Fitting of second degree and exponential curves.

1) