The Straight Line That Fits, Then the Parabola

Last timeThe Two Questions Are One Question

Near a point, a rule can be replaced by a straight line that matches its value and rate, then by a parabola that also matches its bend, with an error that shrinks sharply.

Every applied statement of the form that one thing is proportional to another for small

changes is the same manoeuvre: a rule has been replaced near a point by a straight

line. A pendulum is only linear for small swings, a spring only obeys its law for small

stretches, a resistor only has a resistance over a small range. The manoeuvre is so

common that it is usually invisible, and it has an exact statement and an exact cost.

This lesson is about which line, what it costs, and when paying more buys something.

FIG 1Pinning down the straight line, and then the parabola

: the approximation being built: first a straight line, then a parabola. It is a polynomial, which means it can be evaluated with arithmetic alone
: the point everything is matched at. The approximation is good near here and makes no claim elsewhere
: the rate of the rule at that point, which the approximation is made to share. Matching this is what makes the error shrink faster than the distance
: the rate at which the rate is changing: the bending. A straight line has none, so matching this is what forces a squared term

Three conditions, imposed one at a time. The first says the approximation passes through the right point, which any number of lines do. The second says it leaves that point at the right slope, which pins the line down completely, since a line has exactly two things to choose. The third asks for the bending to match too, which a line cannot do at all, so a squared term has to be added, and the squared term brings exactly one new thing to choose. Each new condition costs one new term, and no condition disturbs those already imposed.

The line, sliding along

Here is the tangent at a point that can be moved. The rule is the one whose rate is

itself, which will reappear in the last lesson of the course.

FIG 2The matching line, at a point you can move

where the line is anchoredc = 0

the rulethe line matching it there

At every setting the line touches the curve and leaves it at the same slope, which is the whole of its construction. Look at how the agreement behaves: close to the anchor the two are indistinguishable, and the gap opens slowly at first and then fast. That slow opening is the content of the approximation. It is not that the line is nearly right everywhere, it is that the error is of a smaller order than the distance, so for a small enough region the line can be used and the error ignored.

Adding the bend

A straight line cannot curve, and the rule above curves everywhere. Matching the

bending requires a squared term, and the coefficient is forced rather than chosen.

The lesson stops here

3 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

The Straight Line That Fits, Then the Parabola

The line, sliding along

Adding the bend

3 more paragraphs to go

This is the reading half

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