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Chapter: Mathematics (maths) - Interpolation and Approximation

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Important Short Objective Question and Answers: Interpolation And Approximation

Mathematics (maths) - Interpolation And Approximation - Important Short Objective Question and Answers: Interpolation And Approximation



1.                 State   the   Lagrange’s.   interpolation   formula

 

Sol:

Let y = f(x) be a function which takes the values y0, y1,……y-corresponding to x=x0,x1,……nx

Then Lagrange’s   interpolation   formula   is

Y = f(x) =  --- --  

 

2.                 What   is   the   assumption   we   make   when   Lagran

Sol:

 

Lagrange’s interpolation formula can be used whether the values of x, the independent variable are equally spaced or not whether the difference of y become smaller or not.

 

3.                 When   Newton’s   backward   interpolation   formu

Sol:

 

The formula is used mainly to interpolate the values of y near the end of a set of tabular values and also for extrapolation the values of y a short distance ahead of y0

 

4. What are the errors in Trapezoidal rule of numerical integration?

Sol:

The error in the Trapezoidal rule is

E< ---- y”

5.Why   Simpson’s   one   third   rule   is   called   a   c

Sol:

Since the end point ordinates y0 and yn are included in the Simpson closed formula.

 

 

6.  What   are   the   advantages   of   Lagrange’sformula?   formul

 

Sol:

 

The forward and backward interpolation formulae of Newton can be used only when the values of theindependent variable x are equally spaced and can also be used when the differences

 

of the dependent variable y become smaller ultimately. But Lagrange’s interpolati be used whether the values of x, the independent variable are equally spaced or not and whether

 

the difference of y become smaller or not.

 

 

7. When   do   we   apply   Lagrange’s   interpolation?

Sol:

 

Lagrange’s formula interpolcation be used when the val not. It is mainly used when the values are unevenly spaced.

 

8. When   do   we   apply   Lagrange’s   interpolation?

Sol:

 

Lagrange’s interpolation formula spacedcanor be us not. It is mainly used when the values are unevenly spaced.

 

9.   What   are   the   disadvantages   in   practice   in   ap

Sol:

1. It takes time.

2. It is laborious

 

10.                    When   Newton’s   backwardformulaisused.   interpolation

Sol:

The       formula   is   used   mainly   to   interpolate

 

values.

 

11.                    When   Newton’s   forward   interpolation   formu

Sol:

 

The formula is used mainly to beginnig interpolate of set of tabular values.

 

 

12.                    When   do   we   use   Newton’s   divided   differenc

Sol: This is used when the data are unequally spaced.

 

13.  Write Forward difference operator.

 

Sol:

 

Let y = f (x) be a function of x and let of the values of y. corresponding to of the values of x. Here, the independent variable (or argument), x

 

proceeds at equally spaced intervals and h (constant),the difference between two consecutive values of x is called the interval of differencing. Now the forward difference operator is defined as


These are called first differences.

 

14.Write Backward difference operator.

Sol:

The backward difference operator is defined as


 

These are called first differences

 

 

 

Part B

 

1.Using   Newton’s   divided   difference and hence formula, find f(4).

X                        0        1        2                 5       

                                                                          

f(x)                      2        3        12               147   

 

 

2.Find the cubic polynomial which takes the following values:

 

X                       0         1        2        3

                                             

f(x)                    1         2        1        10

                                             

3.The following values of x and y are given:

 

X                       1         2        3        4

                                             

f(x)                    1         2        5        11

                                             

Find the cubic splines and evaluate y(1.5) and y’(3).

 

4.Find the rate of growth of the population in 1941 and 1971 from the table below.

Year X              1931   1941  1951  1961  1971

Population        40.62  60.8   79.95 103.56        132.65

Y                                                    

5.Derive            Newton’s   backward   difference   formula

 

6.Using             Lagrange’s   interpolation     formula   fin

 

(0,-12),(1,0),(3,6),(4,12).

7.Using   Newton’s   divided difference   formula

                                                                 

X                       0         1        2        4        5       

f(x)                    1         14      15      5        6       

 

8.Obtain the cubic spline approximation for the function y=f(x) from the following data, given that y0” 3=”=0y.

 

X                       -1        0        1        2

Y                       -1        1        3        35

 

9.The following table gives the values of density of saturated water for various temperatures of saturated steam.

 

Temperature0 C 0         100    150    200    250    300   

                                                                          

                                                                          

Density hg/m3    958    917    865    799    712   

Find by interpolation, the density when the temperature is 2750 .

 

10.Use Lagrange’s10656 method,giventhatlog 10to654 =2find.8156, log log 10 658 =2.8182 , log 10 659 =2.8189 and log 10 661 =2.8202.

 

11.Find f’(x) at x=1.5 and x=4.0 from the fol differentiation.

 

 

x                       1.5      2.0     2.5     3.0     3.5     4.0

Y=f(x)               3.375  7.0     13.625        24.0   38.875        59.0

 

12.If f(0)=1,f(1)=2,f(2)=33 and f(3)=244. Find a cubic spline approximation, assuming M(0)=M(3)=0.Also find f(2.5).

 

13.Fit a set of 2 cubic splines to a half ellipse described by f(x)= [25-4x2]1/2. Choose the three data points (n=2) as (-2.5,0), (0,1.67) and (2.5 , 0) and use the free boundary conditions.

 

14.Find the value of y at x=21 and x=28 from the data given below

 

x                       20       23      26      29

y                       0.3420 0.3907        0.4384        0.4848

15. The population of a town is as follows:

 

x year                1941   1951  1961  1971           1981  1991

                                                                          

y population     20       24      29      36               46      51

(thousands)                                                        

                                                                          

Estimate the population increase during the period 1946 to1976.






Tutorial Problems

 

Tutorial 1

 

1.Apply Lagrange’s formula inversely to obtai f(0)=-4,f(1)=1,f(3)=29,f(4)=52.

 

Ans: 0.8225

 

2.Using Lagrange’s   formula   of   interpolation   ,

X                       7         8        9        10

Y                       3         1        1        9

Ans: 3.625

 

3.Use Lagrange’s   formula   to   find   the   value   of

x                       3         7        9        10

y                       168     120    72      63

Ans: 147

 

4.If log(300)=2.4771,log(304)=2.4829,log(305)=2.4843,log(307)=2.4871find log(301) Ans: 2.8746


 

 




 



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