How do you find #f^8(0)# where #f(x)=cos(x^2)#?

Answer 1

# f^((8))(0) = 1680 #

It doesn't matter how we get the power series—it could come from the Binomial Theorem, a Taylor series, or a Maclaurin series—we know that the power series for a function is unique.

As we are looking for #f^((8))(0)# let us consider the Taylor Series for #f(x)# about #x=0#, ie its Maclaurin series.

By definition:

# f(x) = f(0) + f'(0)x + (f''(0))/(2!)x^2 + (f^((3))(0))/(3!)x^3 + ... + (f^((n))(0))/(n!)x^n + ... #
The (well known) Maclaurin Series for #cos(x)# is given by:
# cosx = 1 - x^2/(2!) + x^4/(4!) - ... \ \ \ \ \ \ AA x in RR#
And so it must be that the Maclaurin Series for #cos(x^2)# is given by:
# cos(x^2) = 1 - (x^2)^2/(2!) + (x^2)^4/(4!) - ... # # " " = 1 - (x^4)/(2!) + (x^8)/(4!) - ... #
And if we equate the coefficients of #x^8# from this derived series and the definition then we have:
# (f^((8))(0))/(8!)= 1/(4!) #
# :. f^((8))(0) = (8!)/(4!) # # " " = (8*7*6*5*4!)/(4!) # # " " = 8*7*6*5 # # " " = 1680 #
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Answer 2

To find ( f^8(0) ), where ( f(x) = \cos(x^2) ), we need to repeatedly apply the function ( f(x) ) eight times to the initial value ( x = 0 ).

Starting with ( f(x) = \cos(x^2) ), we find:

[ f^2(x) = f(f(x)) = \cos\left((\cos(x^2))^2\right) ]

[ f^3(x) = f(f^2(x)) = \cos\left(\cos\left((\cos(x^2))^2\right)^2\right) ]

[ \vdots ]

[ f^8(x) = f(f^7(x)) = \cos\left(\cos\left(\cos\left(\cos\left(\cos\left(\cos\left(\cos\left(\cos(x^2))^2\right)\right)^2\right)\right)^2\right)\right) ]

Then, to find ( f^8(0) ), we substitute ( x = 0 ) into ( f^8(x) ) as follows:

[ f^8(0) = \cos\left(\cos\left(\cos\left(\cos\left(\cos\left(\cos\left(\cos\left(\cos(0)^2\right)\right)^2\right)\right)^2\right)\right)^2\right) ]

Evaluating this expression will give us the value of ( f^8(0) ), which represents the result of applying the function ( f(x) = \cos(x^2) ) eight times to ( x = 0 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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