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Thread: Understanding what is meant by the Taylor Series polynomial problem listed

  1. #1
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    Understanding what is meant by the Taylor Series polynomial problem listed

    I need help understanding the wording of this problem. I think it wants me to write a Taylor Series polynomial for the arctanx....then im not really sure even where to go after that. With writing the polynomial...what does all the p(j) f(j) stuff getting at?

    Give the unique polynomail p of degree <= 2 such that p(j)(1)=f(j)(1) for j=0,1,2.
    determine a reasonable constat C such that abs(f(x) -p(x)) <= C(abs(x-1)^3 for all x that belongs to [1/2,3/2].
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  2. #2
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    Re: Understanding what is meant by the Taylor Series polynomial problem listed

    I am not sure. I do not have your book, and the notation is ambiguous. I would suspect it means the following:

    $p^{(j)}(1)$ is the value of $j$-th derivative of the Taylor polynomial that approaches the function $\arctan x$ at $x=1$.

    $f^{(j)}(1)$ is the value of the $j$-th derivative of $\arctan x$ at $x=1$.

    So, you have:

    $$p^{(0)}(1) = a(1)^2+b(1)+c = f^{(0)}(1) = \arctan(1) = \dfrac{\pi}{4}$$

    $$p^{(1)}(1) = 2a(1)+b = f^{(1)}(1) = \dfrac{1}{1+(1)^2} = \dfrac{1}{2}$$

    $$p^{(2)}(1) = 2a = f^{(2)}(1) = -\dfrac{2(1)}{(1+1^2)^2} = -\dfrac{1}{2}$$

    This gives $a = -\dfrac{1}{4}$, $b = 1$, and $c = \dfrac{\pi-3}{4}$. (That is, if my understanding of the problem is correct).
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    Re: Understanding what is meant by the Taylor Series polynomial problem listed

    I think that is correct...so when putting together the Polynomial how would you write it? In form m((x-1))^2 etc...
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    Re: Understanding what is meant by the Taylor Series polynomial problem listed

    Do the same thing but instead of $ax^2+bx+c$ use $p(x)=a(x-1)^2+b(x-1)+c$.

    Now, you have:

    $$a(1-1)^2+b(1-1)+c=\dfrac \pi 4$$
    $$2a(1-1)+b=\dfrac{1}{2}$$
    $$2a=-\dfrac 1 2$$

    So $a=-\dfrac 1 4, b=\dfrac 1 2, c=\dfrac \pi 4$.
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    Re: Understanding what is meant by the Taylor Series polynomial problem listed

    Quote Originally Posted by SlipEternal View Post
    Do the same thing but instead of $ax^2+bx+c$ use $p(x)=a(x-1)^2+b(x-1)+c$.

    Now, you have:

    $$a(1-1)^2+b(1-1)+c=\dfrac \pi 4$$
    $$2a(1-1)+b=\dfrac{1}{2}$$
    $$2a=-\dfrac 1 2$$

    So $a=-\dfrac 1 4, b=\dfrac 1 2, c=\dfrac \pi 4$.
    Hello,
    What is the meaning of "determine a reasonable constat C such that abs(f(x) -p(x)) <= C(abs(x-1)^3 for all x that belongs to [1/2,3/2]." You have not amswered that.
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    Re: Understanding what is meant by the Taylor Series polynomial problem listed

    Quote Originally Posted by Vinod View Post
    Hello,
    What is the meaning of "determine a reasonable constat C such that abs(f(x) -p(x)) <= C(abs(x-1)^3 for all x that belongs to [1/2,3/2]." You have not amswered that.
    My goal was not to provide a complete solution for the OP, but to offer a base point from which to continue. Since I have not heard a follow-up question from the OP, I hope he/she was able to proceed and complete the problem without further assistance.
    Thanks from topsquark
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