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 4) Approximating cos x with a Maclaurin series.
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- We the last video we took the Mclaurin series of
- representation of e to the x.
- Now let's do it with a couple of other functions and we'll
- see in a few videos it all fits together like a giant puzzle.
- Let's do cosine of x.
- Let's set f of x, f of x is equal to cosine of x.
- What's f prime of x?
- What's the first derivative of cosine of x?
- Well that just equals minus sine of x.
- Minus sine of x.
- What's the second derivative?
- Well that's just minus times derivative of sine of x.
- So the derivative of sine of x is cosine x, it's
- minus cosine of x.
- And what's the third derivative?
- f 3 of x.
- The derivative of cosine x is minus sine of x, we already
- have a minus here so it becomes positive sine of x.
- What's f 4 of x?
- The fourth derivative of x?
- It equals cosine of x again.
- As we keep taking derivatives we'll keep repeating and the
- pattern will go on, right?
- The fifth derivative of x, the fifth derivative of this
- function, the fourth is the same as a function, so the
- fifth is going to be the same as the first derivative.
- cosine of x is minus sine of x.
- Hopefully you see the pattern.
- We're going to do the Macloren representation, which is a
- specific example of the Taylor series where we figure out the
- values of the derivatives at x is equal to zero.
- So let's do that right now.
- So f of zero, let me do it in to another color
- to fend off monotony.
- f of zero.
- What's cosine of zero?
- Cosine of zero is 1.
- f prime of zero is equal to sine of-- well not minus sine
- of zero, but what sine of zero?
- Sine of zero is zero, so minus zero is still
- zero, so this is zero.
- f prime prime of zero.
- Cosine of zero we already know is one.
- We have a negative sine here, so it becomes a minus one.
- The third derivative at x is equal to zero.
- Sine of zero is zero.
- So this is zero.
- I think you might start to see a pattern emerging.
- The fourth derivative at zero.
- Cosine of zero is equal to 1.
- And then the fifth derivative, this is hard to read but you
- get the point is just zero again.
- So what's the pattern as we take the derivatives?
- 1, zero, minus 1, zero, 1, zero.
- So it alternates between zero and 1.
- So 1, zero, minus 1, zero, positive, zero, negative,
- zero, positive.
- So every other number is zero and in between them we
- alternate between a positive 1 and a negative 1.
- So now let's use that information to figure
- The Maclaurin series representation
- out them the Macloren series representation.
- So we proved, hopefully, we didn't prove it definitely
- converges over the entire domain of the function.
- You have to take my word for it.
- We'll experiment a little bit with a graphing calculator
- in a few videos.
- We said that this representation-- and it should
- make intuitive sense, because when you take the infinite
- Maclaurin series, when you take that infinite sum, you're
- essentially creating a function where that function is equal to
- your original function at the point you chose.
- In the case of a Maclaurin we're picking x equals zero, and it
- equals every derivative of this function.
- Just intuitively it seems, well if a function equals something
- at a point and every one of its derivatives is also equal to
- the function at that point, maybe those functions are
- equal to each other.
- I haven't proven that to you yet.
- We know that the representation is a sum from n is equal to
- zero to infinity of the nth derivative evaluated at zero.
- A Macloren series is a specific case of a Taylor series.
- We actually haven't done anything with Taylor series, I
- was hoping to get there later.
- But the Macloren series is a really cool one because it's
- going to show us all these relationships between e and
- cosine and sine and eventually i and pi and you will
- find it exciting.
- The Macloren is that times x to the n over n factorial.
- That's what we said it was.
- So if this is our f of x, f of x is cosine of x, what
- does this turn into?
- Well, f of x is equal to, it equals f of zero times x to
- the zero over zero factorial, that's just one, right?
- Plus, now we're at n equals 1.
- It's the first derivative at zero.
- f prime of zero, well that's just equal to zero.
- And who cares what that-- that would be x to the
- first over 1, right?
- Now we're at the second derivative.
- The second derivative at zero is minus 1.
- Minus 1 times x squared over 2 factorial plus the
- third derivative at zero.
- The third derivative at zero we figured out was zero.
- Zero who cares what that is.
- It would have been x to the third over 3 factorial.
- And then what's the fourth derivative?
- The fourth derivative at zero is just equal to 1.
- So we have times 1 and then we're at x to the fourth
- over 4 factorial.
- Let me see if I can write this a little bit neater.
- The next one, the fifth derivative at zero times x to
- the fifth over 5 factorial.
- We'll keep going.
- Let me write this, clean this up and hopefully the pattern
- merges if it hasn't emerged already.
- f of x is equal to cosine of x is equal to-- let met get rid
- of the zeros-- 1 and then we have minus x squared
- over 2 factorial.
- This term, this goes away.
- This is a zero term.
- And the next one is a positive.
- Plus x to the fourth over 4 factorial.
- And the fifth term goes away.
- But then the cycle continues.
- The next one is going to be minus.
- Because we had minus 1 plus 1.
- It's going to be minus x to the sixth over 6 factorial.
- You could take the sixth derivative.
- You'll see that the derivative of minus sine of x is minus
- cosine of x, that's where we get the minus 1 from.
- And they we're going to go plus.
- So we're just taking all the even terms. x to the eighth
- over 8 factorial minus x to the 10th over 10 factorial.
- We could just keep going on and on and on.
- And so we have a situation where we can rewrite cosine of
- x is equal to the sum, if you believe that this Macloren
- series actually does converge to cosine of x over the entire
- domain of x, that's kind of an assumption we're making.
- Hopefully one day we will have the tools set to actually
- prove that as well.
- From n is equal to zero.
- So what's happening?
- We're taking all of the even powers.
- So we could say x to the 2n, that ensures that no matter
- what value of n I put in here I get an even numbers.
- So we'll go to the zeroth power then the second
- power, over 2n factorial.
- So that takes care of going from 1 to x squared over 2,
- to x to the fourth over 4 factorial, 6 over 6
- factorial, et cetera.
- But now we have to make it switched signs like that.
- Well let's just multiply it negative 1.
- Let's see what we can do.
- Negative 1 to the-- so we want the first term to be positive,
- the second term to be negative.
- So we could say times minus 1 to the n plus 1.
- Let's see if that works.
- When m is zero what's negative 1 to the n plus 1?
- zero, it would be minus 1.
- And then when it's 1-- When it's zero-- no, no it's just
- going to be negative 1 to the n.
- Because when it's zero, negative 1 to zero is 1.
- When it's 1, negative 1.
- So this will work out.
- So it's negative 1 to the n
- You could try it out.
- This is the n is equal to zero.
- We need to switch colors.
- That's n is equal to zero and here we get x to the zero over
- zero factorial, which is 1.
- We have negative 1 to the zero is 1, so that becomes 1.
- When n is equal to 1, this becomes x squared over 2
- factorial, we have negative 1 to the 1 power, so that's
- where you get the negative 1.
- And then when n is equal to 2, the negative 1 squared
- becomes positive again.
- So the negative 1 is what provides the
- alternating numbers.
- So pretty neat.
- We just figured out
- another way to represent cosine of x
- And it might be looking a little bit intresting to you
- that this representation kind of resembles part of the
- representation of e to the x.
- What's the difference between this and the e to the x? e to
- the x had the odd exponent terms and it didn't
- switch signs.
- But other than that, they're pretty much the same.
- So in the next video we'll do sine of x and then we'll
- try to put it all together.
- I'll see you soon.
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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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