Question

please help with #1 a and b. thank you

Use the following data for all problems: The data below defines the sea-level concentration of dissolved oxygen for fresh water as a function of temperature: T°C omg/L 0 8 16 14.621 11.843 9.870 24 32 40 8.418 7.305 6.413 a. Approximate o(19) using a 1st order Newton's interpolating polynomial. b. Approximate o(19) using a 2nd order Newton's interpolating polynomial. 2) Use linear algebra to find the coefficients of the 2nd order polynomial equation that passes through the data points at T = 8, 16, and 24. Use MATLAB to solve the system of equations. Plot the complete data set and the fit line on the same graph. Include axis labels and a legend. Please help with #2

Answer 1

We have solved all the problems and plotted the required graph in MATLAB.

0(19) = ? 16 <19 < 24, det, To=16, T = 24. 0 = 9.870 , 0, = 8.418. for the interval [To,T) 1st order Newton's interpolation polynomial ing is given by, 0 = bo+b, (T-To) - 0 where, bo = 0. = 9,870 b = 0, - Oo Ti-To 8.418 - 9.870 24-16 1.452 8 = -0.1815 .: 0 = 0(T) = 9.870 - 0.1815 (T - 16). .: 0(19) = 9.870 -0.1815(19-16) = 9.870 -0.5445 =9.3255. (Am).

O Let us tarke three pointo, To=8, T,=16, .: O. = 11.843 0, = 9.870 T2=24 02 = 8.118 for the interval [T and order Newton's interpolation polynomial is given by, 0 = bo + b, (T-T.) +62 (T-T.) (7-T) where, bo= 0o = 11.843. b, = 0,-0. - 9.870 - 11.843. Ti-To 16-8 = -0.246625, b.= 02-0, 0.-Oo Tz-T, Ti-To Ti-To 8.418-9.87 8816 9.87 – 11.843 8 x16 = 0.00 40703125 : 0.00407 : .: 0(1) = 11.843 -0.2466.25 (T- 8) +0.00407(7-8) (T-16) i O(19) =9.2644 (Am

② let, and order polynomial o 0 = a + QT +Q₂T² - 0 Howe, 0 contains T=8, 16, 24. ². Putting T=8,0 = 11.843 in ① we get, 11.843 = a +8 a, + 6402 Putting T=16 , O=9.87 in 6 , we get 9. 87 =a + 16a, +256 0₂ Putting T=24, 0 = 8.418 in ③ we get, 8.418 =a +24 a +576 02 3 above aytem lineon equation can be written an, How, 641 111.8431 1.9. 87 18:418) 256 5761 la I 24 Solving this in MATLAB, we get, ho = 14.427 = -0.35931, a = 0.004539 .: Required and order polynomial is, O = 14.427 -0.35931 T +0.00453911?

15 XO Y 14.621 - Actual data ---- Quadratic line X 8 Y 11.843 o(mg/l) X 16 Y 9.87 X 24 Y 8.418 X 32 Y 7.305 X 40 Y 6.413

MATLAB script:

A=[1 8 64; 1 16 256; 1 24 576];
b=[11.843; 9.84; 8.418];
a=A\b

T=[0 8 16 24 32 40];
O=[14.621 11.843 9.87 8.418 7.305 6.413];
t=0:0.1:40;
o=a(1)+a(2)*t+a(3)*t.^2;
plot(T,O,'-o',t,o,'--');
>> xlabel('T^o C')
ylabel('o(mg/l)')
legend('Actual data','Quadratic line')