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Mon 4/5 Section 5.1 Read 5.2; Begin On-line homework for 5.1 (due 4/12).
- Definition:
- All antiderivatives are the same up to a constant:
- Differential Equations:
- A differential equation is an equation linking a function and its derivative(s).
- The simplest differential equations look like this:
- Usually an initial condition (or conditions) will also be provided -- so that the solution (y(x)) has value y=y0 at x=x0.
- Our job then is to find the function y(x) which satisfies both of those conditions.
- Examples:
- #81, p. 247
- A spring satisfies Hooke's law:
Newton said that
If we put those two laws together, we get
For initial conditions, we should give initial position and initial velocity.
- Gravity is essentially constant at the Earth's
surface: we can use a differential equation to deduce
the formula for the height h of an object as a
function of time:
Newton said that
If we put those two laws together, we get
Once again, we should give initial position and initial velocity.
- Notation:
- The integral is an operator that also is linear:
- A couple of important special cases:
- Power rule:
- Trig rules:
- Power rule:
- Examples:
- #3, p. 242
- #29, p. 243
- #48, p. 243
- #63, p. 244
- Dirt!
- 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
- Assumes continuity of underlying speed function
- We might fit straight line segments between data points: this gives rise to the so-called "Trapezoidal rule".
- Suppose that you clock your speed at each minute along a trip, in
mph:
- Other useful rules are rectangular: Left/Right/Midpoint rules:
- relax continuity! Use rectangles...
- also called rectangle rules: lrr, rrr
- Estimates:
- RRR: d = 45.5*10 mph*minutes = 7.58 miles
- LRR: d = 49*10 mph*minutes = 8.17 miles
- Trapezoidal rule = 47.25*10 mph*minutes = 7.88 miles
- Notes:
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- Whenever you have two estimates, you have a third:
- the trapezoidal rule is the average of left and right rectangle rules.
- 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.
- Examples:
- #5, p. 258 (also trapezoidal)
- #9, p. 258
- 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:
that is, that
- Examples:
- #8, p. 258
- 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: