Question

MATH 220 Project 1 Polynomial Curve Fitting It desired to fit a polynomial curve through evenly spaced (x-direction) points. The general form of a polynomial is: f(x) = 4,x" +47-1X2-1 + + ax + ao If one wishes to fit a curve through, say 4 points, one would need a 3rd degree polynomial (n = 3) such that 4 unknown constants could be evaluated. In the absence of availability of many wathematical programming tools (Matlab, etc., Mathematica is available as an online program for student use), I would suggest that the polynomial coefficients can be determined with the appropriate simultaneous equations solved with a Linear Algebra technique. These coefficients can then be used in Excel to plot the fitted curve, along with the given points. Attached to this is one Excel graph for a similar data set to that requested for this project. (Simply using curve fitting provided by Excel is not acceptable.) I would suggest forming a team of 2-3 individuals to spread the work out - there will be more projects where collaboration will be useful. A single submission from each team is fine, including the names of each member. The submission should include a description of the problem, your approach and results, including the fitted curve. The data set for your curve fitting is: х 0 1 2 у 1 3 3 2 4 2 4 5

Answer 1

I've solved the linear equations unsing row transformations. And found the fitting cure for the given data. it can be cross-checked by putting values of x in f(x).

We'll find coefficients of fit cure given, 6 points so, we'll need 5th degree polynomial Let the filting curve is f(x)= 9 x + aux qx 't q2x7 9,x + 90. Y pulting in given data in fox) ។ + Х lorda sewwe ។ 5 2. we get 6 equations in 6 variable. X-ot. tto aos x=0 r=1 x=1 Y=3 95 +94 +93 +9+a, +90= 3. X=2 Y=3 3295 + 1694 + 893+492+ 29, +90 - 3 X2 3 Y=2. 24395 + 8194+2793+9 % + 3a,+90=2. x=4 Y=4. 102495 +25694+6493+ 1692 +49 +90=4. x=5 3125 95+62594 +12593+ 2592 +59 +96=2. we want to solve for 90,91). 95. Y= 2 95 O In matrix form. I 1 13 32 243 1024 16 8 81 27 256 64 625 125 25 O- Joe au 93 92 91 - ♡ ♡doro Lao 3235
cont...
in this matrix form. by a sequence of now applying operations, this mateix can be transford to following matrix called reduced row echelon form. a5 au 93 оооо r - 7160 31/24 +4/ 137/24 -13/60 92 ai 01 190 which gives ag=-7/60 qy = 31/24. 93= -14/3. 92=8137/24 9,= -13160 90 = 1. so, the filting curve is. FC)= -7/601 2 3 х 9 x + 1 137/24 + -13%ot X + -43% + -7%607s + 3/4

Also attaching fitting curve graph in Excel.

Untitled spreadsheet File Edit View Insert Format Data Tools Add-ons Help A Share Last edit was 15 minutes ago 100% % .O .00 123 Default (Ari... - 10 BI $A . na fx I. Chart editor Х A B с D E F G H Setup Customize 1 X Y 2 0 Data range 3 1 1 3 3 2 A1:B7 4 2 3 5 Y vs. X X-axis 6 4 4 4 7 5 2 123 X 8 9 3 U Aggregate 10 11 12 2 Series 13 123 Y 14 15 1 Add Series 16 17 18 0 Switch rows / columns 19 0 2 3 4 5 OS ✓ Use row 1 as headers 20 х 21 ✓ Use column A as labels 22 + = Sheet1 -