- Calculus lab opens next week.
- I hope that you had a chance to do your homework for today. Any issues that you've encountered or would like to share?
I really adore Devlin's Letter to a calculus student, and his discussion of what I call "the most important definition in calculus" - the limit definition of the derivative.
- Let's return to our specific problem from last time:
- What is the area under the graph of "urine fertilized
cabbages":
- What could this area mean? (Think about units of area....)
- What special challenge(s) does this graph and problem present?
- What is the area under the graph of "urine fertilized
cabbages":
- Let's consider another problem, which doesn't seem to have anything to do with area:
Suppose that you clock your speed at each minute along a trip, in mph:
Estimate the distance travelled.minute 0 1 2 3 4 5 6 7 8 9 10 speed (mph) 0 30 45 30 70 65 70 70 45 30 35 How? Dirt! And rectangle rules....
- If you were going at a constant rate of speed, you might use the ol' dirt formula:
- We could imagine that for each short time segment, the given speed is fixed. This leads to a discontinuous speed function. But it's easy to calculate the total distance! It's just a bunch of little dirt calculations.
- Alternatively, we might fit straight line segments between data points: this gives rise to the so-called "Trapezoidal rule". Then we'd have a continuous speed function.
- One thing for sure: since we're dealing with sampled data, we're probably wrong -- we hope that we're close.
- If you were going at a constant rate of speed, you might use the ol' dirt formula:
- There are several useful rectangular rules, such as
Left/Right/Midpoint rules:
These are especially handy when one has a formula, rather than
data. They may even allow us to provide an exact answer, using limiting
processes (such as those described by Devlin!).
- Okay, now let's get to our answer(s), estimates for the distance
travelled with the given data.
Another way of thinking about what we're doing is using (some of) the speed data to generate an average rate for the interval. If we do that, using these two methods, we get these results:
- RRR (Right Rectangle Rule): d = 49 mph * 10 minutes = 8.17 miles
- LRR (Left Rectangle Rule): d = 45.5 mph * 10 minutes = 7.58 miles
- Trapezoidal rule = 47.25 mph * 10 minutes = 7.88 miles
- Notes:
-
- Whenever you have two estimates, you have a third:
- the trapezoidal rule is the average of left and right rectangle rules.
- Notice the unit conversions that occurred.
- Each of these estimates corresponds to computing a rectangle height (an
average speed), and then multiplying by the total time (10
minutes). Hence we're using the "dirt" formula, and computing the
average (the "area under the curve", even though there's no curve!) in
several different ways.
- Basically, every function can be used to create a table like this, and
then we can use these strategies.
- Example:
- Often (as in the table above) we have data located at equal intervals (
). When you have a curve (or a formula) for f, you can take the time intervals shorter and shorter, until you get as close to the area as you need:
(notice that
that is, that
)
Using "Sigma" notation makes the formulas look a lot nicer:
Then, in fact,
- Examples:
- #1, p. 293
- Generalizations?
- 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 bywe 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.
- If the size of the intervals goes to zero, then the approximations get better and better, until they're perfect!
- The area can be calculated by finding an average value on the
interval, times the length of the interval:
- Some more examples:
- #6
- #15