Lesson from September 17

Question of the Day:

Find the equation of the horizontal asymptote for the given function:

Factor the x^4 out of the numerator.

Multiply the entire equation by 1 by mutliplying the numerator and denominator each by 1 divided by the highest degree of x in the denominator.

Distribute the 1/(x^2). Also, recognize that the (2/x^4) in the numerator has been replaced by 0 because it approaches 0 as x gets closer to infinity.

Recognize that (x^2/x^2) in both the numerator and the denominator are equal to 1. Also, recognize that (4/x^2) has been replaced by 0 because it approaches 0 as x gets closer to infinity.

Simplify.

Therefore, the horizontal asymptote is y= -1.

Now onto the lesson. We learned the Intermediate Value Theorem and it's Corollary, and to do so, we performed the following instructions.

-On a graph, draw a function between two endpoints

-On that line, choose two points 'a' and 'b', and label them.
-Find f(a) and f(b).

-Define any point W between f(a) and f(b).

Intermediate Value Theorem:

Hypothesis:
-Suppose f is continuous on the closed inverval [a,b].

-W is any number between f(a) and f(b).

Conclusion:

-There exists as solution within [a,b] such that f(c)=W.

-There is at least one x value for W, your chosen y value.

-If f(x) is continuous, it can't skip any values.

Corollary to the Intermediate Value Theorem:

Hypothesis:

-Suppose f is continuous on a closed interval [a,b].

-f(a) and f(b) have opposite signs.

-[This is true of the above graph].

Conclusion:

-There is at least one number within [a,b] such that f(c)=0.

-The function must cross the x axis at least once, so there is at least one zero.

So, that's about it. Hurrah for finally figuring out how to work this thing. If anybody has any questions, come talk to me and I'll try to explain it better. Good luck studying for the quiz.

The Work of the Day (September 13, 2007)

Hey guys we sure missed you today! I hope that you both are feeling better and i hope you get this blog in time to make use of it. I know you both are dying to get this info... Love you guys and see you tomorrow! = )
-Max Chellemi

Here is the Question of the day!

True or false...sorry Rachel ; ) hehe

The function f(x) = cos(x) - sin(x) is increasing in


Lets analyze the question. We need to be able to visualize the graphs of sin(x) and cos(x). We also need to know that we are looking to see that the value of the output {f(x)} is increasing in value between the given parameters. Remember when the function is said to be increasing there may be no decreasing values in the output of the function between the given parameters.

Ok we know that the graph of sin(x) looks like

We also know that the graph of cos(x) looks like








Ok sorry for the layout issues but you don't know how hard this is...or how slow i am haha but i am trying. Ok so now that we have the graphs we can start tho think about this.

alright. We don't care about any part of the function that has an x less than zero or greater than π/2. We observe that the two functions cross at pi/4 and are the farthest apart as we approach π/2.

Graphically we can already see that the function doesn't increase the entire time but now we must examine it numerically to solidify our argument. Lets pick two easily identifiable points and check our hypothesis with a table.

x	f(x) = cos(x) - sin(x)
0	1
π/4	0

We need go no further ladies. As clearly indicated in the table above we increased the "x" value and the "y" value decreased, ergo the function is not solely increasing between

Today’s lesson talked about the Corollary to the Theorems that we received yesterday concerning continuity (the three things). The theorem reads: suppose “g” is continuous at “a” and “f” is continuous at g(a), then the function is continuous.

I am having some real trouble conjuring up the graph right now but i will try and get it up in a little while along with an explanation.

I hope this was helpful for you guys! Get well! I mean it!
Night
-Max Chellemi

Limits: Are They Your Friends?

One of the purposes of the blog is to provide a medium through which you might express your concerns, anxieties, and triumphs, free from the time pressure of our 55-minute slot. After ample time for reflection, you may find questions or understandings crystallizing in a manner that they often would not do during class.

We are now well under way in our opening foray into calculus. I am interested to hear how you have been coming to grips with limits and our four-fold analysis of them. Please respond to the following prompts but also feel free to add other thoughts that may be of help or interest to your classmates.

In regard to limits:
With what ideas, if any, have you been struggling? Describe any confusion and how you have been making sense of it.

Which representation most appeals to your understanding, numerical, graphical, or analytical? Explain.

Share any "Aha!" that you have had or connections that you have made with your previous mathematical understanding.

Ask a question, if you have one, about something for which you still need clarification.


Please comment before Tuesday's class.