Day 3, Mat227

• We started by considering the problem of calculating distance travelled. We were given a set of speeds, recorded at known times.

  If speeds are constant, we know how to compute distance travelled, using the dirt method:

  

  But what if the data suggest that the speeds are changing?

  After a little thought, we might come up with these two strategies:

  1. Average the speeds, and use that one value for the duration of the trip;
  2. Assume that speeds are constant over the subintervals between which they are measured, and use data to representatives for each sub-interval.

  Both strategies are good. They give different values. Each rectangular method we discussed last time gives a different average of the speeds. A straight average of the speed data does not correspond to any of our methods used so far.

  Our objective is to find the strategies that give the best estimates in general. For that reason, we're going to focus on the second strategy, the "rectangle methods".

  Last time five different rectangle based methods were mentioned:

  • Right Rectangle Rule -- RRR -- using right endpoints
  • Left Rectangle Rule -- LRR -- using left endpoints
  • Each of those two rules ignores a data point, however, which seems foolish; so we apply my general rule that
    Whenever you have two estimates, you have a third estimate

    and create the Trapezoidal rule: the average of the previous two methods:

    

  • Then there's the midpoint method -- MID -- but this one is only useful if one can compute midpoints (that is, one needs to be estimate the value at the midpoint of each subinterval). For discrete data, e.g. the speeds given every minute, we're stuck with the data we have.
  • Finally, the best rule we will study or use is Simpson's rule, which works by taking a weighted average of the Midpoint and Trapezoidal rules:
    

• Example:
  • A Mathematica review of the methods (source)
  • #15, p. 294 (here's a handout)

(notice that that is, that )

Using "Sigma" notation makes the formulas look a lot nicer:

Then, in fact, the area may be obtained as a limit (in many cases):

Now our book focuses on the RRR (see Definition 2, p. 289), but as you can see we could use any of our methods (including the trapezoidal or Simpson's) and get the same result. The area is a limit of these methods as the subinterval size goes to zero.

• #6, p. 293

• The area can be calculated by finding an average value on the interval, times the length of the interval: Area under the graph of f from a to b = where by we mean an average value of f.

• The area can be calculated by chopping the interval [a,b] up into smaller and smaller chunks, and then using rectangular approximations to the small areas.
  • In our work so far, the intervals into which the interval is chopped have been equal-sized: no reason to stick with that.

    Look at Figure 13 on p. 289. Where would you like smaller rectangles, and where could you get away with larger rectangles?

    Can you come up with a general rule?

  • If the sizes of all the subintervals go to zero, then the approximations get better and better, until they're perfect. Let's take a look at Example 3 in the text, p. 290 (corresponding to problem #29, p. 295).
    

    #29, p. 295 (we'll finish it off)