Question

Please answer everything fully and show steps!

3. The set of the following data points is given days 14 22 30 38 46 X 320 490540 500 480 (a) Determine the natural cubic splines that fit the data (b) Determine the interpolated value of y for r 27.5 (c) Plot of the data points and the interpolating polynomials

Answer 1

4 38 2 46 22 30 (4 M90 S40 4-80 320 heeawice palynaminl f dere 0), (14)b (2-14)t (-14)+d, Mer 222 ,)a, (2-22)t(-22)C(7-22) +d2,22xe 30 2 is Courider a S s)2 (-30)t b-30)(2-3 j+d, 301e 28 2 +c Shr-S114)= d 220 S1%)22) |29,t64b,tcd, - 490 2(2 d=410 Sm = S129, t64 bt80std, -o 2 500 S4(4)-SA32)= S4 (05)-446)S1204 t64bt Sce (%)() =1929, t16bit Ge d4 480 8 7es 19292t16b2t 3 19292t 16b3 t3=+ Next Condition is 489,4261=2b2 S14.3) )e S(30)-(30) 4ea2t2b22 s'epse 13

48a2t2b3 - 2b4 (14 Natveal cube tine Comditen: S,) = S o IS S(48)=4841254 ale -3s 2735d 22-0 1 2 7168 b21065 276 95 a 2- 35 ,d2 490 RIL 295 7168 das S4o 16 -25 Qr -275 -13 (4 276 4 7168 (A75)=S(A) 0,(415-24) ,-20re 30865885 7841 alumad is tlted Date Poiuts aud teelating j uotle Paegramining Hutol

restart;
with(Student[NumericalAnalysis]); with(plots);
a[1] := -355/7168;
a[2] := 95/7168;
a[3] := 395/7168;
a[4] := -135/7168;
b[1] := 0;
b[2] := -1065/896;
b[3] := -195/224;
b[4] := 405/896;
c[1] := 2735/112;
c[2] := 835/56;
c[3] := -25/16;
c[4] := -275/56;
d[1] := 320;
d[2] := 490;
d[3] := 540;
d[4] := 500;
T1 := piecewise(x < 22, a[1]*(x-14)^3+b[1]*(x-14)^2+c[1]*(x-14)+d[1], x < 30, a[2]*(x-22)^3+b[2]*(x-22)^2+c[2]*(x-22)+d[2], x < 38, a[3]*(x-30)^3+b[3]*(x-30)^2+c[3]*(x-30)+d[3], x > 38, a[4]*(x-38)^3+b[4]*(x-38)^2+c[4]*(x-38)+d[4]); P1 := plot(T1, x = 14 .. 46, numpoints = 1000, discont = false);
xy := [[14, 320], [22, 490], [30, 540], [38, 500], [46, 480]];
P2 := pointplot(xy);
display(P1, P2);

500- 450 400- 350- 45 15 20 25 30 35 40 x