(1)
x | y | ||||||||
1 | 0.5 | 0.5 | 1 | 1 | 1 | 0.5 | -0.69315 | -0.69315 | |
2 | 1 | 2 | 4 | 8 | 16 | 4 | 0 | 0 | |
3 | 1.5 | 4.5 | 9 | 27 | 81 | 13.5 | 0.405465 | ||
4 | 2 | 8 | 16 | 64 | 256 | 32 | 0.693147 | ||
5 | 3.5 | 17.5 | 25 | 125 | 625 | 87.5 | 1.252763 | ||
total | 15 | 8.5 | 32.5 | 55 | 225 | 979 | 137.5 | 1.658228 |
To fit a linear curve. n=5
We need to fit the curve of the type:
So the normal Equations are given as
We have to solve for a and b by substituting the values
On solving by Cramer's rule, we get
and
Thus the linear curve is
Now to fit a second-degree polynomial
Now the normal equations are ,
We substitute the values and solve for a,b and c22
We write the systen as Augmented matrix and solve by gauss elimination method for a ,b and c.
Perform
Peform and
Now, Perform
Now perform
Perform
Perform
Thus
Thus the required second-degree polynomial is
To fit an exponential curve :
It will be of the type
Consider
Now put
thus
Now the normal equations are
we put the values and solve for A and B
on solving we get
Perform
perform
perform
perform
Thus
2)
n=5
to Fit a straight line means to fit at curve of the type
So the normal Equations are given as
Similarly solving by Cramer's rule we get
thus we get.
3)
Required equation is
on solving by cramers rule
b= = 0.05095 and a ==2560.13543
4 problem can be done in same manner as exponential done in part 1
o fit the best Curve fit among the following formulas for the given datain >C: 2...
Exercise 6: Given the table of the function f(x)-2" 2 X 0 3 2 f(x) 1 2 4 8 a) Write down the Newton polynomials P1(x), P2(x), Pa(x). b) Evaluate f(2.5) by using Pa(x). c) Obtain a bound for the errors E1(x), E2(x), Es(x) Exercise 7: Consider f(x)- In(x) use the following formula to answer the given questions '(x) +16-30f+16f,- 12h a) Derive the numerical differentiation formula using Taylor Series and find the truncation error b) Approximate f'(1.2) with h-0.05...