One of our fundamental concepts is the limit of a function at a point. But soon we’ll need to consider what happens as we let the input to a function grow without bound.
So let’s consider a function defined for , where this interval means the set . It really doesn’t matter here what is, just that we’ve got some point where is defined for all larger numbers. We want to come up with a sensible definition for .
When we took a limit at a point we said that if for every there is a so that implies . But this talk of and is all designed to stand in for neighborhoods in a metric space. Picking a defines a neighborhood of the point . All we need is to come up with a notion of a “neighborhood” of .
What we’ll use is a ray just like the one above: . This seems to make sense as the collection of real numbers “near” infinity. So let’s drop it into our definition: the limit of a function at infinity, is if for every there is an so that implies . It’s straightforward to verify from here that this definition of limit satisfies the same laws of limits as the earlier definition.
Finally, we can define neighborhoods of as leftward rays . Then we get a similar definition of the limit of a function at .
One particular limit that’s useful to have as a starting point is . Indeed, given we can set . Then if we see that , establishing the limit.
From here we can handle the limit at infinity of any rational function . Let’s split off the top degree terms from the polynomials and . Divide through top and bottom by to write
Now every term in has degree less than , so each is a multiple of some power of . The laws of limits then tell us that they go to , and the limit of the denominator of is . Thus our limit is the limit of the numerator.
If we have a positive power of as our leading term, which goes up to or down to (depending on the sign of . If , all the powers are negative, and thus the limit is . And if , then all the other powers are negative, and the limit is .
So if the numerator of has the higher degree, we have . If the denominator has higher degree, then . If the degrees are equal, we compare the leading coefficients and find .
We showed that all convex functions are continuous. Now let’s assume that we’ve got one that’s differentiable too. Actually, this isn’t a very big imposition. It turns out that a result called Rademacher’s Theorem will tell us that any Lipschitz function is differentiable “almost everywhere”.
Okay, so what does differentiability mean? Remember our secant-slope function:
Differentiability says that as we shrink the interval down to a single point the function has a limit, and that limit is .
So now take . We can pick a between them and points and so that . Now we compare slopes to find
so as we let approach and approach we find
And so the derivative of must be nondecreasing.
Let’s look at the statement a little more closely. We can expand this out to say
which we can rewrite as . That is, while the function lies below any of its secants it lies above any of its tangents. In particular, if we have a local minimum where then , and the point is also a global minimum.
If the derivative is itself differentiable, then the differential mean-value theorem tells us that since is nondecreasing. This leads us back to the second derivative test to distinguish maxima and minima, since a function is convex near a local minimum.
Yesterday we defined a function defined on an open interval to be “convex” if its graph lies below all of its secants. That is, given any in , for any point we have
which we can rewrite as
or (with a bit more effort) as
That is, the slope of the secant above is less than that above , which is less than that above . Here’s a graph to illustrate what I mean:
The slope of the red line segment is less than that of the green, which is less than that of the blue.
In fact, we can push this a bit further. Let be the function with takes a subinterval and gives back the slope of the secant over that subinterval:
Now if and are two subintervals of with and then we find
by using the above restatements of the convexity property. Roughly, as we move to the right our secants get steeper.
If is a subinterval of , I claim that we can find a constant such that for all . Indeed, since is open we can find points and in with and . Then since secants get steeper we find that
giving us the bound we need. This tells us that within we have (the technical term here is that is “Lipschitz”, which is what Mr. Livshits kept blowing up about), and it’s straightforward from here to show that must be uniformly continuous on , and thus continuous everywhere in (but maybe not uniformly so!)
John Archibald Wheeler diad last night. This will take some getting used to, since for years whenever I’ve been reminded of him I’ve thought, “surely he’s not still kicking around, is he?” A truly singular individual.
(I accidentally wrote this as a page, I just noticed. This new WordPress interface takes a bit of getting used to.)
Evidently my Brazilian readers (if I have any) will be readers no longer, now that a Brazilian court has ordered all ISPs to block WordPress. Officially it’s about one well-known and powerful lawyer there being insulted by something one WordPress blogger said. Me, I think it’s all part of a cunning plan on Isabel‘s part to dominate the Brazilian blathosphere.
Hat tip to Frank Pasquale at Concurring Opinions.
The exponential function is, as might be expected, closely related to the operation of taking powers. In fact, any of our functions satisfying the exponential property will have a similar relation.
To this end, consider such a function and define a positive number . Then we can calculate , , and so on. Since we see that , and similarly for all other rational numbers.
So we have a function defined on all real numbers, and we have a function defined on all rational numbers, and where both functions are defined they agree. Since the rationals are dense in the reals (the latter being the uniform completion of the former) there can be only one continuous extension of to the whole real line. We’ll discard the function and just write for this extension from now on. In particular, the function gives us a special number , and we write .
Like we saw before, we can use the exponential function to give all the other exponentials . We know that for some constant , but which? If we take the natural logarithm of both sides we see that . In particular, setting we find . That is, given any positive real number we can define the exponential as
The exponential property is actually a rather stringent condition on a differentiable function . Let’s start by assuming that is a differentiable exponential function and see what happens.
