Use Newton’s Divided Difference Formula to interpolate f (1), if the polynomial has given points (−1,0), (2,1), (3,1), (5,2). Hint: Construct a divided difference table first.

They are functions that describe relationships between two variables... a) Quadratic b) Linear c) Polynomial

Express the GCD of (210,162) as a linear combination of 210 and 162.

3. Determine the Newton's Divided-Difference Interpolating Polynomial P,(x) written in Newton's Divided-Difference form for the polynomial interpolating the data points (-3, 9), (0, -5), and (3, 2). You must show your work.

Using Newton's Divided Difference, the interpolating polynomial of the points (xo.Yo) = (- 2,1), (x1,Y1) = (- 1,3), (x2,y2) = (0,2), (x3.Y3) = (1, – 2), and (x4.Ya) = (2,– 1) can be represented by p(x) = co + C1P1(x)+ c2P2(x) + C3P3(x) + C4P4(x) . Now, compute P4(3). А 40 60 C 20 120 (B)

Q2. (a) Compare and contrast the following four interpolated polynomials. Lagrange Polynomial. (i) (ii) Newton Divided-Difference Polynomial. (iii) Hermite Polynomial. (iv) Cubic Spline Polynomial.

Please show complete steps and solution, use excel or manual computation. 1. Find the Lagrange form of the interpolating polynomial for the following table of values: x| 1/3 | 1/4| 1 y 2 -1 7 2. Find the Newton form of the interpolating polynomial for the following table of values: x | 1/3 | 1/4 | 1 y|2 |-1 | 7 3. Construct a divided differences table and write out the Newton form of the interpolating polynomial for the following table of values: 1 | 3/2 | 0 | 2 f(x) 3 13/4 3 5/3 4. Given the data: 1.6 2 2.5 3.2 4.5 f(x) 14 15 2 Calculate f(2.8) using Newton's interpolating polynomials of order 1 through 3. Choose the sequence of the points for your estimates to attain the best possible accuracy.

Using Newton's Divided Difference, the interpolating polynomial of the points (xo,Yo) = (- 2,1),(x1,yı)=(-1,3), (x2,y2) = (0,2) , (x3,Y3) = (1, – 2) , and (x4,Y4) = (2, – 1) can be represented by p(x) = co + C1P1(x)+ c2P2(x)+ C3P3(x) + C4P4(x) . Now, compute P(3). A 17.8 22.3 c) 20.6 15.4

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