Asking which function had this derivative, and why the answer is a whole family

0:00Here is a function. The cosine. Now here is a different kind of question about it. Not what is its rate of change -- you can do that. The other way round. WHICH function has the cosine as its rate of change? That question has a name. The answer is called an anti derivative, and finding it is called integration. And the first thing to notice about the question is that it does not have an answer. It has answers, plural, and the whole of this video is about how many.

0:36Differential calculus grew out of one problem: the tangent to a curve, the slope at a point. Integral calculus grew out of two, and they look unrelated. The first is the one I just asked. You are handed a rate of change at every point of a stretch, and you have to recover the function it came from. The second is area. You are handed a curve and a piece of the axis, and you have to say how much region sits between them.

1:07Those two problems lead to the two kinds of integral, and this video is entirely about the first one. The area problem gets answered later, and when it is, the connection between the two is the surprise, not the starting point. Start with three you already know, and read each of them from right to left. The rate of change of the sine is the cosine. So the sine is an anti derivative of the cosine.

1:39The rate of change of a cube over three is the square. So a cube over three is an anti derivative of the square. The rate of change of the exponential is the exponential. So it is an anti derivative of itself. The checker behind this video does not read any of that off a list. It measures. It takes a difference quotient from each side of an input, shrinks the increment until the reading settles, and then asks whether what it settled on is the rule we were handed.

2:14All three pass, at five inputs apiece, none of them whole numbers. Now add a constant. Any constant. The sine plus seven. A cube over three, minus two. The exponential plus a third. Differentiate again and nothing has changed, because the rate of change of a constant is nought. The checker measures that too, and it measures both halves of it: the values MOVE, by exactly seven, by exactly minus two, by exactly a third -- and the rates do not move at all.

2:52So take one anti derivative and you can make more. Forty-one constants a quarter apart were added to one answer, and all forty-one passed the rate test, sitting at forty-one different heights. It could have been forty-one thousand. There is a whole line of answers. And here is where almost every telling of this goes quietly wrong. It presents the constant as bookkeeping. A little plus C you tack on at the end so the marker does not take a point off you.

3:27But look at what we have actually established, and what we have not. We have established that the set of answers is AT LEAST a line's worth. We have shown that a certain construction produces more answers. We have not established that those are all of them. Nothing so far rules out some completely different function, of some entirely different shape, that happens to differentiate to the cosine as well.

3:56A line's worth is a lower bound. The word we want is ALL, and we have not earned it. Here is the argument that earns it, and it is four steps long. Suppose two functions have the same rate of change everywhere on a stretch. Subtract one from the other. The difference has a rate of change of nought at every point, because the two rates cancel. A function whose rate of change is nought everywhere on a stretch is constant on it.

4:29Therefore the two functions differ by a constant. Not by something; by a NUMBER. That is what turns a line's worth into all of them, and it is usually told as an aside, tucked in after the interesting part. It IS the interesting part. The checker put it to five pairs that are not obviously the same function. The square of one more than the input, against the square plus twice the input. The square of the sine, against minus the square of the cosine. Twice sine times cosine, against the sine of twice the input. The exponential of twice the input, against the square of the exponential. And a cube of a linear expression over six, against the same thing multiplied out.

5:20For each pair it measured both rates, from both sides, at five inputs, and found them equal. Then it read the DIFFERENCE at sixteen other inputs -- somewhere new -- and asked how many numbers it held. One number, every time. And the numbers were one, one, nought, nought and a sixth. Two of the five pairs turn out to be the same function written two ways, and the same measurement says so without being told.

5:52Now the part that gets skipped. That argument was stated ON AN INTERVAL. On one unbroken stretch. And that is not decoration. It is the whole thing. Watch what happens without it. Take a domain made of two separated pieces -- say from nought to one, and then from two to three, with a gap in between where the function simply is not defined. On that domain, consider two functions. The first is nought everywhere. The second is nought on the left piece and one on the right piece.

