Inversing the Derivative
So we learned how to find the rate of change, but now we are being taught to go backwards, finding the parent function from the first derivative. Or finding the first derivative from the second derivative.
Strange eh? In life we are always told to focus on now and the future, never to worry about the past. Now all of a sudden we are taught to go back. But then again it is calculus :P
Alright Ladies and Gentlemen. Have you read Ara’s scribe about Thursday’s lesson back at the AP Calculus blog? If you haven’t shame on you! Now go read it.
Pretty much this scribe will be about Thursday and Friday’s lesson. The beginning of Chapter 3 or shall I say half of Chapter 3. If you look forward on the textbook I think there are only 4 sections in this chapter. It will go by very fast. I think so anyways. So when the chapter is done, I’ll sum up the whole chapter again.
Here we go:
Velocity = change in distance / change in time
We were taught to find velocity that way. But now we want to find out the change in distance over the given change in time, how do we find it?
Change in distance = velocity x change in time
Drawing this image might help you when you’re trying to figure out either variables. I like saying “Very dumb things” in other words “v=d/t”.
So graphically what exactly are you doing? Given a velocity graph, How do you find the change in distance?
You take the area under the graph, or maybe I should be putting, the area occupied between the function and the x-axis. Why? Because someday we will be dealing with functions that have certain intervals that are beneath the x-axis. Why do I say change in distance? Because of intervals beneath the x-axis, when we do calculate them, you’ll see why it’s a “change” in distance.
Within the two days: We were taught about Riemann Sums.
The function being showed is monotonic decreasing, in other words in the shown diagram the function is ONLY decreasing. If it was said to be monotonic increasing, it’ll ONLY be increasing. Many functions are not monotonic. But if you look at it in segments, in certain interval, they can be monotonic.
Once again the idea of “many” is brought up in this chapter. In the past chapter we used “many” in the number of zeroes after the decimal. As “h” gets infinitely close to 0. Now in this chapter, we need to have “many” subintervals. The more we have, the smaller the width of a subinterval, and the smaller it is, the more the rectangles become looking like the area under the graph. We have seen that with our calculators.
Right Hand Sums and Left Hand Sums doesn’t give the exact change in distance, but it gives you a range of where it may be. Finding the average of the two sums or in other words taking the trapezoid sum gives you a much better estimate. We were also taught about midpoint sums, using the value of t in between the subinterval.
I know this isn’t the best explanation I’ve given at all. It’s rather a rushed explanation. So sorry. But really you guys should read Ara’s scribe. She did a good job. Till sometime this week. I’ll summarize the whole chapter for you, and hope it helps for the chapter test. Till then. L8r days.