. If it is lim x -> c+, then you describe the right side behavior. If it is lim x -> c-, then you describe the left side behavior.

Continuity is an important idea in calculus because our main tools are only guaranteed to work when we have a continuous function. (In fact, at some point in the near future, we will worry about the notion of differentiability. This includes continuity but is in fact more restrictive).

Continuity is defined in calculus for a function at a point. Any math student can intuitively look at the graph of a function and determine whether it is continuous or not. If it has any holes, breaks, or gaps in it, it isn't continuous at that point. You will mainly be concerned at being able to cite the reason why the function is not continuous.

Notes from today are below.

August 23 - Limit sin x / x as x->0

We spent some time talking about the true/false problems from the homework on Friday. I think that the issue there is getting comfortable with If –, then – statements perhaps more than the actual limits concepts involved.

The main topic for today was the limit as x->0 of sin(x) / x. See the diagram below:


Angle theta is drawn in standard position on the unit circle. Thus the dashed red line has length sin(theta) (since the hypotenuse is 1) and the thick green arc has length theta (Arc Length = Radius * Central Angle and the radius = 1). Clearly the arc has greater length than the triangle side, but as the angle theta gets smaller, the side of the triangle and the length of the arc get closer to being the same length. Or alternatively, if their lengths are growing similar, the ratio of the lengths is approaching 1. Thus an argument that as theta->0, sin(theta) / theta -> 1

We worked some examples to illustrate how to work problems with trig limits. Those are copied below:

August 20 - Limits continued

We started class today with a quiz on solving trig equations.

I covered a number of topics related to limits today and I'm hoping that most if not all of this was review. We worked some examples about limit properties (basically limits behave in a pleasing, intuitive way) and then talked about algebraic techniques for evaluating limits.

As a general rule, the first step in evaluating a limit is to do direct substitution. If this yields a value, then often that is the limit (nice functions). More often however, direct substitution gives a result of 0/0 which is an example of an indeterminate expression. For the student, this implies there is more work to do in order to evaluate the limit or to conclude that the limit doesn't exist.

We worked two examples - one showing the factor/cancel algebra and the other showing the multiplying by the conjugate algebra.

Finally, I reminded students that there are two trig limits with which they should be familiar. They are:

lim ((sin x) / x) = 1
x->0

and

lim (1 - cos x) / x = 0
x->0

We will talk about proofs of these on Monday.

There will be a limits quiz on Monday.

Aug 19 - Limits introduction

We talked about limits today in a very informal way. The main idea that you need to remember is that the statement

lim f(x) = L
x->c

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

What "close to" means is not well defined here. There is analysis that we will talk about that makes more specific the intuitive idea of "close to," but really for our purposes in calculus, the intuition should be fine.

Given the graph of y = f(x) here (click for an enlargement) :


Evaluate the limit of f(x) for each of the following:
x->0, the limit is 1
x->-3, the limit is 3
x->1, the limit is 2. this limit problem is somewhat bothersome b/c of the presence of the point at (1, 1). understand that the limit only refers to points "near" x = 1, not necessarily "at" x = 1.
x->-1, the limit does not exist due to different left/right behavior

Note that if you were asked for f(1), you could say f(1) = 1. If you were asked for f(-3), you would say that f(-3) is undefined.

Recall that there are three reasons for which a limit does not exist

• different left/right behavior
• unbounded behavior
• oscillating behavior

We will investigate these more tomorrow.