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Sheet # 1

1- Find a polynomial of order 2 that interpolates the table

x 0.2 0.4 0.6

y -0.95 -0.82 -0.65

2- Determine a polynomial of degree 3 that interpolates the data

x 1.2 2.1 3.0 3.6

y 0.7 8.1 27.7 45.1

3- Determine the Lagrange polynomial that interpolates the data in the

following table

x 0 2 4 6

y 1 -1 3 4

4- Let 2( ) 2 1xf x x e . Construct a Lagrange polynomial of degree

two or less using 0 10, 0.5x x and 2 1x . Approximate (0.8)f

5- Determine the polynomial of degree 5 , using Newton's divided-

differences, that interpolate the data in the following table

x 1.0 2.0 3.0 4.0 5.0 6.0

( )f x 14.5 19.5 30.5 53.5 94.5 159.5

Use the resulting polynomial to estimate the value of (4.5)f .

Compare with the exact value, (4.5) 71.375f .

6- To investigate the relationship between yield of potatoes, y, and

level of fertilizer application, x, an experimenter divided a field into

5 pots of equal size and applied differing amounts of fertilizer to

each. The data recorded for each plot are given by the table (in Kgs).

x 1.0 2.0 3.0 4.0 5.0

y 22 23 25 30 28

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a) Find an interpolation polynomial for this table.

b) According to the interpolating polynomial, approximately how

many Kgs would you expect from a plot to which 2.5 Kgs of

fertilizer had been applied?

7- Construct a first degree-spline to approximate the ( ) xf x x e at

0.4x , using the data.

x 0.1 0.3 0.5 0.7

( )f x 0.110517 0.404957 0.824360 1.409627

8- Construct a first degree-spline ( )S x to approximate the function

2

3( )

x

f x e by using the value given by ( )f x at 0, 0.02x and

0.04 . Find an approximation for 0.04

0( )S x d x . Compare the result

to the exact value 0.04053811.

9- Given the table

x 1.4 1.8 2.2

( )f x 3.12 2.81 1.70

use natural cubic spline interpolation to approximate (1.6)f .

10- Given the table

x 2.0 2.1 2.2 2.3

( )f x 1.5 2.0 3.8 5.1

approximate (2.05), (2.15)f f and (2.25)f using natural cubic

spline approximation.

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11- Construct a natural cubic spline that interpolates the function

2( ) 1/(1 )f x x at 2, 1, 0, 1x and 2. Compare the interpolated

values with the true values at 1.5, 0.5, 0.5x and 1.5.

Problems solved by MATLAB program.

12- Consider the graph of the function

2

1( )

1f x

x

a- Determine the polynomial 10 ( )P x of 10th degree that

interpolates the points with absicca 5, 4, ..., 4, 5 , all

spaced one unit apart.

b- Plot both ( )P x and ( )f x in the interval [ 5, 5] . Show

that the instabilities in 10 ( )P x occur near the ends of the

interval over which the interval is used by finding error

10| ( ) ( ) |E f x P x at 4.2, 4.4, 4.6x and 4.8 .

13- Generate a table of values by taking

1, 2, , 106,i ix i , 2

1

1i

i

yx

Find a natural cubic spline that interpolates this table. Compare

the values of the spline with the values of 21/ (1 )y x at

points between the data points (midpoints).

14- A biologist who is studying the growth of a bacteria culture,

recoded the following data:

t 0 3 6 9

( )b t 2.0 3.0 4.0 6.0

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where ( )b t denotes the number of bacteria at time t (minutes).

Use natural cubic splines to estimate the number of bacteria at

5t .