Graphical and Algebraic method of Interpolation: Solved Example Problems

Example 5.13

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

Solution:

To find the population for the year 1905 (i.e) the value of y at x = 1905

Since the value of y is required near the beginning of the table, we use the Newton’s forward interpolation formula.

= 98,752 + 46946.2 + 639.8 + 584.36 + 1390.23

= 1,48,312.59

= 1,48,313

Example 5.14

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

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

Solution:

Since we apply the forward interpolation formula, last four values of f(x) are taken into consideration (Take the values from x = 3).

The forward interpolation formula is

Example 5.15

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

Solution:

Let x be the marks and y be the number of students

By converting the given series into cumulative frequency distribution, the difference table is as follows.

Example 5.16

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

Solution:

Let x be the weight and y be the number of students.

Difference table of cumulative frequencies are given below.

Let us calculate the number of students whose weight is below 70. For this we use forward difference formula.

Number of students whose weight is between

60 and 70 = y(70) −  y(60) = 424 − 370 = 54

Example 5.17

The population of a certain town is as follows

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

Solution:

Here we find the population for year1946. (i.e) the value of y at x=1946. Since the value of y is required near the beginning of the table, we use the Newton’s forward interpolation formula.

= 20 + 2 - 0.125 + 0.0625 − 0.24609

= 21.69 lakhs

Example 5.18

The following data are taken from the steam table.

Find the pressure at temperature t = 175⁰

Solution:

Since the pressure required is at the end of the table, we apply Backward interpolation  formula. Let temperature be x and the pressure be y.

Example 5.19

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

Solution:

Since the required value is at the end of the table, apply backward interpolation formula.

Example 5.20

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.

Solution:

Let age = x and premium = y

To find y at x = 63. So apply Newton’s backward interpolation formula

Example 5.21

Find a polynomial of degree two which takes the values

Solution:

We will use Newton’s backward interpolation formula to find the polynomial.