6:29Both have a rate of change of nought at every point where they are defined. Every point. There is no input at which they differ in rate, because the place where the step happens is not in the domain -- it is in the gap. So they have the same rate of change. And their difference is not a constant. It is nought on one piece and one on the other.

6:53The checker built twelve of these -- four different two-piece domains, three different step heights -- and measured all twelve. Twelve agreements of rate. Twelve differences holding two numbers rather than one. Adding constants to an anti derivative was never going to reach these. They are not on the line. So why does the argument work at all? Because on an interval there is nowhere to hide the step. Run exactly the same construction on one unbroken stretch. Nought below some point, one above it, and the point itself now inside the domain rather than in a gap.

7:38The checker put the join at five different places. At every other input the rate is still nought -- six readings each time, thirty in all. At the join itself, the difference quotient does not settle. Not at a tenth, not at a hundredth, not at a ten-thousand-millionth. It grows without bound as the increment shrinks, because the function jumps and the increment does not. So there is no rate there at all. Five joins, five refusals.

8:10And note carefully what that is NOT. The function is perfectly well defined at the join -- it has a value there. What it does not have is a rate. The step needs a gap in the domain to hide in, and an interval has no gap. That is the whole hypothesis, doing visible work. Now we can draw the answer, and the picture is the useful thing here. One curve. Then the same curve slid up. And down. And up again. A whole stack of copies, one for every real number.

8:48At any input you like, every curve in the stack has the same slope. They differ only in height. Nine copies, half a unit apart, read at four inputs: thirty-six slopes, and every one of them is the same value the rule takes there. Nine heights at each of the four inputs, all different. One slope everywhere, nine heights everywhere. And the integral sign is a name for that stack. Not for a curve in it. For the stack.

9:20When you write the integral of the cosine and get the sine plus C, the answer is not a function with a decoration on it. The answer is a family, and the C is how the family is written down. A short piece of vocabulary, because these words get used interchangeably and two of them should not be. The thing under the integral sign is the integrand. The letter the d is attached to is the variable of integration.

9:52An anti derivative of a function is a function that differentiates back to it, and the words primitive and an integral of both mean the same object. Integration is the process. The arbitrary constant, or the constant of integration, is the real number you may add freely. And the family of anti derivatives is the whole set of answers, one for each of those numbers. That is a glossary. It is not a set of rules, and nothing in it is a theorem. Read it once and move on.

10:30Applications want one function, not a family. So they hand you one extra fact. Here is the shape of it. The rule is four times a cube, less six. An anti derivative is a fourth power less six times the input, plus a constant. And you are told the answer takes the value three at nought. The usual move is to substitute and solve. The checker does something more informative: it searches the family.

11:01Four hundred and one members, laid out an eighth of a constant apart, each one asked whether it takes the value three at nought. Exactly one does. Its constant is three. And that phrase -- exactly one -- is two separate facts, so both were counted. Six different families were each given one prescribed value, and every one of the six left exactly one member standing. One condition is not too few.

11:33Four of the six were then given a SECOND value, one the first already forces. The same member survives. So the second condition does no work. Move that second value by one, so the two cannot both hold, and nought members survive. Four times out of four. One condition is exactly enough, in both directions. Here is a question that offers four candidates and asks which one differentiates to four cubes less three over a fourth power, and vanishes at two.

12:10There are two different ways for a candidate to be wrong, and mixing them up is how the mark gets lost. The checker put all four through the rate test at five inputs, and then asked each of them to vanish at two. Three of the four have exactly the right rate of change. Only one of those three vanishes at two. So two of the candidates fail with a perfectly correct derivative. They are members of the right family, sitting at the wrong height. A reader who checks only the derivative passes three of these four.

12:46The one that works carries the constant minus a hundred and twenty-nine eighths, because at two the rest of the expression comes to sixteen and an eighth, and that is what has to be cancelled. Some anti derivatives you can simply recognise, by running a derivative you know backwards and adjusting. For the sine of twice the input: minus half the cosine of twice the input. The half is there because differentiating the inside produces a two.

