Hi!

Attached is a homework problem AND I also attached a similar problem with answers to help you answer it. Step by step solutions wold be greatly appreciated.

to clarify:

the problem I need help with is #2

the problem meant as a reference is #3

Thank you!!!

2. Data interpolation has many applications. Suppose we have been given the task of nding the degree 3 polynomial P(x) = m3+bx2+cx+d which satises P'(l) = 3 and passes through the points (1,1), (0,1), and (1,3). (a) (3 points) Show that the coefcients of P satisfy the system of equations 3a2b+c=3 a+bc+d= 1 (i=1 a+b+c+d=3. a Write a matrix equation of the form A 2 = r so we could solve for a, b, c, and d. d (b) (5 points) Find 24*, and use it to nd 11,1), c, and d. Write the formula for P(a:). (c) (2 points) Your boss suddenly changes the problem at the last minute. You have now been asked to nd the degree 3 polynomial 3' which satises f'(1) = 13 and passes through the points (1, 3), (0,2), and (1, 3). Use your work from parts (a) through (c) to nd the formula for f. 3. Data interpolation has many applications. In this problem, we will answer the question: What is the degree 3 polynomial P[:1'] = on + 1212 + 0.2122 + 532:3 which passes through the points [2. E1], [1,-4}, (1,6), and [2, 22}? [a] [3 pts] Show that since PEZ} = E, P[l} = 4. 13(1) = 6, and FEE} = 22. our problem is really the problem of solving the system of equations aZG1+4E2Ea3=, o1+2e=4. o+1+2+s=. no + 2E1 +4113 + 35.3 = 22. [h] [3 pts} The system in part [a] is the matrix equation 124s an s 1111 a1_4 1111 rigs 1242 as 22 23 82 llSEl The inverse of the shove 4 x 4 matrix is E 2 _2 _2 2 [you don't have to shotsr this}. 1 2 2 1 Use 211 to solve the system of equations in [a]. Solution: {'1} Flt2] =51 IIIo+i|:2l+2ll3lg +2[2}3=. fin-21 +4I12-32 =15- P[l} = 4, on +a1[l] +m[l}2+o3[l}3 =4, [tna1 +122 a.3 =4. P[l}=, o+1[l}+2[l]2+3[l}3=, g+a1+g+ag =15. H2} = 22, on + a1[2]+ 122(2)? + 1231:2113 = 22, an + 21:1 + 45.2 +3113 = 22. The student doesn't have to show all those steps fully. but there should at least demonstrate somewhere how they know that the :1 points on the graph correspond to the 4 equations. can 2 s s 2 e 2 (h]v=A1b. a1 =i 1 s s 1 4 _g 1