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Numerical Differentiation using Newton's Forward, Backward Method
1. From the following table of values of x and y, obtain `(dy)/(dx)` and `(d^2y)/(dx^2)` for x = 1.2 .

x

1.0

1.2

1.4

1.6

1.8

2.0

2.2

y

2.7183

3.3201

4.0552

4.9530

6.0496

7.3891

9.0250

2. From the following table of values of x and y, obtain `(dy)/(dx)` and `(d^2y)/(dx^2)` for x = 2.2 .

Numerical Interpolation using Forward, Backward Method

1. The population of a town in decimal census was as given below. Estimate population for the year 1895.

Year

1891

1901

1911

1921

1931

Population
(in Thousand)

46

66

81

93

101

2. Let y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10. Find y(4) using newtons's forward difference formula.

3. In the table below the values of y are consecutive terms of a series of which the number 21.6 is the 6th term. Find the 1st and 10th terms of the series.

X

3

4

5

6

7

8

9

Y

2.7

6.4

12.5

21.6

34.3

51.2

72.9

4. The population of a town in decimal census was as given below. Estimate population for the year 1895.

X

0.10

0.15

0.20

0.25

0.30

tan(X)

0.1003

0.1511

0.2027

0.2553

0.3073

Find (1) tan 0.12 (2) tan 0.26

5. Certain values of x and log10x are (300,2.4771), (304,2.4829), (305,2.4843) and (307,2.4871). Find log10 301.

6. Find lagrange's Inerpolating polynomial of degree 2 approximating the function y = ln x defined by the following table of values. Hence find ln 2.7

X

2

2.5

3

ln(X)

0.69315

0.91629

1.09861

7. Using the following table find f(x) as polynomial in x