Fall+Week+03

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Monday 8/31
First item on the agenda was to go over our test from last Thursday. The test and the key are posted on the Tests page. Many of us need to learn the formal delta epsilon definition of a limit more thoroughly and practice finding delta in terms of epsilon algebraically! Also, some of us were guilty of working limit problems without writing the correct limit notation; as promised ahead of time, such answers were treated harshly and awarded no credit. Next up for discussion was the evaluation of limits for their own sake, not necessarily in the context of a rate of change. Finding the limit of a continuous function at an interior point of the domain presented no issues at all; the limit of the function at x = c will be f(c). For example:

and

Where we have to be more creative is when the function is discontinuous due to either a removable discontinuity or asymptotic behavior. In the first instance, the form of the limit is often 0/0 and some algebraic manipulation of the function allows us to calculate the limit. In the second instance (asymptotic behavior) the form is often k/0 where k is non-zero and these limits are often infinite. We were given a worksheet of limit problems and this worksheet, along with the solutions will be posted under Tuesday's class heading.

Tuesday 9/1
Problem of the day:

We had to remind ourselves of the Exponent Properties

Our discussion then turned to a more formal presentation of the limit theorems which we had been using intuitively without having really defined them! The theorems are shown in the slide-show below. media type="custom" key="4298033" Our practice problems and homework came from Limits Worksheet attached here. Solutions to most of these problems should appear shortly as each student in the class is responsible for posting a solution to one of the problems. ]



16. (Rob Bruce)

Nathan Vassey Problem 18





Sara Cohen #33 Sofia Bonilla

Sean Gannon #30 -

Wednesday 9/2
During class today, we worked some more with limits, going over the limit properties again and working some of the problems from the homework sheet. We also used the limit properties to evaluate limits when the function is presented graphically and you can see the homework problems in the slide-show below. media type="custom" key="4313039"

Thursday 9/3
First things first- our problem of the day. We next turned our attention to evaluating trigonometric limits. Since the sine and cosine function are continuous at all values of is domain, the limits of these functions as x approaches c will always be f(c). Also, since the sine and cosine functions oscillate between -1 and 1, the limits of these functions as x approaches infinity do not exist. Two limits have to be memorized and the derivations of these limits, along with a few other problems are shown in the graphics which follow,

media type="custom" key="4313041" The first slide shows a geometric derivation of comparing areas of two triangles and a sector of a circle. the second slide uses those limit to find. The final slides show some limit problems using these two basic limits; of course, we had many of these to do for homework.

Friday 9/4
Homework was the first order of the day; the solution to the problems we did can be seen below. media type="custom" key="4316743" Following this, we had the option of either working on some more of the trig. limits or "playing" Limits Bingo. Limits Bingo was originally written by Mrs. Kay Fenton; our thanks go the her for her efforts and generosity in sharing the activity with students other than her own. A second fun activity is "What did one math book say to the other?" Maybe we will get some answers to these questions on Tuesday!