Using Newton’s formula for interpolation estimate the population for the year 1905 from the table:

Year	1891	1901	1911	1921	1931
Population	98.752	1,32,285	1,68,076	1,95,690	2,46,050

The values of y = f(x) for x = 0,1,2, ...,6 are given by

x	0	1	2	3	4	5	6
y	2	4	10	16	20	24	38

Estimate the value of y (3.2) using forward interpolation formula by choosing the four values that will give the best approximation.

From the following table find the number of students who obtained marks less than 45.

Marks	30-40	40-50	50-60	60-70	70-80
No. of Students	31	42	51	35	31

Using appropriate interpolation formula find the number of students whose weight is between 60 and 70 from the data given below

Weight in lbs	0-40	40-60	60-80	80-100	100-120
No.of.students	250	120	100	70	50

The population of a certain town is as follows

Year : x	1941	1951	1961	1971	1981	1991
Population in lakhs:y	20	24	29	36	46	51

Using appropriate interpolation formula, estimate the population during the period 1946.

Evaluate \(\Delta \)\(\left[ \frac { 5x+12 }{ { x }^{ 2 }+5x+6 } \right] \) by taking ‘1’ as the interval of differencing.

The following data are taken from the steam table

Temperature C0	140	150	160	170	180
Pressure kg f / cm2	3.685	4.854	6.302	8.076	10.225

Find the pressure at temperature t = 175⁰

Calculate the value of y when x = 7.5 from the table given below

x	1	2	3	4	5	6	7	8
y	1	8	27	64	125	216	343	512

From the following table of half- yearly premium for policies maturing at different ages. Estimate the premium for policies maturing at the age of 63.

Age	45	50	55	60	65
Premium	114.84	96.16	83.32	74.48	68.48

Find a polynomial of degree two which takes the values

x	0	1	2	3	4	5	6	7
y	1	2	4	7	11	16	22	29

Estimate the production for 1964 and 1966 from the following data

Year	1961	1962	1963	1964	1965	1966	1967
Production	200	220	260	-	350	-	430

Evaluate Δ\(\left[ \frac { 1 }{ (x+1)(x+2) } \right] \) by taking ‘1’ as the interval of differencing

Find the missing entries from the following

x	0	1	2	3	4	5
y = f(x)	0	-	8	15	-	35

Using Lagrange’s interpolation formula find y(10) from the following table:

x	5	6	9	11
y	12	13	14	16

Using Newton’s forward interpolation formula find the cubic polynomial.

x	0	1	2	3
f(x)	1	2	1	10

The population of a city in a censes taken once in 10 years is given below. Estimate the population in the year 1955.

Year	1951	1961	1971	1981
Population in lakhs	35	42	58	84

In an examination the number of candidates who secured marks between certain interval were as follows

Marks	0-19	20-39	40-59	60-79	80-99
No.of.candidates	41	62	65	50	17

Estimate the number of candidates whose marks are lessthan 70.

Find the value of f (x) when x = 32 from the following table

x	30	35	40	45	50
f(x)	15.9	14.9	14.1	13.3	12.5

The following data gives the melting point of a alloy of lead and zinc where ‘t’ is the temperature in degree c and P is the percentage of lead in the alloy

P	40	50	60	70	80	90
T	180	204	226	250	276	304

Find the melting point of the alloy containing 84 percent lead.

Find f(2.8) from the following table.

x	0	1	2	3
f(x)	1	2	11	34

Using interpolation estimate the output of a factory in 1986 from the following data

Year	1974	1978	1982	1990
Output in 1000 tones	25	60	80	170

Use Lagrange’s formula and estimate from the following data the number of workers getting income not exceeding Rs. 26 per month

Income not exceeding (Rs)	15	25	30	35
No. of workers	36	40	45	48

Using interpolation estimate the business done in 1985 from the following data

Year	1982	1983	1984	1986
Business done (in lakhs)	150	235	365	525

Using interpolation, find the value of f(x) when x = 15

x	3	7	11	19
f(x)	42	43	47	60

Find the missing figures in the following table

x	0	5	10	15	20	25
y	7	11	-	18	-	32

Find f(0.5) if f(−1) = 202, f (0)= 175, f(1) = 82 and f(2) = 55

Using Lagrange’s interpolation formula find a polynomial which passes through the points (0, –12), (1, 0), (3, 6) and (4,12).

From the following table obtain a polynomial of degree y in x

x	1	2	3	4	5
y	1	-1	1	-1	1

If u₀ = 560, u₁ = 556, u₂ = 520, u₄ = 385, show that u₃ = 465

From the following data find y at x = 43 and x = 84

x	40	50	60	70	80	90
y	184	204	226	250	276	304

The area A of circle of diameter ‘d’ is given for the following values

D	80	85	90	95	100
A	5026	5674	6362	7088	7854

Find the approximate values for the areas of circles of diameter 82 and 91 respectively