13:21For the cosine of three times it: a third of the sine of three times it. For the exponential of twice it: half that same exponential. For a linear expression squared: its cube, over three times the coefficient inside. And for a difference of two of them, take the two answers and subtract. All five were measured rather than recognised. Five answers, five inputs each, none of them whole. Now a sentence you will meet, and it is more careful than it looks.

13:57There is an integrand -- the exponential of the negative of a square -- which, it is said, cannot be done by inspection, because no familiar function has it as a derivative. That is a statement about a vocabulary. And a vocabulary is a thing you can write down and search. So the checker wrote one down. Nineteen familiar rules -- powers, reciprocals, sines, cosines, exponentials -- and twelve coefficients, taken one at a time and two at a time. Twenty-four thousand, eight hundred and fifty-two candidates, fixed before any target was named.

14:39Every candidate is screened on a difference quotient at one input, and every survivor is then confirmed at four more through the same rate test as everything else. Put to the eight integrands this video has already answered, the search finds an anti derivative for every single one. Eight for eight. Put to the exponential of the negative of a square, it finds nothing at all. That matters BECAUSE the search succeeds elsewhere. A search that never finds anything proves nothing about the one case where it fails.

15:19And here is the sentence that must not be upgraded. Not in the catalogue is not the same as does not exist. Those are different claims, and only the first one has been shown. So let us show the difference, by producing a function whose rate of change IS that integrand. Take the area under the graph, accumulated from nought up to wherever you are standing. Where the graph only falls, that area is trapped between two staircases of rectangles: one built from each strip's left-hand height, one from its right. At eight strips, then sixty-four, then five hundred and twelve, each trap sits inside the one before it, and the gap between the staircases falls by very nearly a factor of eight each time. So the area is a number.

16:12And the rate of change of that accumulated area is trapped the same way, between the graph's own two heights a little either side. At a hundred-millionth either side, at three different inputs, that trap holds the integrand's own value and is narrower than a millionth. So the function exists. You just cannot write it with those nineteen rules and twelve coefficients. The catalogue has no such function. That is a fact about the catalogue.

16:46One more habit worth breaking, because it costs people whole questions later. There is a product rule for derivatives. It is tempting to expect the reverse: that to integrate a product, you integrate each factor and multiply. Eight pairs of rules were taken, each with an anti derivative already checked, and the two answers were multiplied together and put to the rate test against the product of the two rules.

17:16It works three times out of eight. And all three are pairs with a nought in them, where both sides come out nought anyway. On the five pairs that carry any content at all, it fails. Integration is the inverse of differentiation statement by statement, not rule by rule. Every technique for integrating a product arrives separately, and has to be earned separately. A small one to finish the mechanics.

17:48The letter under the d is not part of the answer. It says which symbol is being varied, and nothing else. The power rule written in one letter and the same rule written in another were built as two separate rules sharing no code, and read at thirteen inputs apiece for five powers. Sixty-five agreements out of sixty-five, exact ones -- not agreements within a tolerance. And to prove that is what was being reported, a third version was built with a lean of a hundredth on it and read the same way. It agrees at five of the sixty-five: the five where the input is nought and the lean contributes nothing.

18:33So the agreement is real. Change the letter and nothing about the statement changes. So: what did we actually get. The question is which function had this rate of change, and the answer is never one function. Adding a constant shows the answer set is at least a line's worth. That is the easy half and it buys a lower bound. The argument that two functions with the same rate on an interval differ by a constant is what turns the lower bound into the answer. It is the reason the word ALL is allowed, and it is not a curiosity attached to the constant -- the constant is the claim, and this is its proof.

19:21The word interval in that sentence is load-bearing, and you can watch it work: give the domain a gap and the extra answers appear immediately. The integral sign names the whole family. One prescribed value picks a member, and one is exactly enough -- not too few, and any second condition that agrees is doing nothing. And when a familiar formula cannot be found, what has been shown is that a vocabulary does not contain one. It has not been shown that nothing does.

19:57The constant is not bookkeeping. It is the answer.