Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this second part--part two of five--we cover derivatives, differentiation rules, linearization, higher derivatives, optimization, differentials, and differentiation operators.
BONUS!
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來自 University of Pennsylvania 的課程
Calculus: Single Variable Part 2 - Differentiation
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從本節課中
A New Look at Differentiation
Think derivatives mean "slopes"? Not anymore... In this module, we will reconsider what a derivative is and means in terms of the asymptotic (or big-O) notation from the previous chapter. This will give us a new language for describing and understanding rates of change and the rules that govern them.
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Robert GhristProfessor
Mathematics and Electrical & Systems Engineering
Welcome to Calculus, I'm professor Grist. We're about to begin Lecture 11, bonus
material. In our main lesson, we covered the basic
rules for differentiation. Well, the question comes up, why spend so
much time on those rules? You might think that it is merely, so
that we can do computations more easily, more efficiently.
And although that's very helpful, there are other reasons why you want to know
these rules. Mathematics, and indeed, the sciences,
are full of interesting patterns, and sometimes you'll find something you
recognize hiding within a very different-looking system.
Knowing the rules helps you recognize patterns.
We're going to take a look at two examples of this from very different
fields. We'll begin by looking at spaces, or
geometric domains on which you might do Calculus.
Examples would include a simple interval. Or, maybe a circular disk.
Now, there's an operator that is not unlike differentiation, but it acts on
spaces as opposed to functions. This is the boundary operator.
And we denote it with a scriptie sort of d.
What does the boundary operator do? Well, it gives you boundary of a domain.
For example, the boundary of an interval is simply the two end points.
The boundary of a circular disc is simply the circle that is at the edge or
boundary of that space. Now, there are other operations that act
on spaces as well. For example, there's a way to multiply
two spaces together in something called the Cartesian product.
For example, a rectangle can be considered as the product of two
intervals. Or, a circular cylinder can be considered
as the product of an interval with a circular disc that would be a solid three
dimensional domain. Now, you might guess that there's some
interesting Mathematics contained inside of this product.
For example, the disc is two dimensional, the interval is one dimensional, and this
cylinder is three dimensional. 2 plus 1.
I wonder if that pattern continues. Well, let's consider what happens when we
compute the boundary of a product. What is the boundary of a rectangle?
Well, it consists of the edges along the boundary of the rectangle of course.
But we can think about that in terms of the product structure.
This boundary is really the boundary of the first interval times the second
interval. But we also have the other edges as well,
which can be viewed as the boundary of the second interval times the first.
We need to add these together. This is done through taking a union of
those two sets of edges. Hmm, it seems as though there's something
I've seen before hiding within that boundary computation.
Le'ts see if the same thing works with the cylinder.
If I look at the boundary of this cylinder, then what does it consist of?
It consists of the two end caps which is the circular disc, cross the boundary of
the interval, but it also has the side that wraps around.
That is the interval times the boundary of the circular disk.
If we take the union of those collections, we get the boundary of the
cylinder. It is a fact that if you have two spaces,
A and B, you take their product and compute the boundary of that.
You can decompose that boundary as the boundary of the first space, times the
second union. The first space times the boundary of the
second. Now, where have you seen something like
that before? That's really a product rule.
But we're not differentiating anything, we're simply computing boundaries.
This gives you a hint that there's some deep relationshiop between
differentiation and boundaries. That relationship will be fully exploited
when you get to the end of multi-variable Calculus.
Let's consider another example. A completely different setting.
This time, working in computer science and looking at lists.
Let's say that we have a list of objects. Let's say they're all the, the same type
of object, so that we call them x. So, a list of five items would be x x x x
x. Let's do some Mathematics.
Let's combine those together and call that list x to the 5th.
That tells us there are five elements. Now, how do you take the derivative of a
list? Well, that doesn't make any sense, but
there are some things that we can do. Consider the following deletion operator,
D, that acts on a list and deletes one of the items.
What is D of this five item list? Well, you could delete the first item in
the list or you could delete the second item, or the third, or the fourth, or the
fifth. Now, each of these is really a a list of
four elements. So, we would call it x to the 4th.
One way to encode this logical or, is to use a formal sort of addition.
We'll just call that plus. That means, or.
So, deleting a five item list gives us one of five different four item lists.
We could, say, more compactly, that D of x to the 5th is 5 times x to the 4th, and
this is something that should look familiar.
I wonder if there's something deeper than we can do with this intuition.
So, let us denote by x to the N, a list of N elements.
What would X to the 0 be? Oh, that would be an empty list.
You have no items in your list. But instead of calling that x to the 0,
let's call it 1, like it ought to be, and consider the collection of all finite
lists. We'll call that L.
Let's do some Calculus. Here's a statement.
Any list is empty, or it has a first entry.
I hope you'll agree that that's intuitively true.
What does it mean in this language that we're constructing?
Well, any list certainly means L. That's our collection of all finite
lists. What does it mean that it could be empty?
Well, that means that it could be 1. What does or mean?
Oh, or is our formal addition. So, L is equal to 1 plus, and here's the
tricky one. What does it mean to say that a list has
a first entry? Well, that means it has an x in it,
followed by something, by some finite list, maybe empty, maybe not empty.
We can rewrite this statement as L equals 1 plus x times L.
And now, let's do some Algebra. See what comes out.
If we move all of the L terms over to one side and factor out that L, what happens
when we solve for L? And get an expression for all finite
lists? Well, as you can clearly see, L is 1 over
1 minus x. What in the world does that mean?
Well, we know what one over 1 minus x really is.
In terms of the geometric series. What does that mean?
Well, in this language we've constructed, you can read this statement, as saying,
that any finite list is empty, or, it is one item, or it has two items, or it has
three items, or 4, or 5 etcetera. And so, we see yet another interpretation
for the geometric series. Now, some students are surprised by this
example that you have derivatives and Taylor series coming up in a more
computer science context. Well, that shouldn't be too much of a
surprise. Mathematics is full of patterns.
Patterns that can describe all sorts of things in the natural world.
If you're interested in learning more about some of these unusual examples.
The first, concerning spaces, comes from the subject of topology.
The second, concerning lists, comes from the subject of analytic combinatorics
