4) (a) Find the function P(x) = a + bcos(m) + c sin(TX), which interpolates the...
(a) Find the interpolating polynomials to the function f(x) = e-* sin(7x) at x = -1, 0, 1 and 2 using. (1) Lagrange's interpolation formula.
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
Given the data points (-3,5),(-2,5),(-1,3), (0, 1) (a) Find the interpolating polynomial passing through these points. (b) Using your polynomial from (a), evaluate P(1). (c) This polynomial interpolates the function f(x) = 24. Find an upper bound for the approximation in part (b).
Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following interpolating polynomials, and use MATLAB to graph both the interpolating polynomials and the data points: a) The piecewise linear Lagrange interpolating polynomialx) b) The piecewise quadratic Lagrange interpolating polynomial q(x) c) Newton's divided difference interpolation pa(x) of degree s 4
Problem 2. Given the data points (xi. yi), with xi 2 02 4 yil 5 1 1.25 find the following...
Question l: Consider the function f(x) = sin(parcsinx),-1 < x < 1 and p E R (a) Calculate f(0) in terms of p. Simplify your answer completely fX) sin(p arcsinx) f(o) P The function fand its derivatives satisfy the equation where f(x) denotes the rth derivative of f(x) and f (b) Show thatf0(n2p2)f(m)(o) (x) is f(x). (nt2) (nti) (I-x) (nt 2 e 0 (c) For p E R-仕1, ±3), find the MacLaurin Series for f(x), up to and including the...
1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r).
1. Consider the polynonial Pl (z) of degree 4...
8396 5101281 5 8 2 0 1 12 ( 4 2 1 ) ) ) 0000 f-000 0246802 (i) Defining fo-f(zo). Л that the quadratic f(x) and f2 f(x2), where Zo-x1-h and x2-xuth, show 2 , f2 - jo 2h2 2h is the quadratic interpolating function for fo, fı and f2 (i.e. show that p(x)-f) 4] (ii) Use the interpolating polynomial p(x) as defined above, with Zo-12, xỉ-1.4 and 22 -1.6 (and fo, fı and f2 given by the table...
5) Consider the polynomial P() z2-z-1. (a) Find two integers n, m E Z, so that P(x) has a zero in [n, m. (b) Use the bisection method twice to get an approximation to the zero of P(x) in n, m] (c) Use Newton's method twice to get an approximation to the zero of P() in n,m (d) Use the quadratic formula to find the actual zero of P() in [n, m (e) Compute the relative %-error for each of...
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation, T1 (f) = f(1) + f(-1), for f(r)dr.
(a) Find the degree 1 interpolating P(x) through the points (a, f(a)) and (b, f(b)) (b) Develop the following formula by using the interpolating polynomial P1 (x), (c) Find the degree of precision of the approximation,...
1.1. Find a constant b (in terms of a) such that the function tx, y(x, y) =fb.e-lacty) 04220 0y< 1.2 Find the expresion for the marginal densities fx lx), and foly). HINT: SI fulzig) dardy = 1, fax) = | Saxlayboy, etc.