Fall+Week+03

Monday 8/29
We began class with a Problem of the Day which can be seen in the graphic below We graphed the function to see whether there were any discontinuities in the graph.. Since the were none, the limit of the function as x approached 0 is equal to f(0). Hence the limit L is 1.

Part b) was much more complicated. L + epsilon and L - epsilon form upper and lower boundaries for the values of f(x). Since L = 1 and epsilon is .1, those boundaries are 1.1 and 0.9 respectively. The need to keep the y values between 0.9 and 1.1 led to the inequality 0.9 < f(x) < 1.1 the inequality we needed to solve. Looking at the graph helped us make the decision to split this inequality into the two written in green and red. The solutions of these two inequalities led to the statement -0.6464 < x < .100167

Lastly we had to chose the correct value of delta to fill in the blank in the following statement:

If x is within _ units of 0, f(x) will be within .1 of a unit of 1.

Next on the agenda was the calculation of limits WITHOUT the use of a calculator. We had done a little of this when working with the definition of the derivative back in chapter 1 but most of those expressions had been polynomials which were factorable. We have since learned that functions with removable discontinuities at x = c will have a limit as x approaches c, even though f(c) is undefined. We can expect this to happen if the form of the limit is 0/0, indeterminate form. This is a clue to "do some algebra" to simplify the expression before using the limit properties we learned last week.Some of the problems we did can be seen below.

Tuesday 8/30
Today we went over the Calculus Problems for a New Century limits worksheet. The solutions are below. media type="custom" key="10316101"

Thursday 9/1
Today we began an investigation into the definition of the continuity of a function at a point. To do this, we used a document on the TiNspire CAS machines and the screenshots from the answer document can be seen in the slideshow below. media type="custom" key="10362340" The last slide !asked us to come up with our own definition of continuity which, you can see, was revised a little during class on Friday

Friday 9/2
The worksheet started during class, with a set of answers, is attached below.