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# AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

## Saturday, October 29, 2005

### In the Beginning - Chapter 3

Previously, we had an incredible road trip and figured out the rates of velocities we have traveled...
and then...
we went backwards.
Now we needed to solve the distance for a specific velocity.
How did we do it?
Here's how:

If in the previous chapter, we spent hours and hours on calculating velocities, slopes or derivatives, this time the DISTANCE is what we will sweat on. It is like reversing the solutions we have done in Chapter 2.

Basically, to acquire the distance, we use the formula:

DISTANCE = VELOCITY * TIME ELAPSED

Obtaining the precise distance is not as easy as multiplying the velocity by the time elapsed, since in a typical graph, the velocity is changing moment by moment. So to have better measure, a LOWER ESTIMATE and an UPPER ESTIMATE must be considered. These estimates are the slowest and largest estimates of the graph, respectively. But having those won't make it any closer to an accurate distance. SMALLER INTERVALS must be used. At a particular point, its LEFT-HAND SIDE and RIGHT-HAND SIDE are to be added and divided into half.

In solving, it is better to first look at how the graph behaves. It is easier to get the distance of a point in a graph that looks like a line - a linear function. This graph is called MONOTONIC since it can only be STRICTLY INCREASING or STRICTLY DECREASING. But sometimes, NON-MONOTONIC graphs can be turned into MONOTONIC.

In an INCREASING GRAPH, the LEFT-HAND side is the LOWER ESTIMATE and the RIGHT-HAND side is the UPPER ESTIMATE. In a DECREASING GRAPH, the LEFT-HAND side is the UPPER ESTIMATE and the RIGHT-HAND side is the LOWER ESTIMATE.

... There you go. My own summary of what I learned in the first discussion of CALCULATING THE DISTANCE TRAVELED.

and... as planned, Prince is the next scribe. (^_^)