Intro to Derivatives
Finally Calculus was introduced to us on Thursday with a fieldtrip. The idea of looking at smaller intervals within the function was taught to us. The smaller the interval was, the more accurate our result was. That is only because we are looking at the instantaneous rate of change. The rate of change at that instant.
On Friday we learned or we were taught in greater depth on certain “new” vocabularies.
What is a tangent line?
A line that results from several secant lines as the change in its x value gets smaller and smaller, closer to zero. In other words, a line that touches the curve at one point, as a result of many secant lines.
What is a derivative?
A derivative is a rate of change. Or to be exact it is the total change in a function within that interval.
There are many ways to find a derivative at a point. We were taught three ways on Friday. (I think it was three. :S)
1) Difference Quotient. Which can be showed in different ways, but mean the same thing.
2) Zooming in at the point till what appears on the screen is a straight line.
3) Table of Values.
We didn’t exactly look at it in a table of values. We actually just calculated it in our calculators. This was when the change in x got smaller and smaller. (1/1000, 1/10000, 1/1000000, 1/1000000000) Our resulting values got closer and closer to a more accurate value.
That’s only a couple of ways to find a derivative. There are more ways to find it. We’ll learn several of it in this chapter. ;-)
Looking at the rate of change also showed us what the “particle” was doing at a certain point. Take for example, the door opening and closing. We saw that if the rate of change or the slope at a point was above 0, the door was opening. If it was below 0, the door was closing. If it was 0, it had stopped. So overall, the rate of change, or the “derivative” shows us what the particle was doing at that point.
I’m not sure if I’m missing anything, and if what I just typed makes sense. Hope you guys can understand it. Well next Tuesday’s scribe will be Steve. Oh yeah, comments are welcome.