Need help with maclaurin series question

[Yura]

its using the Maclaurin series, i have already worked out the equations:

cos x = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + (x^8)/8! - (x^10)/10! + ...
e^x = 1 + x + (x^2)/2! +(x^3)/3! + (x^4)/4! + (x^5)/5! + (x^6)/6! + ...

how do i use these two results to obtain the first 6 terms on the maclaurin series for e^x.cos x?

i've been working at this question for ages and cannot get it.
thanks heaps if anyone can help me out.

 

[BobG]

You want the first terms of $$cos x e^x$$?

You do it the same way you did each term individually. You find the first derivative (you need the product rule), the second derivative (even longer), the third derivative (not so bad - a couple of terms in the second derivative canceled out), and so on.

Evaluate each of those derivatives and apply them the same way you did when you found each individual term.

Edit: Your first 6 coefficients should wind up being 1,1,0,-2,-4, and 0 if you find your derivatives correctly.

 

[Yura]

ok i'll try that, thanks for that. i didnt think to try it
thanks again

 

[shmoe]

You can also multiply the series together, keeping as many terms as necessary:

e^x.cos x=(1 + x + (x^2)/2! +(x^3)/3! +...)(1 - (x^2)/2! + (x^4)/4! -...)
=1(1 - (x^2)/2! + (x^4)/4! -...)+x(1 - (x^2)/2! + (x^4)/4! -...)+x^2/2!(1 - (x^2)/2! + (x^4)/4! -...)+...

Distribute over the brackets and collect terms. How many terms you need from each bracket depends on how many terms in the final series you want.

 

[Yura]

hmm.. i just tried it in maple 6 but it didn't come out too well because i can't really work it.. i think i'll try it manually for now. i'll try work it your method as well shmoe.
thanks

 

[shmoe]

code that works in maple V:

series(exp(x)*cos(x),x=0,6);

Change the 6 if you want more or less terms.

But do it by hand too!

 

[Yura]

ah k, thanks

 

[Yura]

sorry about this but I've done the code for maple and it worked but i was trying to do it manually as well but i can't seem to get past the third differentiation.
i was using maple 6 to solve it but it was only giving me partial differentiations and i couldn't complete it.

this is what i have (im not sure if its right) so far:
f(x) = e^x*cosx
f^1(x) = e^x*ln(e)*cosx - e^x*sinx
f^2(x) = e^x*ln(e)^2*cosx - 2e^x*ln(e)*sinx - e^x*cosx

and that's about how far i got. i tried getting to the next one but i keep messing it up and getting it all wrong. could someone please help me do this. i don't know what I am going to do because i need ssix terms and if some of the terms wind up like (sin 0) then i'll need to do a lot of differentiations...
is there a shortcut rule to differentiating this?
help is really appreciated.

 

[shmoe]

is there a shortcut rule to differentiating this?

No shortcut I can think of. This is probably why you were given the series for e^x and cos(x), multiplying the series is less work and less error prone (to me anyways).

to differentiate a function 6 times in maple you can use diff(f(x),x,x,x,x,x,x); Change the number of x's to change the number of times you differentiate.

 

[Data]

Don't forget that $$\ln{e} = 1$$!

 

[SomeRandomGuy]

sorry about this but I've done the code for maple and it worked but i was trying to do it manually as well but i can't seem to get past the third differentiation.
i was using maple 6 to solve it but it was only giving me partial differentiations and i couldn't complete it.

this is what i have (im not sure if its right) so far:
f(x) = e^x*cosx
f^1(x) = e^x*ln(e)*cosx - e^x*sinx
f^2(x) = e^x*ln(e)^2*cosx - 2e^x*ln(e)*sinx - e^x*cosx

and that's about how far i got. i tried getting to the next one but i keep messing it up and getting it all wrong. could someone please help me do this. i don't know what I am going to do because i need ssix terms and if some of the terms wind up like (sin 0) then i'll need to do a lot of differentiations...
is there a shortcut rule to differentiating this?
help is really appreciated.

Why are you trying to differentiate the terms? The method your doing is like trying to shovel snow with a spoon... it's way to hard. You know the series for cos(x) and e^x, just multiply the 2 series together term by term like someone said.

 

[Yura]

thanks, i think i have the series now:
f(x) = 1 + x – (x^3)/3 – (x^4)/6 – (x^5)/30 + (8x^7)/7!

but I am not sure why they left the 2nd derivative out in maple 6 but not the 6th one

 

[shmoe]

In the answer you're quoting from maple the last term isn't 0*x^6, it's O(x^6). This is the usual Big-Oh notation.

 

[Yura]

ah, ok thanks!