Question

5b only

Interpolatn F0X) - For alan Change is no to ONS Part 1: Name: Answer any 4 questions. Omit one the Chebyshev 21 ) Find the exact value of the) 4 zeroes of the the 4th degree Chebyshev polynomia TO) -70) 736)-70) at the Taylor 7.60). Totx)=1 T(n = x T2(x)=2x.1-110 72(x) = 2xA)-10) Tix) = 2277 Cx) - cos TiM = 2x Ch(x)-1(x). • 736x) = 2x (2x74) -(0) To(x) = 4x3 3x Tu(x) = 2X (Tsaxo - (T2x) Tufy) = 2x (4x2-3x)-)(2x²-)) (14(X) = 8x4 -8x²+1) b) Suppose P(x) interpolates tox) - eat the 4 zeroes of Tax). Find an upper bou shev poly wer degre on the in SI? for les - P(x) =) on the interval -I SXS 1. Ty 888"_8x2+1 XK = Cos/(2k +1 r

Answer 1

T4(x) = 8 x 7 - 8 x + 1. het us take x=2 . Then the above equation becomes - 1 T4 (2) 820-82+1. We have to find the root of the quation 820 82+1=0 This is a quadratic equation in 2. the roots are given by 2 +8 + 164 - Ax8*4 : 2x8 - 8 + 7 64 - 32 Since its E 2J2i = 21, 22, San H; Xsa, and in 22 - 2- Vai = 2+J2i or, 4x² = 2-Jai or, 4x = 2 + Tail | 0 1 - 2 - or x = + 2+521 Hence the roots are + 1.527 Jai