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J0 3. Determine cı and cz in f(x) dac1f(0) + cf(1) so that it is exact...
2. Determine cı and c2, 1, and 2 such that the integration formula gives the exact...
2. Determine cı and c2, 1, and 2 such that the integration formula gives the exact result whenever f(x) is a polynomial of degree three or less.
4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree...
4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree of precision of this rule, that is, find the highest degree of polynomial for which the above rule is exact. (10 marks) OC
4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree of precision of this rule, that is, find the highest degree of polynomial for which the above rule is exact. (10 marks) OC
2. (4) Determine if each of the following is a subspace of P2[x] (the set of...
2. (4) Determine if each of the following is a subspace of P2[x] (the set of all polynomials of degree no more than 2). (a) All polynomials in P2[x] that satisfy f(1) = f(0) + 1; (b) All polynomials in P2 [x] that satisfy f(2x) = f(-x). (Hint: use the condition to find an equation of the coefficients of the polynomial f(x).)
Suppose the density for a random variable is given by the following: f(x) = Cz" for...
Suppose the density for a random variable is given by the following: f(x) = Cz" for 1 < x < 2, and f(x) = 0 otherwise. Find the value of C, and then find the mean of this random variable.
Problem 4 Let f(x) be a cubic polynomial defined on interval [−1, 1]. Determine a Gaussian interg...
Problem 4 Let f(x) be a cubic polynomial defined on interval
[−1, 1]. Determine a Gaussian intergration formula with minimal
number of nodes such that the integral formula Xn i=0 f(xi)wi is
exact for cubic polynomials.
Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,. Determine a Gaussian intergration formula with minimal number of nodes such that the integral formula is exact for cubic polynomials.
Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,....
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using...
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of degree 0 and 1,
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of...
1. (Taylor Polynomial for cos(ax)) For f(x)cos(ar) do the following. (a) Find the Taylor polynomials T(x) about 0 for f...
1. (Taylor Polynomial for cos(ax)) For f(x)cos(ar) do the following. (a) Find the Taylor polynomials T(x) about 0 for f(x) for n 1,2,3,4,5 (b) Based on the pattern in part (a), if n is an even number what is the relation between Tn (x) and TR+1()? (c) You might want to approximate cos(az) for all in 0 xS /2 by a Taylor polynomial about 0. Use the Taylor polynomial of order 3 to approximate f(0.25) when a -2, i.e. f(x)...
3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0,...
With explanation!
3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where...
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where P2(x) is the polynomial of degree at most two which interpolates f at 0, 1/2, and 1. (a) Write P2(x) in Lagrange form and prove that Q[F] o [s0 f(0) + 4f 45 (2) +scn)] (2) (b) Consider now a general interval [a, b] and the integral só f(x)dx. Do the change of variables x = a + (b − a)t to transform the...
1. For the function f(x)V1+x, let o 0, 0.6, and2 0.9. Construct inter- or the function...
1. For the function f(x)V1+x, let o 0, 0.6, and2 0.9. Construct inter- or the function t( polation polynomials of degree at most one and at most two to approximate f(0.45 and find the absolute errors.
2. Determine cı and c2, 1, and 2 such that the integration formula gives the exact result whenever f(x) is a polynomial of degree three or less.
4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree of precision of this rule, that is, find the highest degree of polynomial for which the above rule is exact. (10 marks) OC
4. Consider the quadrature rule +s0) 2 (F'0) +35() + 3fj f (x)dx Determine the degree of precision of this rule, that is, find the highest degree of polynomial for which the above rule is exact. (10 marks) OC
2. (4) Determine if each of the following is a subspace of P2[x] (the set of all polynomials of degree no more than 2). (a) All polynomials in P2[x] that satisfy f(1) = f(0) + 1; (b) All polynomials in P2 [x] that satisfy f(2x) = f(-x). (Hint: use the condition to find an equation of the coefficients of the polynomial f(x).)
Suppose the density for a random variable is given by the following: f(x) = Cz" for 1 < x < 2, and f(x) = 0 otherwise. Find the value of C, and then find the mean of this random variable.
Problem 4 Let f(x) be a cubic polynomial defined on interval
[−1, 1]. Determine a Gaussian intergration formula with minimal
number of nodes such that the integral formula Xn i=0 f(xi)wi is
exact for cubic polynomials.
Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,. Determine a Gaussian intergration formula with minimal number of nodes such that the integral formula is exact for cubic polynomials.
Problem 4 Let f(z) be a cubic polynomial defined on interval [-1,....
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of degree 0 and 1,
3. Let Po(x) respectively. Find the monic Legendre polynomials of degree 2, 3 and 4 using the orthogonality relation f P(x)Pm(x)dx = 0, mn and m,n E N. 1 and P1(x) = x be two Legendre polynomials of...
1. (Taylor Polynomial for cos(ax)) For f(x)cos(ar) do the following. (a) Find the Taylor polynomials T(x) about 0 for f(x) for n 1,2,3,4,5 (b) Based on the pattern in part (a), if n is an even number what is the relation between Tn (x) and TR+1()? (c) You might want to approximate cos(az) for all in 0 xS /2 by a Taylor polynomial about 0. Use the Taylor polynomial of order 3 to approximate f(0.25) when a -2, i.e. f(x)...
With explanation!
3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where P2(x) is the polynomial of degree at most two which interpolates f at 0, 1/2, and 1. (a) Write P2(x) in Lagrange form and prove that Q[F] o [s0 f(0) + 4f 45 (2) +scn)] (2) (b) Consider now a general interval [a, b] and the integral só f(x)dx. Do the change of variables x = a + (b − a)t to transform the...
1. For the function f(x)V1+x, let o 0, 0.6, and2 0.9. Construct inter- or the function t( polation polynomials of degree at most one and at most two to approximate f(0.45 and find the absolute errors.
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