Question

alpha = 3

beta =2 Can you solve it in a hour please Thank you very much.

2. (20p.) Consider the cubic spline for a function f on (0,2] defined by S(z) = { (-- 1)327 ct an 10 Fritisa52 where r,c and d are constants. Find f'O) and f'(2), if it is a clamped cubic spline.

Answer 1

Soln Given that *= 3, 8=2, and the given spline S (21) = 2n²+ antraal (n.) ² + c (0)² + d(0-1) aß otall • ILALL Putting values of & and ß we geh 2a²+ z n't rati SME olan 3. (n-1)² + c (n.) + d (0-1) + 2 Isner Since s is a clamped cubic spline, we have that f'(o)= s' (on and f'(2) = s'( 21 I clamped boun boundary s And since sis a cubic spline, so s'ands" must be continuous at aal and and we geh the foll following: aj) Continuety of S(A) at nal. lim Son lim S(w) -) (1-12 (1) (1-1) +2 = 2.1+ 3.1+7.1 H 2= 2+ 3+ N +1 a rth=0 =) re-4

(ii) contime timely of s' (m) at nal lim 3! (m) alim slim lit um [3 (n.) + + 20 (0-1)+d ) + d] - = lim [62²+ bn to d = 6+6+r 2 646-4 T:04) = d = 8 (nil Continuity ontinuity son) at nal lim S" (n) & lim S" (n) 9-1 =) lim [ 6(N-1) + 2 c] a lim [ 129 +6] 2c = 12 + 6 = 2 c =18 =) 2-9. putting the values of rad in the spline s we get. 22² + 3n_4041 S (1) ? OLALI ( wa (ment 8 (nl) + 2 14222

from o We get, slo) d dal 2ut zur - w ty [ Cur en ru] ro londe or4 and sest [ 601922 m (ou as co-)me =[ { 18 (4) 48 care 2-3 (2-1) + 18 (2-1) +8 3+18+8 = 29 Therefore f'(o)=-4 I (2) = 29. And