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osnarf
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Question from spivak's calculus - 3rd edition - chapter 3, question 6(a).
If x1, ..., xn are distinct numbers, find a polynomial function fi of degree n - 1 which is 1 at xi and 0 at xj for j =/= i. Hint, the product of all (x - xi) for j =/= i, is 0 at xj if j =/= i. (This product is usually denoted by see below, the symbol (capital pi) playing the same role for products that sigma plays for sums.)
n
[tex]\prod(x - xj)[/tex]
j = 1
j =/= i
I really don't even know where to start, the question is really confusing. If someone could just explain the question better that would be really appreciated. I'm confused by just about every word:
-x1,...,xn <--- Are these just random numbers? or are they increments along the x axis? or ?
-why does the function have a subscript (...find a polynomial function fi)?
-which x is xi?
-which x is xj?
-when he says (x - xi), I would assume x is not a distinct number, but the variable x, and that xi is meant to represent one of each of the distinct x's each time and that they are multiplied together, but later he asks for a specific function value at xi. I don't understand that.
-why did he need to write j =/= i twice, about 5 words apart (not trying to be sarcastic I'm sure there's a reason i just don't know why).
-in response to the capital-Pi eqation: so xj is x1?
Thanks for helping, I'm sure I can figure out the problem once i understand what it is. If someone could just explain it to me I would really appreciate it.
Homework Statement
If x1, ..., xn are distinct numbers, find a polynomial function fi of degree n - 1 which is 1 at xi and 0 at xj for j =/= i. Hint, the product of all (x - xi) for j =/= i, is 0 at xj if j =/= i. (This product is usually denoted by see below, the symbol (capital pi) playing the same role for products that sigma plays for sums.)
Homework Equations
n
[tex]\prod(x - xj)[/tex]
j = 1
j =/= i
The Attempt at a Solution
I really don't even know where to start, the question is really confusing. If someone could just explain the question better that would be really appreciated. I'm confused by just about every word:
-x1,...,xn <--- Are these just random numbers? or are they increments along the x axis? or ?
-why does the function have a subscript (...find a polynomial function fi)?
-which x is xi?
-which x is xj?
-when he says (x - xi), I would assume x is not a distinct number, but the variable x, and that xi is meant to represent one of each of the distinct x's each time and that they are multiplied together, but later he asks for a specific function value at xi. I don't understand that.
-why did he need to write j =/= i twice, about 5 words apart (not trying to be sarcastic I'm sure there's a reason i just don't know why).
-in response to the capital-Pi eqation: so xj is x1?
Thanks for helping, I'm sure I can figure out the problem once i understand what it is. If someone could just explain it to me I would really appreciate it.