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
Polynomial approximation of functions (part 1) Using a polynomial to approximate a function at f(0).
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- Let's say I have this bizarre-looking function.
- It's just some arbitrary function.
- And we'll call that f of x.
- So this function right there is f of x.
- And what we're going to do in this video is, it's not an
- experiment, but we're going to play around a little bit, and
- we're going to try to approximate this function
- using a polynomial with some coefficients.
- And this polynomial we're going to do, we're going to keep
- adding terms to the polynomial, so that we can better and
- better approximate this function.
- And that's actually called a power series.
- And we'll do more about that later, but we're going to
- specifically try, in this case, to approximate the function
- around x is equal to 0.
- So around this point.
- So the easiest way to approximate it is to say, well,
- the simplest polynomial is just a constant, right?
- Let's say this is my polynomial, let me call
- my polynomial p of x.
- The simplest polynomial is just a constant, and it would just
- be a horizontal line someplace.
- So if I just wanted this one term polynomial, what would be
- my best approximation for this function, at least
- at this point?
- Well, I could just set p of x is equal to f of 0.
- And in that case, p of x would just look like a horizontal
- line going through f of 0.
- It would just look like that.
- I'm going to erase that now, just so I don't dirty up
- this picture too much.
- But that's, you could say, a very rough approximation
- of f of x, right?
- So that's a start.
- Well, what would be one way to approximate it even more?
- Well, what if not only does p of x equal f of 0 at x is equal
- to 0, so that horizontal line we did, we got p of 0 is equal
- to f of 0, so we knew that at x equals 0, at least that
- horizontal line is the same value of f of x, that's a
- very rough approximation.
- But what if we set it up so that the derivative of p of 0
- is equal to the derivative of the function at 0?
- All of a sudden, this could be a little bit more interesting.
- So how could we set it up like that?
- Well, what if we set p of x, and I'm doing it very general,
- and we're going to do specific examples, and actually, the
- first example we're going to do is probably the coolest one.
- So what if p of x is equal to, well, the constant term is f of
- 0, and then it's plus the derivative of this function, so
- the slope of this function at that point, f prime
- of 0 times x.
- Let's say I'm defining, so this is a polynomial.
- I just added a first degree term here.
- I had a constant, and now I'm adding a first degree term.
- And let me confirm that this will have the same derivative.
- So let's see.
- First of all, let's confirm that p of 0 is equal to f of 0.
- Well, p of 0 is equal to f of 0 plus f prime of 0 times 0.
- Well, this last term just goes to, is nothing, right?
- Times 0.
- So that checks out.
- At x is equal to 0, the two functions are
- equal to each other.
- Now let me confirm that their derivative, their first
- derivatives are the same.
- So what's the first derivative of p?
- p prime of x is equal to, well, the derivative of a
- constant term is 0, right?
- Plus, and what's the derivative of a next term, of a
- first degree term?
- Well, it's just f prime of 0.
- So this checks out.
- My new polynomial that I've defined right here is equal to
- the function f at x is equal to 0, and its derivative is equal
- to the function f at x is equal to 0.
- So what would it look like?
- Well, it would intersect, at x is equal to 0, the two
- functions would overlap.
- And also, they would have the same slope at that point.
- So whatever f of x is doing, that function's
- going to be doing.
- So it's going to look something like, I'm going to try my best
- to, it's going to look something like that.
- And so that is a better approximation, if we had to use
- a line, that's as good as any, especially around 0,
- of what f of x is.
- f of x might have been some really crazy weirdo function,
- but we were able to approximate it reasonably well with this
- very simple linear equation.
- Well, that's all good, but let's approximate it with a
- quadratic equation, with adding another x squared term.
- And we're going to do that way, but we're going to say, well,
- we said, when at x is equal to 0, the functions
- equal each other.
- That's what we did here.
- Then we said, the derivatives equal each other, and
- so we added this term.
- And now I'm going to say, what happens when their second
- derivatives equal each other?
