Sal's old Maclaurin and Taylor series tutorial
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Polynomial approximation of functions (part 1)
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Polynomial approximation of functions (part 2)
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Approximating functions with polynomials (part 3)
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Polynomial approximation of functions (part 4)
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Polynomial approximations of functions (part 5)
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Polynomial approximation of functions (part 6)
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Polynomial approximation of functions (part 7)
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Taylor Polynomials
Approximating functions with polynomials (part 3) A glimpse of the mystery of the Universe as we approximate e^x with an infinite series.
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- I had dinner between the last video and this one.
- So I might have forgotten what I just did.
- I think I was about to-- if what I see on my board makes
- sense-- I was about to use the Taylor series, or in this
- specific example the Maclaurin series approximation, to figure
- out a polynomial version, a sum of polynomial terms to
- approximate e to the x.
- And remember let me write here what the definition of
- the Maclaurin series was.
- That we said that f of x is equal to the sum from n equals
- 0 to infinite of the nth derivative of f evaluated at
- 0-- I don't know if I remembered to put it evaluated
- at 0 last time I wrote this down-- times x to the
- n over n factorial.
- And hopefully that makes sense to you.
- This might seem really confusing and strange.
- But now that we're going to apply it to e to the x, this
- should maybe be a little bit more concrete.
- And I think at the end of the last video I said that if f of
- x is e to the x, f of 0 is e to the 0, which is 1.
- And f prime of x, any derivative of e to the x
- is equal to e to the x.
- So you take any derivative of at 0, and it equals 1 for e to
- the x for this particular case of f of x.
- And that's really neat.
- That means that the rate of change of y with respect to x
- is for every 1 you move in x, you move 1 in y at e to the 0.
- But that also means that the rate of change of the
- rate of change is also 1.
- At the rate of change at the rate of change at the
- rate of change is also 1.
- So at either the 0 or x equals 0 of e to the x, the slope of
- the slope of the slope of the slope of the slope of the slope
- of the slope, they're all 1.
- Which to me tells me something mysterious is happening.
- It's another reason why you should just sit and ponder e.
- But anyway, back to what we were trying to do.
- So how would we do this?
- How would be write the approximation?
- Let me write the approximate.
- I'll call that p of x.
- Because it's going to be a polynomial.
- Well in this particular case, what's the derivative of any
- derivative evaluated at f of 0?
- Well that term is 1.
- We wrote that down right here.
- f of 0 is 1.
- The first derivative at 0 is 1.
- The second derivative at 0 is 1.
- Right?
- That's what's special about e to the x.
- So all of these terms are going to equal 1.
- So this polynomial simplifies to the sum from n equals 0 to
- infinite of x to the n over n factorial.
- That to me is very neat.
- Remember these are all 1 in every term.
- So that's why I took it out.
- So what does that mean?
- Well that tells us that e to the x can be approximated.
- And actually, I don't prove it here.
- But it actually turns out that we take the infinite sum that
- the Maclaurin series not only approximates e to the
- x at x equals 0.
- When you take the infinite series, it actually
- equals e to the x.
- So when you take a Maclaurin series at 0, and the resulting
- function, the resulting polynomial actually converges--
- and that's something we'll learn a little bit more
- rigorously hopefully later when we start doing analysis-- but
- it can actually converge to the function at all points.
- And it actually is the case with e to the x.
- So we can actually say that e to the x is equal.
- I didn't prove this.
- But you can take my word for it.
- And you can even test it out with some numbers.
- It equals this sum.
- Well what is this sum?
- It's x to the 0 over 0 factorial plus x to the 1 over
- 1 factorial plus x-squared over 2 factorial.
- And you keep going.
- Of course that's equal.
- So e to x is equal to-- x to the 0 is 1.
- 0 factorial, I said in the last video is 1.
- So it's 1 plus this is just x, plus x-squared over 2 factorial
- plus x to the third over 3 factorial plus x to the
- fourth over 4 factorial.
- And you just keep going on forever.
- And that's e to the x.
- And to me that is amazing.
- Because this strange number e, this 2.7 whatever, whatever,
- that we got from compound interest, it can be written as
- an infinite polynomial, this polynomial series or this
- Maclaurin series, that actually has a certain beauty to it.
- This number is kind of ugly when you write it out,
- 2.7 whatever, whatever.
- But when you write it to an exponent power as an infinite
- sum, it kind of has a nice rhythm to it.
- It's very patterned.
- Who would have guessed that you could have written it
- in such a simple form.
- And even more, what happens when x is equal to 1.
- Right?
- So what's e to the 1?
- Well then we set x equal to 1 on both sides.
- And I think I have space to do it here, e to the 1,
- which is equal to e.
- We just set all the x's to 1.
- So we get that's equal to 1 plus 1 plus 1 over 2 factorial
- plus 1 over 3 factorial plus 1 over 4 factorial plus
- 1 over 5 factorial.
- That to me, once again, is amazing.
- That the number e, and we've just stumbled on another
- definition of e, e is equal to the sum from n equals 0 to
- infinite of 1 over n factorial.
- That is amazing.
- So now we have two definitions for e.
- We have this one that we stumbled on, and of course we
- had the ones from compound interest that I will
- do in magenta.
- The limit as n approaches infinite, 1 plus 1
- over n to the n.
- That also is equal to e.
- This is starting to give me chills.
- Because this very strange, bizarre number is popping out.
- This might not seem so natural to you.
- But it's neat.
- And it comes out in compound interest, and continuously
- compounding interest.
- But this is even simpler.
- I just keep picking one over the factorial of numbers and
- I add them all together.
- And if I take every number really in existence, and I
- sum them all up, I get e.
- That to me is amazing.
- 1 over n factorial of essentially every integer
- from 0 to infinite.
- If I sum them up, I get the number e.
- You hopefully are getting chills right now.
- Well anyway, let's do the Maclaurin series for a
- couple more functions.
- And then we'll get to something that is even more mind-blowing.
- I'll see you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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