A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 5 years ago

The Taylor polynomial of degree 100 for the function f about x=3 is given by \[p(x)= (x-3)^2 - ((x-3)^4)/2! +... + [(-1)^n+1] [(x-3)^n2]/n! +... - ((x-3)^100)/50!]/ What is the value of f^30 (3)?

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

    Plug in \(n=15\) to the expression \((-1)^{n+1}/n!\)

  2. watchmath
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Because \(f^{30}(3)\) is the coefficient of \(x^{30}\)

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

    So each term is\[((-1)^{n+1}/n!)(x-3)^{2n}\] and \[(f^{(n)}(3)/n!)(x-3)^n\] so can I say \[(f^{(30)}(3)/30!)(x-3)^{30}=((-1)^{30+1}/30!)(x-3)^{2*30}\] I don't really get where the 15 comes from

  4. watchmath
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    remember that the exponent on \((x-3)\) is \(2n\). And we want this \(2n=30\). So we need to take \(n=15\). Since we are looking for the coefficient of \((x-3)^{30}\)

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

    so \[1/15! = f^{(30)}(3)/30!\]

  6. watchmath
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Ok, let me make this more clear. We want fo select the \(n\) so that we know the coefficient of \((x-3)^{30}\). We know that the coefficient of \((x-3)^{2n}\) is \((-1)^{n+1}/n!\). So in order to find the coefficient of \((x-3)^{30}\) we need to choolse \(n=15\). In that case the coefficient would be \((-1)^{15+1}/15!=1/15!\). So \(f^{(30)}(3)=1/15!\)

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

    Why is the coefficient of \((x-3)^{n}\), \(f^{n}(3)\) and not \(f^{n}(3)/n!\) Isn't each term \((1/n!)(f^{(n)}(3))(x-3)^n\)?

  8. watchmath
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Ah you are right! :)

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

    Thank you so much, I never would have gotten there in the first place :P

  10. 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.