We calculate the derivative as usual by taking the limit of the difference quotient
Then the exponential property says that our derivative is
So we have a tight relationship between the function and its own derivative. Let’s see what happens for the exponential function . Since it’s the functional inverse of we can use the chain rule to calculate
Showing that this function is its own derivative. That is, this is the exponential function with .
Since a general (differentiable) exponential function is a homomorphism from the additive group of reals to the multiplicative group of positive reals, we can follow it by the natural logarithm. This gives a differentiable homomorphism from the additive reals to themselves, which must be multiplication by some constant . That is: . How can we calculate this constant? Take derivatives!
So our constant is the derivative from before. Of course we could also write
And since is invertible this tells us that . That is, every differentiable exponential function comes from by taking some constant multiple of the input.
By the usual yoga of inverse functions we can then see that every differentiable logarithmic function (an inverse to some differentiable exponential function) is a constant multiple of the natural logarithm . That is, if satisfies the logarithmic property, then
We’ve defined the natural logarithm and shown that it is, in fact, a logarithm. That is, it’s a homomorphism from the multiplicative group of positive real numbers to the additive group of all real numbers. Now I assert that this function is in fact an isomorphism.
First off, the derivative of is , which is always positive for positive . Thus it’s always strictly increasing. That is, if then . So no two distinct numbers ever have the same natural logarithm, and the function is thus injective.
Flipping this around tells us that we definitely have some nonzero values for the function. For example, we know that . Now, since the real numbers are an Archimedean field, no matter how big a number we pick, there will be some natural number so that , where the latter inequality follows from the logarithmic property.
That is, no matter how large a number we pick, takes values at least that large. But because is continuous on a connected interval there must be some number with . Similarly, if then there will be some with , and thus . Thus the natural logarithm is surjective.
So, since our function is one-to-one and onto, it has an inverse function. We will call this function the “exponential” (denoted ), and define it to be the unique function satisfying
for all positive real and all real .
From here it’s straightforward to see that must be the inverse homomorphism. That is, given two real numbers and we know there must be (unique!) positive real numbers and with . Then we calculate
And it’s clear from here that . A homomorphism from the additive reals to the multiplicative positive reals like this is said to satisfy the “exponential property”, which is just the reverse of the logarithmic property from last time.
Whoops.. Between preparing my exam, practicing my rumba, and adapting to the new WordPress interface, I forgot to actually post today’s installment :-/
Yesterday we defined the natural logarithm as the function
on the interval . This function is differentiable everywhere in this interval, and its derivative is at each point .
We call this function a logarithm because it satisfies the “logarithmic property”. Simply put, it’s a homomorphism of groups from the group of positive real numbers under multiplication to the group of all real numbers under addition.
That is, since the real numbers are an ordered field they are a fortiori a group if we just throw away the multiplication and order structures. Also, if we get rid of that pesky noninvertible element, they’re a group under multiplication, and the positive elements are a subgroup. The logarithm takes elements of this group and sends them to the additive group, and the homomorphism property reads: . In particular, we must have .
So is our “natural logarithm” a logarithm? First off, we can easily check that
As for the other property, let’s write
Now let’s take the second term on the right here and perform a change of variables, setting . Then we have , and as runs over the new variable runs over . That is, we have
and the logarithmic property holds.
Before this little break, we defined a function on the interval of integration. We proved some properties about the functions we get like this, lining them up against the Fundamental Theorem of Calculus. In particular, integrating like this can construct antiderivatives.
Now let’s consider some of the most basic functions of one variable — monomials — and their derivatives. We know that the derivative of is whenever is an integer. Let’s try running this backwards by using Riemann integration.
First for we know that is defined everywhere, so we can consider the function defined for any real by
Whatever function this is will have as its derivative. We can see that has this derivative, and we know that any two antiderivatives differ by a constant. That is, for some real constant . But we can also tell that because in that case we’re integrating over a degenerate interval of zero width. This tells us that , and we’ve determined our constant.
How about for ? Now our integrand has an asymptote at so we can’t integrate across it. Let’s start at and define a function for all positive real by
Again we know that the derivative will be , and that is such an antiderivative. We also know that , which tells us that so our constant of integration is . That is, we’ve defined the function on the interval .
Now what happens when we take this exact same procedure and apply it to the function ? There is no monomial whose derivative is a scalar multiple of , so the above procedure breaks down. Still, there’s some function out there. Indeed, consider the integral
For any positive real number the integrator is of bounded variation on (in fact it’s monotone), and is continuous for positive , so the integral is indeed defined. Since the integrator is differentiable for all positive values, the integral must be as well, and .
That is, this procedure has defined for us an antiderivative of on the interval . We call this function the “natural logarithm” and denote it . Tomorrow we’ll start exploring some of its properties.
As a side note, those of you who have been paying close attention will notice that I have yet to use any function more complicated than a rational power of the variable yet. I’m following the pattern of “late transcendentals” in presenting the calculus. The alternative — “early transcendentals” — is to give a hand-waving (but not rigorous) definition of exponentials and logarithms early on to get more examples into the students’ hands. I advocate that position for college-level calculus classes for a number of reasons, but ultimately delaying the transcendentals makes for less unlearning later on.