Maclaurin and Taylor series
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Maclaurin and Taylor Series Intuition
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Cosine Taylor Series at 0 (Maclaurin)
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Sine Taylor Series at 0 (Maclaurin)
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Taylor Series at 0 (Maclaurin) for e to the x
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Euler's Formula and Euler's Identity
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Complex number polar form intuition
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Multiplying and dividing complex numbers in polar form
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Powers of complex numbers
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Visualizing Taylor Series Approximations
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Generalized Taylor Series Approximation
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Visualizing Taylor Series for e^x
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Error or Remainder of a Taylor Polynomial Approximation
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Proof: Bounding the Error or Remainder of a Taylor Polynomial Approximation
Taylor Series at 0 (Maclaurin) for e to the x Taylor Series at 0 (Maclaurin) for e to the x
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- Now, let's do something pretty interesting, this
- in some degree will be one of the easiest functions to find
- the maclaurin series representation of
- let's try to approximate e^x
- f(x) is equal to e^x
- and what makes this really simple
- is when you take the derivative
- and frankly this is one of the amazing things
- about the number e
- is that when you take the derivative
- of e^x, you get e^x
- so this is equal to f ' (x)
- this is equal to f 2nd derivative of x
- this is equal to f 3rd derivative of x
- this is equal to the nth derivative of x
- it's always equal to e^x
- that's the first mind-blowing thing about e
- you can just keep taking it's derivative
- the slope at any point on that curve
- is the same as the value of the point on that curve
- that's kind of crazy
- with that said
- let's take its maclaurin representation
- so we have to find
- f(0), f ' (0), the 2nd derivative of 0
- and when we take e^0
- e^0 is just equal to one
- and so this is going to be equal to f(0)
- this is going to be equal to f ' (0) it's going to be
- equal to any of the derivatives evaluated
- the nth derivative is going to be valued at zero
- and that's what makes the Maclaurin series
- fairly straightforward
- if I want to approximate
- e^x using a maclaurin series
- so e^x
- and I'll put a little approximately over here
- and we'll get closer and closer
- to the real e^x and as we keep adding more and more terms
- especially if we add an infinite number of terms
- it would look like this:
- f(0) and we do it in
- what colors did I use for cosine and sine?
- pink and green
- I'll use the yellow here
- f(0) is 1
- plus f ' (0) times x
- f ' (0) is also one
- plus x
- plus this is also one
- so x^2 / 2!
- all of these things are going to be one
- this is one
- this is one
- when we're taking about e^x
- so you go to the third term
- this is one
- you just have x^3 / 3!
- plus x^3 / 3!
- and I think you see the pattern here
- we just keep adding terms
- x^4 / 4!
- plus x^5 / 5! plus x^6 over 6!
- and something pretty neat is starting
- to emerge
- this is just really cool
- that e^x can be approximated by
- 1 plus x plus x^2 / 2!
- plus x^3 / 3!
- once again, e^x is starting to look like
- a pretty cool thing
- this also leads to other interesting results
- like if you want to approximate e
- you just evaluate this as x is equal to one
- so this is, well, so you just say e is e ^ 1
- and this is going to be approximately equal
- to this polynomial evaluated at 1
- if x is 1 here, then we make x equal to one
- over here
- so it will be, one plus one
- plus one over 2!
- plus one over 3!
- plus one over 4!
- and so on and so forth all the way into infinity
- and you could also do this as one
- over 1 ! as well
- but what's really cool is that
- this is another way to represent e
- it shows that e
- once again shows up as
- kind of 2 + 1/2 + 1/6
- if you keep doing this, you get close
- to the number e
- but that by itself isn't the only
- fascinating thing
- if we look back at the Maclaurin Representation
- of these other functions.
- cosine (x)
- let me copy and paste cosine(x)
- so, cosine (x) right up here
- let me do my best to copy and paste the whole thing
- so that is cosine(x) and let's do the same thing
- for sine(x) that we did last video
- so the sine of x
- let me copy and paste that
- so do we see any relationship between
- these approximations?
- so before, you probably would have
- guessed there's some relationship
- between cosine and sine
- but what about e^x?
- so what you see here
- is that cosine(x) looks a lot like this term
- plus this term, although we probably want
- to put a negative out front here
- so we have a negative version of this term
- right here
- plus this term right here
- plus a negative version of this term
- right over here
- and sine (x) looks just like
- this term plus a negative version of
- this term
- plus this term
- plus a negative version of the next term
- so if we could somehow reconcile the
- negatives in some interesting way
- it looks like e^x
- at least the polynomial representation
- of e^x
- is somehow related to a combination
- of the polynomial representations of
- cosine of x and sine of x
- so this is starting to get
- really really really cool
- we're starting to see a connection
- between something related to
- compound interests
- or a function whose derivative
- is always equal to that function
- and these things that come out of the unit circle
- and oscillatory motion and all kinds of things
- there starts to seem a kind of pure
- connectiveness there. but i'll leave you
- there on that video and in the next video
- I'll show you how to reconcile these three
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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