- So what has to happen for their second derivatives
- to equal each other?
- Well, let's try out this, and I think you'll start
- to see the intuition here.
- Let me define my new p of x, so let me add another term.
- p of x, the first terms are going to be the same.
- They're going to be f of 0.
- Remember, this is just a constant term.
- Plus f prime of 0, the first derivative at 0,
- the slope at 0 times x.
- Plus f prime prime, the second derivative of the function at
- 0, times x squared over 2.
- Now you're probably saying, why are you multiplying
- it by 1/2 here?
- And you'll see, and maybe you'll even realize it, when
- you take a second derivative, what happens, right?
- You multiply the expression by the exponents so you can have a
- 2 come down, it's going to cancel out with the 1/2.
- And that's why I put the 1/2 there.
- So that when I take the derivative, that 2 and the 1/2
- cancel out, and I'm just left with the second derivative
- of the original function.
- So let me confirm that.
- So p of 0 is equal to f of 0 plus, well when x is equal to
- 0, that's 0, this term is 0, and when x is equal to 0,
- that term is 0, right?
- So that checks out.
- What's the first derivative of p?
- The first derivative of p is going to be, up here, this
- was the first derivative of p at 0, right?
- So what's the first derivative of p?
- Well, the constant term becomes 0, plus-- oh, actually, no,
- this was actually of x, sorry.
- I shouldn't go back on my work, I know it best when I'm doing
- it the first time around.
- Anyway.
- The first derivative of p of x, this is my current p of x, this
- constant term, derivative of 0.
- This x term, the derivative is f prime of 0.
- And then, what's the derivative of this term?
- Well, we take the exponent, multiply it by the expression.
- We have 2 times 1/2, that cancels out.
- So we're just left with f prime prime of 0 x.
- Right?
- You take the exponent, multiply by the whole thing, and then
- decrement the exponent by 1.
- So what is p prime of 0?
- What is the derivative at 0?
- Well, it equals, this is nothing.
- It equals f prime of 0 plus, and, well, this
- term's going to be 0.
- So that checks out.
- And so what's the third derivative?
- Let me clean up some of the stuff on the top.
- Since this is our current f of x anyway, I can clean
- up all of this stuff.
- Let me clean up all of this.
- So what is the third derivative of this p that I defined here?
- This is our current p.
- Well, the third derivative is going to be, so p prime prime
- prime of x, we could have also written p3 of x, is equal
- to the derivative of this.
- Oh, sorry, we're not on the third, we're only on
- the second derivative.
- p, and I'll write prime prime, I was going to write a 2 there.
- p prime prime of x.
- That equals what?
- That's the derivative of this.
- So there was a 0 here, that goes to nothing.
- This is now a constant term, that becomes nothing.
- And then we take the derivative of this term.
- Well, we just, it's a constant times x.
- Remember, this might look like a function or some variable.
- It's just a constant.
- Because we evaluated this curvy function, it's
- second derivative 0, so we just got a number here.
- So this derivative is just this number.
- So it equals f prime prime of 0.
- And so our current p of x has the same value when x is equal
- to 0 as f of x, it has the same first derivative at xis equal
- to zero as f of x, it has the same second derivative.
- And I don't, this is getting beyond my visualization
- ability, especially for an arbitrary function like this,
- but I could guess that maybe it looks something like this.
- I don't know.
- Maybe it looks, maybe our new function will curve, and it'll
- approximate it a little bit better, and then maybe
- it'll come down like that.
- I don't know.
- I'm just guessing.
- But around x is equal to 0, it's going to be a better
- approximation of f of x.
- Well, we could just keep doing this, and actually, we will
- keep doing this, and you know, just saying, well, the zeroth
- derivative, or at the value, is the same the first derivative
- is the same at 0, the second derivative is the same at 0,
- we'll say the third derivative, the fourth derivative, and
- we'll keep doing that.
- And I only have 20 seconds left in this video, so we will
- continue that in the next video.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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