Today in class, we started out by walking through an investigation about the Instantaneous Rate of Change of a Function with a partner for the beginning of class.
This investigation begins by giving us an example of a door closing automatically, and asking us to examine the rate at which the door closing, and the relationship between the rate of the door closes and the time that it takes.
The class worked on this investigation for a while, until many people got caught on #6, because of the idea of instantaneous rate of change.
Mr. O'Brien stopped the class, and explained that instantaneous rate of change is another way of saying, "the d word", or Derivative. We learned that derivative is the instantaneous rate of change of a function.
To show us this, we started a quick class activity in Geogebra.
We had graphed the investigation, and had also made a table of values:
We found the instantaneous rate of change (the derivative!) at t = 1 by using shorter and shorter time intervals between t = 1 and t = 1 + h where h is a very small value. We then graphed the line through the point (1, d(1)) with a slope equal to the derivative and noted that it was the tangent line to the function d at the point t = 1. Coolness!
We then represented the derivative with a new notation: the limit. When working with a limit, it will always have a function in it. There are three parts to a limit: the letters lim, a function, and the input variable with an arrow telling you what number the input value is approaching.
We wrote the derivative as:
We noted that the expression following the limit can be any function- i.e. it does not need to be a slope expression. In other words, while limits are necessary to calculate derivatives, they are used in expressions other than derivatives.
We used WolframAlpha to evaluate the above limit as:
However, Mr. O'Brien explained that we wouldn't be able to do this ourselves algebraically until later in the year, so we will want to come back to look at it then.
To gain an intuitive understanding of a limit and the three parts of the limit definition, we followed a link through the blog, explaining to us what a limit is.
http://archives.math.utk.edu/visual.calculus/1/limits.16/tut1-flash.html
As stated by the website:
If f is a function and a and L are numbers such that
- If x is close to a but not equal to a, then f(x) is close to L
- As x gets closer and closer to a but not equal to a then f(x) gets closer and closer to L
- We can make f(x) as close to L by making x close to a but not equal to a.
We then went over several example problems of what we will be going over on the homework for tonight. These problems were simple limit problems, designed to help us get a grasp of the concept before we dive in tonight.
These problems examined the idea that you can input the function into your grapher, and in the table option on your TI-84,89 etc. you can input values very close to the value the input is going towards. Then, by inputing these numbers, you can see where the gap it, and make an accurate estimate of what the limit is.
Overall, today was a good day of reinforcement on what exactly derivatives are (and what they do), along with some introduction about limits, which when understood, are less complicated then they seem.
Update: During the lab period, we learned that some limits can be solved algebraically by factoring out a FUFOO and then just evaluating the resulting function. The reason that this works is that the original function and the factored function are identical at *all* points except the value which makes the factor zero. Since in a limit it doesn't matter what happens *at* the value of the input (only what happens close to the value of the input), the limits have the same value.
If you still don't understand limits and derivatives after this class, check out this discussion:
http://mathforum.org/library/drmath/view/53398.html
IW #2
- p. 66/1, 3, 15, 19, 25, 35, 51-54, 75, 76
- Quiz on trig values next class
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