A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 5 years ago

calc II: find the taylor series (at x=0) of cos(x^2) can anyone help me i need reassurance to make sure my answers is correct ?

  • This Question is Closed
  1. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    whatcha got so far?

  2. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    didn't do it yet im doing it now i just did the first problem which was cos(x)

  3. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    wouldnt x=0 be a Maclaurin series?

  4. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    can you explain in your way of understanding when i would use Maclaurin series and taylor and which would be easier because my professor said to find it using taylor series

  5. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    as well as the answer

  6. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Maclaurin series is just s special case of taylor series when it's centered at 0.

  7. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    its been awhile, but i think when f(a), that taylor series is used; but when a=0; then f(0); and the Maclaurin is used...

  8. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    because ive got a final tomorrow and im freaking out been studying and idk everything that im supposed to yet also yes i think thats what he said

  9. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    cos(0) -sin(0)-cos(0)+sin(0)+cos(0)....like that rght?

  10. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    forgot the factorials....

  11. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    cos(x^2) = 1- sin(0)x -cos(0)x^2+sin(0)x^3+cos(0)x^4 -------- -------- --------... 2! 3! 4!

  12. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    right?

  13. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    every other one goes to zero since sin(0)=0... which leaves us with: 1 -1 +1 -1 +1 -1 +1 -- -- -- -- -- -- ... right? 2! 4! 6! 8! 10! 12!

  14. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    x^2...x^4....x^6....

  15. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i believe so

  16. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i havent finished yet

  17. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I think the point here is the derivatives. I have does up to 8th derivatives, all are zeros at x=0 except the fourth and the eighth.

  18. amistre64
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i got no idea what to do after that :)

  19. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Oh I got an idea. We can derive the Taylor series of cos(x^2) from that of cosx. Since tge expansion of cosx is well known

  20. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So the Taylor series of cos x centered at 0 is: \[\cos x=\sum_{n=0}^{\infty}(-1)^n{x^{2n} \over (2n)!}\] Then:\[\cos(x^2)=\sum_{n=0}^{\infty}(-1)^n{(x^2)^{2n} \over (2n)!}=\sum_{n=0}^{\infty}(-1)^n{x^{4n} \over (2n)!}\]

  21. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Do you know how to find the Taylor series of cos x at x=0?

  22. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes i did alread

  23. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    here

  24. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Good. So, I just used that expansion and replaced x by x^2.

  25. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    F^(1) (x^2)=-sin x^2?

  26. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    f2 = -cosx^2?

  27. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    What's that?

  28. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    derivatives of the function

  29. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    of cos(x^2)?

  30. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    then your evaluate them then move on to the generate a formul

  31. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    part

  32. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yeah. But the first derivative is not -sin(x^2). You should use chain rule here. It should be: -2xsin(x^2)

  33. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    kk i didnt know that

  34. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    And for the second derivative, you should apply both the product rule and chain rule.

  35. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I did it and found that all derivatives are zeros except those who are multiple of 4. I mean the 4th, 8th, 12th.. derivatives.

  36. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    kk sec....... im going to try to do it now and thanks for this your a life saver

  37. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what did you get for the second derivative

  38. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    -4x^2cos(x^2)-2sin(x^2)

  39. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the third derivative

  40. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and you said the fourth derivative should be the one that isnt zero when evaluating it right?

  41. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Yep. You should not bother yourself with finding these derivatives. I am pretty sure the question wants you to use the method I used.

  42. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hey buddy also as i try to comprehend and understand this can you take a look at this Let f(x) = the cube root of x. Find the area of the surface generated by revolving the curve y = f(x) around the x-axis, for x ranging from 0 to 1.

  43. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok explain to me why it really doesnt matter to find the derivatives

  44. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that is if you can

  45. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You told me you know the Taylor series of cosx, then you just have to substitute for each x in that expansion of cosx by x^2. That's all.

  46. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes so the ans should be 1-1/2x^4+1/24x^8 then whats the nxt one

  47. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    -x^12/720

  48. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok now explain to me why not to worry about the derivative part because i must know this for tomorrow

  49. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is there any tricks to getting the correct answere fast4r then going through each an evaluating it

  50. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    because after you find this i must know if it converges or diverges

  51. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I mean you shouldn't worry about it in this particular problem, since you can find the Taylor series without doing any derivatives.

  52. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh ok

  53. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    can you help me figure out if it now converges or diverges?

  54. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You know how to use the ratio test?

  55. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is there an easy way to know it so i know? or no

  56. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    because i dont understand exactly bc my professor did a poor job explaining it

  57. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hmm I am not sure if there is a simpler way. All I know that you need to know the ratio test and the root test to find the radius and interval of convergence for power series such as Taylor series.

  58. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is that*

  59. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    like ive got the definitions i front of me but explain why it converges or diverges

  60. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ill talk to you on the other page this one i need for this page

  61. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.