Sine Taylor Series at 0 (Maclaurin)

Calculus

Sequences, series and function approximation

Maclaurin and Taylor series

Maclaurin and Taylor Series Intuition
Cosine Taylor Series at 0 (Maclaurin)
Sine Taylor Series at 0 (Maclaurin)
Taylor Series at 0 (Maclaurin) for e to the x
Euler's Formula and Euler's Identity
Complex number polar form intuition
Multiplying and dividing complex numbers in polar form
Powers of complex numbers
Visualizing Taylor Series Approximations
Generalized Taylor Series Approximation
Visualizing Taylor Series for e^x
Error or Remainder of a Taylor Polynomial Approximation
Proof: Bounding the Error or Remainder of a Taylor Polynomial Approximation

Sine Taylor Series at 0 (Maclaurin) Sine Taylor Series at 0 (Maclaurin)

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Sine Taylor Series at 0 (Maclaurin)

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In the last video we took the Maclaurin Series of Cosine of x
we approximated it using this polynomial
and we saw this pretty interesting pattern.
Let's see if we can find a similar pattern if we try
to approximate sine of x using a Maclaurin series.
And once again a Maclaurin series is really the same thing
as Taylor Series where we are centering our approximation
around x is equal to 0 .so this is a special case of a Taylor Series
so let's take f of x in this sitiuation to be equal to sine of x
f of x is now equal to sine of x and lets's do the same thing
that we did with cosine of x . Let's just take the different
derivatives of sine of x really fast. so if you have the first
derivative of sine of x is just cosine of x . The second derivative
of sine of x is derivative of cos x which is negative sine of x
the third derivative is going to be the derivative of this
so I will just write 3 in parentheses there in stead of doing
all the prime prime prime. so the third derivative is
derivative of this which is negative cosine of x . the fourth
derivative the fourth derivative is the derivative of this
which is positive sine of x again. so you see just like cosine of x
it kind of cycles after you take the derivative enough time.
and we care in order to the Maclaurin series we care
about evaluating the function and each of these derivatives at x is equal to 0 .
so let's do that. so for this let me do this in a different
color not that same blue. so i'll do it in this purple color.
so f that's hard to see I think. so let's do this in the
other blue color. so f of 0 in this situation is 0 and f the
first derivative evaluated at 0 is 1. cosine of 0 is 1
negative sine of 0 is going to be 0. so f prime prime
the second derivative evaluated at 0 is 0.
the third derivative evaluated at 0 is negative 1.
cosine of 0 is 1 you have a negative out there it is
negative 1 and the fourth derivative evaluated at 0 is
going to be 0 again. we could keep going once again seems like
there is a pattern 0 1 -1 0 then you are going to go back
to positive 1 so on and so forth . so let's find it's
polynomial representation using Maclaurin Series.
just a reminder this one up here this was approximately
cosine of x and you will get closer and closer
to cosine of x I am not rigorously showing you how close
and that is definitely the exactly the same thing as cosine of x
but you get closer and closer and closer to cosine of x
as you keep adding terms here and if you to infinity
you are going to be pretty much at cosine of x
now let's do the same thing for sine of x . so i'll pick
a new color. this green should be nice. so this is
our new p of x. so this is approximately going to be
sine of x as we add more and more terms
and so the first term here f of 0 that's just going to be 0
so we are not even going to need to include that. the next term
is going to f prime 0 which is 1 times x . so it's going to be x
then the next term is f prime the second derivative at 0
which we see here is 0. let me scroll down a little bit
it is 0 so we won't have the second term
this third term right here the third derivative of sine of x
evaluated at 0 is negative 1 so we are now going to have
a negative 1 . let me scroll down so you can see this
negative 1 this is negative 1 in this case times x the third
over 3 factorial. so negative x the third over 3 factorial
and then the next term is going to be 0 because that's
the fourth derivative . that's the fourth derivative evaluated
at 0 is the next coefficient. we see that that's going to be 0
so it's going to drop off and what you are going to see here
and actually maybe I haven't done enough pep terms
for you. for you to feel good about this let me do
one more term right over here just so it becomes clear
f of fifth derivative of x is going to be cosine of x
again. the fifth derivative let me do that in the same color
just so that it's consistent. the fifth derivative
the fifth derivative evaluated at 0 is going to be 1
so the fourth derivatives evaluated at 0 is 0. then you
go to the fifth derivative evaluated at 0 is going to be
positive 1 and if I kept doing this it would be positive 1 times
I would have to write 1 as a coefficient times x to the fifth
over 5 factorial so there is something interesting going
on here and for cosine of x I had 1 essentially 1 times
x to the zero then I don't have x to the first power
I don't have x to the odd powers actually then I just
have x to all the even powers and whatever power it is
I am dividing it by that factorial and then the sign
keeps switching and this is ,I shouldn't say this is an
even power because 0 really isn't , well I guess you
can view it as an even number cuz it.. I won't go into
all of that but it's essentially 0 2 4 6 so on and so forth
so this is interesting specially when you compare to
this . this is all of the odd powers this is x to the first
over 1 factorial I didn't write it here there's x to
the third over 3 factorial plus x to the fifth over
5 factorial. ya zero would be an even number . anyway
I don't.almost. my brain is in a different place right now
and you could keep going if we kept this process up
you would then keep switching signs x to the seventh
over 7 factorial plus x to the ninth over 9 factorial
so there is something interesting here you once
again see this kind of complimentary nature between
sine and cosine here. you see almost this..they kind of
they are filling each other's gaps over here cosine of x
is all of the even powers of x divided by that power's factorial
sine of x when you take it's polynomial representation
is all of the odd powers of x divided by it's factorial
and you switch signs. In the next video I'll do e to the x
and what's really fascinating is that e to the x starts
to look like a little bit of a combination here. but not quite
and you really do get the combination when you involve
imaginary numbers and that's when it starts to get

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