Numerical analysis how many of the following are correct ? In the following ques
ID: 3123535 • Letter: N
Question
Numerical analysis
how many of the following are correct ?
In the following questions assume that P_n(x), T_n(x) and h_2n+1(z) denote Lagrange, Taylor and Hermit polynomials respectively of a function f(x) with some set of nodes. Assume also that by osculating polynomials we mean osculating polynomials of the same function f(x). How many of the following assertions are correct? (i) An osculating polynomial can be considered as a limit of a sequence of Lagrange polynomials with nodes condensing to the nodes of the osculating polynomial. (ii) The largest degree which the Lagrange polynomial for the data with 10 nodes can have is 9. (iii) Taylor polynomial can be considered as a limit of a sequence of Lagrange polynomials all nodes of which condense to the single node of the Taylor polynomial. (iv) Osculating polynomials are a special case of Lagrange polynomials. (A) 4 (B) 0 (C) 3 (D) 1 (E) 2Explanation / Answer
(ii) Correct. In general through (n+1) distinct point, we can construct a unique polynomial of degree less than equal to n because of the requirement of (n+1) interpolating conditions on interpolating polynomial.
(iii) Incorrect. As for taylor polynomial, we require the existence of derivative of the function.
(iv) B). From osculation polynomial, we can obtain Lagranges's polynomial by taking mi=0 (i=1,2,...,n). That is without considering the derivative conditions.
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