Homework 2 MATH 455 Instructions 1 Interpolation Written assignments are graded based on quality of work and fully worked out solutions Points will be deducted for incomplete reasoning and disorganized work even if your answers are correct a Use Lagrange interpolation to nd a polynomial that passes through the given points i 0 1 2 3 3 0 ii 1 0 2 1 3 1 5 2 with the polynomials found in problem 1 a i 0 1 2 3 3 0 ii 1 0 2 1 3 1 5 2 b Use Newton s divided di erences to nd a polynomial that passes through the given points Do these agree c Find the 2nd degree interpolating polynomial P2 x through the given points and address the questions that follow i 0 0 2 1 0 A Calculate P2 4 which should approximate sin 4 B Give an error bound for the approximation provided above C Compare the actual error to your error bound ii 1 0 2 ln 2 4 ln 4 A Use P2 x to approximate ln 3 B Give an error bound for the approximation provided above C Compare the actual error to your error bound d Use Newton s divided di erences to approximate f 0 3 given the following data Suppose it is discovered that f 0 4 was understated by 10 and f 0 6 was overstated by 5 By what amount should the approximation for f 0 3 be changed x 0 0 0 2 0 4 0 6 f x 15 21 30 51 1 u G i L x x 2 X 3 10 2110 3 h X X 0 X 3 2 0 2 3 b ii I g Go I I 2 a 13 z 42 2 3 5 az 2 Y G3 24 Ls X x 0 X 2 13 03 21 won t contribute x 2 x 3 X Pg X Go 9 x Xd 22 X X Gy X X X Xy X Xz X Xs Pg x 0 z x 1 z x 1 x 2 cy x 1 x 2 x 3 i A Go g G Es 92 2 2 4 2 2 T O P2 4 3 4 Pa X Go 9 x X 22 X Xo X X P x X X x 3 B To do this we use our Lagrange Error Formula 18 X P x x X x x x Xz New With X 4 f x Sin x And n 2 Note that gc 2 11 When f x Sin X 18 N2 P2 4 3T1 01 y l y 3 4 4 N 4 18 2 P 4 0 2422 2 C Actual Error is Sin H 314 0 042920 2422