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 first part--part one of five--you will extend your understanding of Taylor series, review limits, learn the *why* behind l'Hopital's rule, and, most importantly, learn a new language for describing growth and decay of functions: the BIG O.
BONUS!
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來自 University of Pennsylvania 的課程
Calculus: Single Variable Part 1 - Functions
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從本節課中
Taylor Series
This module gets at the heart of the entire course: the Taylor series, which provides an approximation to a function as a series, or "long polynomial". You will learn what a Taylor series is and how to compute it. Don't worry! The notation may be unfamiliar, but it's all just working with polynomials....
與講師見面
Robert GhristProfessor
Mathematics and Electrical & Systems Engineering
Welcome to Calculus. I'm Professor Greist, and we're about to
begin Lecture 5, Bonus Material. In our lesson, we saw two important
series that have convergence constraints. These are the geometric series for 1 over
1 minus x, and the binomial series for 1 plus x to the alpha.
Both of these are guaranteed to converge, only when x is less than 1 in absolute
value. Let's put these series to work.
In a problem in Electrostatics. An electric dipole is a pair of equal and
oppositely charged particles separated by a short distance.
Now, this is a calculus class, not a physics class.
Don't worry, you're not going to need to know this for the exam, but let's learn a
little bit and have some fun with it. The electrostatic potential gives the
potential energy of this dipole, this pair of charges, as a some of point
charge potentials. Now what do I mean by that?
Well, each charge has a potential energy associated to it.
The pair of them together yields a potential energy that depends on your
position. So let's say that you're at some location
away from the dipole. You want to know, what is the
electrostatic potential. It's given by the following formula, V is
equal to kq over d plus, minus kq over d minus.
What does that mean? Well, k is a constant, the Coulomb
constant. Q is the amount of charge on each element
of this dipole. And d plus and d minus are the distances,
respectively, to the positive and the negative charge in the dipole.
Alright, that's the physics that we need. Let's do some math.
Let's compute what this electrostatic potential V is, in the case where you are
standing at a distance d, away from the positive charge, and where d is much
greater than r, the separation distance between the two charges in your dipole.
Now I'm going to assume that we are orthogonal to the dipole, that is we're
at distance d from the positive charge but perpendicular to the orientation of
the dipole. That means that the distance to the
negative charge is, by Pythagoras's theorem, the square root of d squared
plus r squared. So that in computing V, we get kq over d,
the positive charge distance, minus kq over square root of d squared plus r
squared. We could stop there, but let's simplify
this as much as possible. If we factor out kq over d, then we get a
one from the left term, from the right term we get 1 over the square root of
quantity 1 plus r over d quantity squared.
Well, at this point we can try to expand that out using the binomial series.
Recall the binomial series is for something of the form 1 plus x to the
alpha. In this case, alpha is negative one half.
We're looking at one over the square root of something, and x is r over d quantity
squared. Because the first two terms in the
binomial series are 1 plus alpha x, we get the first two terms in the series
applied here to be 1 minus one half times r over d quantity squared.
Simplifying, we see that the ones cancel and we're left with kq over d times one
half, are over d quantity squared. Everything else is of higher order term
in r squared over d squared. Let's remember that one half r over d
quantity squared. In fact, we'll set it over here, right
next to the location. And consider what happens when we look at
distance d away from that positive charge, but in a different direction.
Instead of being orthogonal to the dipole, let's look parallel to it.
Well, in this case, d plus remains the same.
It is this distance d. But now d minus is a bit simpler looking.
It is d plus r, the separation distance in the dipole.
Now again, factoring out a kq over d, we get a term on the right hand side that is
of the form, one over one plus r over d. We can expand this out using, if we like,
the binomial series with alpha equal negative one, or what that really is, the
geometric series. For 1 over 1 minus x, with x being equal
to negative r over d. Writing out the first two terms of that
series, as well, we get 1 minus r over d. The ones cancel, just as before.
And we see that up to the leading order term, our electrostatic potential is kq
over d times r over d. Now, notice how different that is from
before. By using different series, we see that
the leading order terms in the electrostatic potential differ depending
on how you are oriented with respect to the dipole.
Inline potential is much stronger than orthogonal.
This is just one example of how we can use different series in order to get a
handle on various physical phenomena by expanding, ignoring the higher order
terms and retaining the leading behavior.
