Class 9/9/10

Today we began class by taking our first homework quiz of the year. The quiz took half an hour and focused on limits (graphically, algebraically, and visually) with some trig values thrown in too. If you weren't here today, that is something you'll need to make up. Don't forget that the first three assignments are due tomorrow - numbered, checked, and corrected.

We began with this function:



and were asked to graph it without using technology, showing all relevant features. If we had to use tech we could, but we're trying to get to a point where we don't need to.

We began by using a little precalculus knowledge and finding the factors (using factoring by grouping for the denominator):





Remember that if x=-1, then (x+1) is a factor. We also used synthetic division (remember that?) to reduce a cubic into a quadratic and then find the factors from there. We found that the factors for f(x) were: .

(If you can't recall synthetic division and how that works, you can have a nice chat with out our old friend Purple Math: http://www.purplemath.com/modules/synthdiv.htm.)

Then we canceled to simplify, remembering that if you cancel you must eliminate the factors from the parts of the equation that you canceled. We noted some important points: f(2) = DNE, f(1) = 0, f(-1) = DNE, f(-2) = DNE, and f(0) AKA the y-intercept = 1/4.

We talked about different types of asymptotes (kissing, crossing, and volcanic) and watched Mr. O'B do some funky hand motions to demonstrate. Remember we find basic asymptotes by evaluating numbers at a little bit above and below the asymptote to determine which way it's going - we just need to know if it's positive or negative. We also discussed end behavior. You can find the end behavior by multiplying the function by a FUFU.
In this particuular case, that FUFU was .

Then we took the new function and put it into limit notation. Then we evaluated it as x approached infinity and found that the limit of this function was 0. Therefore we knew that the end behavior of the asymptotes was that they approached 0 - a horizontal asymptote.

If you are confused about asymptotes you can check out this basic asymptote tutorial: http://www.freemathhelp.com/asymptotes.html.

If you are a little shaky on graphing polynomial functions, check out our Google Doc from last year's Red class: https://docs.google.com/Doc?docid=0AcOzB06hYe_EZGZqNXNqNjVfOTNkc3RnaDNmYg&hl=en&authkey=CPfBwZsF.

Then we drew in our asymptotes on the graph and graphed our function (remember that there is a hole at (-1, 2/3) that you can't see on the graph).

Sorry about the tiny numbers - they are just labels of each feature of the graph.

We were then asked to find several limits:
a) , which = 2/3

b) , which Does Not Exist (because of the non-removal discontinuity at 2)

c)

d)

e)

Then we were asked to find these example limits:

infinity

negative infinity

One way to do this is to use your calculator. Another is to use your understanding of the rational function - dilate it and shift it, and then look at the graph.

More examples:

(we multiplied the first part by a FUFU to solve the rest of this one and find that the limit is 4/7)


(same idea with the FUFU, but the answer was positive infinity)

If you're having a hard time understanding the concepts of infinity and limits, you might want to look at this helpful site: http://www.sosmath.com/calculus/limcon/limcon04/limcon04.html.

Next we moved on to look at a pendulum displacement function:



First we were asked to find . The limit is 50 - and in real life the 50 means the stopping point or equilibrium of the pendulum when it is 50 cm away from the measurement point.

Then we asked, "true or false?: the larger t gets, the closer d(t) gets to 50". The function oscillates back and forth from 50 so the statement is not really true.

Don't forget that the first 3 homeworks are due tomorrow! The homework for tonight is not going to be collected tomorrow though - you can find that on the iCal.

UPDATE: OCTOBER 20, 2010
It's been a while since I posted this... post. But the ideas of limits and discontinuities are still important! When reading back over this post I realized that we are still using "Algebra 1" concepts like factoring and FuFOOs. We originally used them to find limits, and now we are using them to find derivatives! Now that we're older and wiser, we know that derivatives are really just limits but seen in a different way- see the example below for an reminder of how this works.













So a little over a month ago, we did understand limits and we had the limit definition forms more or less under our belt. Now, however, we also understand how limits apply to derivatives.

And if you're looking for a good overview of basic concepts on limits and discontinuity you can check out this website! The link I gave starts at slide 21 since the ones before that deal with more sequences and series than limits and discontinuity... http://www.people.vcu.edu/~mikuleck/courses/limits/sld021.htm

- Annie