which of the following functions is represented by the Maclaurin series sum from n=0 to infinity [(-9)^n x^(4n)]/(2n)! a) cos(3x^2) b)sin(3x^2) c)cos(9x^4) d)arctan(3x) e)e^(-3x^2)

Question

myininaya · Answer

So we know that the Maclaurin series is a taylor series expansion of a function about 0.
That is 
$$f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+...+ \frac{f^{(n)}(0)}{n!}x^n+...$$

myininaya · Answer

We are given the expansion and we are asked to find the function 
The expansion is:
$$1+\frac{-9}{2}x^4 +\frac{81}{4!}x^{8}+\frac{-729}{6!}x^{12}+...$$

myininaya · Answer

You could eliminate some of these by evaluating f(0)
If you don't get 1 then throw it out

myininaya · Answer

Are you there?

anonymous · Answer

yes sorry!

myininaya · Answer

Which choices can we go ahead and eliminate?

myininaya · Answer

We are given f(0)=1
which of the functions given satisfy that.

anonymous · Answer

a c and e

anonymous · Answer

so we can eliminate b and d

myininaya · Answer

yep

myininaya · Answer

now looking at a can you find it's fifth term which is 
$$\frac{f^{(4)}(0)}{4!}x^4$$

myininaya · Answer

we are trying to see if this is the same as 
$$\frac{-9}{2}x^4 $$

myininaya · Answer

you are going to need the 4th derivative

anonymous · Answer

okay ill calculate that

anonymous · Answer

I dont think that a will be the answer because these derivatives are getting super long

anonymous · Answer

i got the fourth derivative but if i plug in 0 i get 0

anonymous · Answer

i did that wrong, now i got -108 after plugging in

myininaya · Answer

$$f(x)=\cos(3x^2)$$
$$f'(x)=-6x \sin(3x^2) $$
$$f''(x)=-6\sin(3x^2)-6x(6x)\cos(3x^2)=-6\sin(3x^2)-36x^2\cos(3x^2)$$
So fall all of these will give us 0 when pluggin in 0 which is a good sign 
since in the other series we don't see another term until x^4
$$f^{(3)}(x)=-36xcos(3x^2)-72xcos(3x^2)+36(6x)\sin(3x^2) \ =-36xcos(3x^2)-72xcos(3x^2)+216xsin(3x^2)$$
still have zero when we plug in 0


ok...I will stop writting what I was writing 
yep f^(4)(0)=-108

myininaya · Answer

so that means the fourth term is?

myininaya · Answer

remember we are trying to compare it to 
$$\frac{-9}{2}x^4 $$

anonymous · Answer

-108/4! X^4

myininaya · Answer

and does -108/4! simplify

myininaya · Answer

or reduce

anonymous · Answer

yes it simplifies to -9/2

anonymous · Answer

so a is the answer!

myininaya · Answer

It looks that way. I would look at the other 4terms from the other functions
but that seems to be a long process

anonymous · Answer

thanks for your help!! I really apprectiate it!

myininaya · Answer

np