**Tuesday, September 25, 2007**

**Wednesday, September 26, 2007**

*specific intervals,*such as depicted above by the

**green triangles**. We determined the average velocities to be 40km/hr and 60 km/hr respectively.

The above image is of a zoomed-in portion of the graph (upper-right portion), which was used to then transition how we would calculate the velocity for time intervals in units of seconds (such as 1 second, or 0.5 seconds). As we may know, seconds are extremely short amounts of time, and calculating this velocity is approximately and yet increasingly similar to determing the instantaneous velocity immediate to that interval. This meaning that since we are calculating the ratio of the change in distance to the change in time, as the change in time decreases, the distance values are ever closer, narrowing into some specific value.

We attempted to absorb the true connection of the secant line and it's importance to the concept of the derivative to a certain extent. Mr. K then altered the distance and time variables of the graph help dignify this connection by explaining how we are determining the rate of change within the graph.

This graph illustrates this explanation. The alteration is visible as a function of the volume of a balloon in terms of it's radius as it is being filled. The graph portrays how we would expect to determine the rate of change of the overall filling process over intervals, on average, or as a cumulative generality. This rate of change was given by the triangles we found earlier, which were formed by specific secant lines. As shown below, and through the above explanation of determining increasingly miniscule time intervals of units in seconds, Mr. K once again elaborated on the secant line and revisited an aforementioned comparison.

From the long magenta line, which is a secant, we can see that as we found smaller and smaller intervals (leading to smaller secant lengths) we achieved larger average velocity values. But as we also outlined, examining these intervals using secant lines (determining the slope/rate of change) with an increasingly insignificant amount of change, we can see how the line then becomes closer and closer to acting as if it were a tangent line. This means that by using secant lines with an ascendingly small change in values, is like determining the instantaneous velocity approximate to that point. As this change approaches an infinitely small amount, the instantaneous rate of change becomes more apparent and more accurate. This, in essence, is the concept of the derivative.

To conclude our class, we shortly discussed what day our first calculus test of the year lands on, which is **friday **in case anyone missed that, and we also covered what questions will be due for homework tomorrow (chapter 1, pages 76-82) and the homework for the weekend on derivatives (exercise 2.1, all odd questions including 6 and 12). I will now conclude this scribe post by conveying the aptly subtle, yet beautiful analogy Mr. K used to contrast the significance and vastness of the concept of the derivative to another profoundly familiar concept. He stated that his son asked him what a decimal was, and Mr. K simply explained to him that a decimal is used to indicate values less than 1 instead of using fractions. He said that decimals have a more universal and prolific usage than just to indicate number less than 1, but to comprehend the subject fully, one must start somewhere and build up their knowledge. This is the same as the derivative, though we can analyze it's uses and it's essential meaning and implications, it's just the birth of our progressively intricate understanding of the derivative.

Enough of that, I think I covered the entire class now, I hope I helped anyone if they didn't understand something in class, or if anyone other than my classmates that require help, have come no further but to seek sufficiently helpful information here at our very own blog. Please, feel obligated to leave any comments, feedback, corrections or questions that you may have. But for now, I bid everyone goodday and have a great night!

**The next scribe will be Mark!**

## 4 comments:

I think Grey-M just finished scribing for this cycle a couple of days ago. Good job on the scribe!

lol jeeze chris you made my heart stop for a second there! think you have to pick another person I've done my second cycle scribe already

Hi MrSiwWy,

You mentioned "enough about the unnecessary" in your intro paragraph. I'm thinking that your intros and conclusions are what make your scribes extra special!

That and your very thorough explanations!

Best,

Lani

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