prove the following Hermite polynomial property H2,(0) = (-1)" (2n)!
with distinct nodes, prove there is at most one polynomial of degree ≤ 2n + 1 that interpolates the data. Remember the Fundamental Theorem of Algebra says a nonzero polynomial has number of roots ≤ its degree. Also, Generalized Rolle’s Theorem says if r0 ≤ r1 ≤ . . . ≤ rm are roots of g ∈ C m[r0, rm], then there exists ξ ∈ (r0, rm) such that g (m) (ξ) = 0. 1. (25 pts) Given the table...
Let f(x) = xlnx. Approximate f(2) by the Hermite interpolating polynomial using x0 = 1 and x1 = e and compare the error.(e ≈ 2.7...)
Prove that P2n(0)= (-1)n ((2n-1)!!/(2n)!!) using the generation function and a binomial expansion. Show that (sqrt(pi)(4n-1)/(2gamma(n+1)gamma(3/2-n))=(-1)n-1((2n-3)!!/(2n-2)!!)(4n-1)/2n
Prove the following integral. ., xPn-1(x)P,(x)dx = 2n (2n-1)(2+1) 2 use (n + 1)Pn+1(x) – (2n +1)xPn(x) + nPn-1(x) = 0, L, Pn(x)Pm (x) dx = 0, S, Pn(x)2 dx = 2n+1
2. Prove that lim (-1)"+1 0. 72-00 n 2n 3. Prove that lim noon + 1 2. 80 4. Prove that lim n-+v5n 0. -7 9 - in 5. Prove that lim n0 8 + 13n 13
1. Prove that 1.3....2n-1 1. Prove that-.-. ...--ㄑㄧ for any n E N 2n V2n+1
A significant outcome from Hermite's equation are the Hermite Polynomials Hn(x). They are the nth-order polynomial solution toHermite's equation(v or v,) multiplied by an appropriate constant cn such that the coefficient of x" is 21. (a) Find the first three even Hermite Polynomials. That is, to find Ho(x) find a constant co to multiply with y1 when α-0 such that the coefficient in front of xo is 20. To find H2 (x) find a constant c2 to multiply with y1...
T47:02 731 VPN 97% 5 TOⓇ + : 4. Find the Hermite interpolating polynomial which interpolates the values f(1) = 4, f'(1) = -3, f(4) = 13, f'(4) = 9 and verify your answer. 5 13
2. (a) Prove the transitive property for polynomial-time mapping reductions (b) Using the transitivity, show that if A Sp B and A is NP-Hard, then B is NP-Hard as well
m2 2. Prove that lim -+0n3 + 1 -=0. 3 5 100 3n2 + 2n - 1 3. Prove that lim = 5n2 +8 cos(n) 4. Prove that lim = 0. n-700 m2 + 17 5. Prove that lim (Vn+1 - Vn) = 0 Hint: Multiply Vn+1-vñ by 1 in a useful way. In particular, multiply Vn+1-17 by Vn+1+vn