Question

Obtain the interpolation value from the data WS in the attachment at the X value below in the given way. х Y 1 68 1.5 2. 6 41 94 4 5 16 28 5.3 5. 6 23 40 95 6.5 7 10 19 11 1. X=1.23, Linear interpolation 2. X=1.67. Lagrangian interpolation 3. X=3.01, Cubic spline interpolation 4. X=4.68, Linear interpolation 5. X=5.25. Lagrangian interpolation 6. X=5.43, Cubic spline interpolation 7. X=5.88, Linear interpolation 8. X=6.12. Lagrangian interpolation 9. X=6.78, Cubic spline interpolation 10. X=8.54, Linear interpolation

to the fifth decimal place.

Answer 1

as per HomeworkLib guidelines, I'm solving only first four subparts, as you've not mentioned which parts you want me to solve:

1.) x=1.23 , linear interpolation:

for linear interpolation of 1.23, points ( x,y) at 1 and 1.5 is considered as 1.23 lies between them:

(x1,y1)=(1,68) , (x3,y3)= (1.5,6), x2=1.23, y2:

Solving for y2 (x2 - xys - yı) + Yi (x3 -X,

y2= 39.48000

2.) x=1.67, lagrangian interpolation:

formula:

(1 – 22)(1 – 13)... (I – In) Yı (11 – x2)(21 – 23)... (21 – In) (2 - x1)(1 – 13) (2 – In) + y2 (12 – 21) (12 – 13)... (22 – In) p() (x – 11)(1 – 12)... (I – In-1) ++Yn (In – 21) (7n – 22)... (In – {n-1)

applying this formula: y(for x=1.67)= -234.46000

3.) x= 3.01, cubic spline:

Points P. (168); P1(1.5 6); P2 (241); P3 (494); P(516); P (5.328); P6(5.7|23); P-640); P: (6.5|95); P. (7|19); P10(10/11) Equation 1.9831 · 102.23 5.9494 . 102.2? + 4.2136 . 10 x +4.3265 . 10, if x € [1, 1.5], -2.1556 . 102. 13 + 1.2675 . 103.72 – 2.3723 · 103 - +1.4401 · 103, if x € (1.5, 2], -8.1748.73 +2.3172. 101.22 +1.1636 - 102 . -2.1901 - 10%, if € (2,4), 8.7578 - 101.73 1.1259 . 103.x2 + 4.7125 . 103.2 - 6.3472 . 103, if € (4,5), -4.2858 · 102 • 23 +6.6165 · 103 • 2? – 3.3999 . 104.2 +5.8172 · 104, if x € (5,5.3], f(x) = 2.9666 . 102.23 4.9149. 103.22 +2.7117. 104.1 - - 4.9799 . 10^, if x € (5.3,5.7], 6.6376 - 101.23 - 9.7694 . 102.2? + 4.6708 - 10% - 2 - - 7.1518 - 10%, if 3 € (5.7,6), -4.5980. 102.23 +8.4943 . 103 -22 – 5.2157. 104 • x +1.0650 - 105, if & € (6,6.5), 3.7973 . 102.73 – 7.8765 . 103.72 +5.4254 . 104. - 1.2405 . 105, if x 6.5, 7], -1.0856 - 107.23 +3.2569 - 102 .22 – 3.1618 - 103 • 2 +9.9168 . 10', if IE (7, 10).

apllying 3rd equation: y( for x=3.01)= 118.24000

4.) x=4.68 , linear interpolation:

for linear interpolation of 4.68, points ( x,y) at 4 and 5 is considered as 4.68 lies between them:

(x1,y1)=(4,94) , (x3,y3)= (5,16), x2=4.68, y2:

Solving for y2 (x2 - xys - yı) + Yi (x3 -X,

y2= 40.96000