September 21, 2010

So this is my first scribe post I have ever done because I was not in pre-AP calc last year, so hopefully I do this right.

Today in class the main things we did were as followed:

1. Warm Up
2. Review Quiz #2
3. Review Assignment #7
4. Preview of the Upcoming Test

Warm Up:

Today to start the class we opened the document Warm Up on our computers and spent about 30 minutes working on it. This warm up worked on continuous and discontinuous functions.

Question one on this warm up asked us to graph

for when k=1. I know that before today I did not know how to put piecewise functions into GeoGebra, so I figured that I would quickly show how to. In the input bar you type "if'" which will give you brackets that you can put the functions into, in the end it should look like this

which gave us a graph that looked liked

Question 2 then asked us what it means for a function to be discontinuous. A very simple way that I found to put it is that a discontinuous function is a function that when graphed will have breaks in it. If there is still any confusion on what it means for a function to be discontinuous click here.

Question 3 then asked:

after substituting 2 into both limits we got

once we got these values we knew that for f to be continuous at x=2

must be true.

For question 4 we needed to find the value for k that would make f(x) be continuous at x=2. This was very simple algebra after doing question 3. We found that k would have to equal (1/3) so that both limits equaled 3.

In question 5 it asked to find out the origin of the word "cusp" and why it is a good word to describe this graph. In the dictionary cusp is defined as a point or a pointed end, and its appropriate when talking about this graph because of the sharp point at x=2.

Question 6 then asked what we would say if someone asked if f(x) is increasing or decreasing at x=2, and because x=2 is such a sharp point it is neither increasing nor decreasing. It then asked what we could conclude about the derivative at x=2, and what we concluded was that the derivative at x=2 does not exist.

The class seemed to understand this warm up well I thought. A good question asked by Lange during class was how can you tell the difference between removable and non-removable discontinuity. A good explanation that I found is here. Basically if you can redefine the function it is a removable discontinuity, and if you cannot then it is a non-removable discontinuity.

Quiz Review:

For this quiz there were not too many questions, but one problem that we went over in class was number 3 which was

and what you had to do to solve it was to split it giving us:

after factoring this we were left with 1 x -1/7, which means the limit is -1/7.

For the rest of the quiz there were not many questions, but one thing that Mr. O'Brien did say was that a few of the problems people got wrong were very similar to homework problems, so people so pay more attention to their assignments if they want to do better.

Assignment #7 Review:

Next we went over a few of the questions on assignment 7 that we did not get around to last class.

We started with 5c which was:

with this function we had to find the limit using the idea of local linearity. Mr. O'Brien then showed us many ways in which we can use local linearity to find the limit. The first way was using GeoGebra. We put both functions into GeoGebra and zoomed in.

The red line in this graph is ln(x) and the purple line is (x^2-1). When examining both lines this close we can conclude that the slope of ln(x) is 1 and the slope of (x^2-1) is 2. Now we know that the limit as x approaches 1 is 1/2.

Next he showed us another way using graphing calculators, which can be done in 4 steps.

1. Click "vars" button
2. Pick "y-vars"
3. Select function
4. Select the values

The next way he showed us was using nderiv, which was not something new this class, but I figured I would include it as well.

1. Click "math" button
2. Pick "nderiv" or "8"
3. Then put nderiv(function, x, 1) into the calculator

and there you go you have to slope.

The next way is a very easy and cool method, using wolframalpha.com. With this website you can pretty much solve any math problem given to you. To find the limit of 5c again we put it into wolfram alpha by typing:

and with barely any work we have the limit.

The last way that I just found is sort of similar to wolframalpha where it calculates the limit value for you. This website is an online limit calculator and it works very well. Click here if you would like to use it. [UPDATE]

Next we went over question 8. Two important things that we learned from this problem was that average rate of change = the slope, and the instantaneous rate of change = the limit. With this problem by the end of class everyone seemed to understand it well, but if you do not understand it fully you can click here to see the work.

Test Preview:

Now with the last 10 minutes or so of class we went over what to expect from the test on thursday, and what we found out was that 1 page will be with calculators, and the last 2 pages will be with out a calculator, so make sure you know how to do everything algebraically. Also, make sure to try on every problem because there will be super corrections on this test.

In addition, Mr. O'Brien gave us assignment 8, which are all of the questions that he made for the test, but decided not to put on. Although it is not really due until Friday it would be extremely smart to have it done by Thursday, so that if you do not understand anything you can clarify any questions you have before it is test day. Also, tomorrow Mr. O'Brien will post the solutions to all of the questions on assignment 8, so you can check your answers.

Here is a link to Mr. O'Brien's schedule for the next few days, so you know when he is free to answer any questions. Schedule.

Also here is a reminder that assignments 4-8 will be due this Friday, so make sure to have them all checked off before class.

Okay so I guess this is the end of my scribe post. All I have left to say is make sure to study a lot for this test because it is a big one, and good luck to everyone.