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Spivak's Calculus - Capital-Pi/Function Proof

  1. Oct 31, 2012 #1
    1. The problem statement, all variables and given/known data
    If [itex]x_1,...,x_n[/itex] are distinct numbers, find a polynomial function [itex]f_i[/itex] of degree [itex]n-1[/itex] which is ##1## at [itex]x_i[/itex] and ##0## at [itex]x_j[/itex] for [itex]j=/=i.[/itex] (This product is usually denoted by [see below] the symbol Capital Pi playing the same role for products that Capital Sigma plays for sums.)


    2. Relevant equations
    ##n##
    [itex]\prod(x-x_j)[/itex]
    [itex]j=1[/itex]
    [itex]j=/=i[/itex]


    3. The attempt at a solution
    This is a question out of Spivak's Calculus (particularly Chapter Three, Question Six). I've looked in the answer book, and he comes to the conclusion that
    [itex]f_i (x) =[/itex]
    ##n##
    [itex]\prod(x-x_j)[/itex]
    j=1
    j=/=i
    ##divided## ##by##
    ##n##
    [itex]\prod(x_i - x_j)[/itex]
    j=1
    j=/=i
    But I'm not satisfied with this answer alone
    I'm almost positive that I'm over-complicating this problem, so I need help simplifying it.
    -- I'm not sure why the function needs to be subscripted as [itex]f_i[/itex]
    -- I'm not sure where the un-subscripted ##x## comes from in the Pi equation; does it rely on anything, or it just any random number?
    -- What CAN [itex]x_j[/itex] and [itex]x_i[/itex] equal? Anything that equals 1 and 0 respectively (in the function)?
    -- What does it mean that ##j=1## , does it mean that it is the first number in the previously mentioned series?
    I suppose I have more questions, but I've confused myself and really need help being walked through it.

    Thanks
     
    Last edited: Oct 31, 2012
  2. jcsd
  3. Oct 31, 2012 #2

    lurflurf

    User Avatar
    Homework Helper

    That is nearly unintelligible, but I think you are talking about
    http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html

    -- I'm not sure why the function needs to be subscripted as fi
    You are not finding a function, but n functions hence the subscript
    -- I'm not sure where the un-subscripted x comes from in the Pi equation; does it rely on anything, or it just any random number?
    x is the variable we are finding functions, not numbers
    -- What CAN xj and xi equal? Anything that equals 1 and 0 respectively (in the function)?
    the xi are distinct numbers
    -- What does it mean that j=1 , does it mean that it is the first number in the previously mentioned series?
    x1 is the first number, though it does not say the numbers are in order, it would not matter.

    Maybe you will understand the general case if you do one specific example
    find 4 cubic polynomials f1,f2,f3,f4 such that
    f1(0)=1
    f1(0)=0
    f1(0)=0
    f1(0)=0
    f2(1)=0
    f2(1)=1
    f2(1)=0
    f2(1)=0
    f3(3)=0
    f3(3)=0
    f3(3)=1
    f3(3)=0
    f4(4)=0
    f4(4)=0
    f4(4)=0
    f4(4)=1

    expressed more briefly as
    fi(xj)=δij
    where i,j=1,2,3,4
    (x1,x2,x3,x4)=(0,1,3,4)
    and
    δij=1 when i=j
    δij=0 otherwise
     
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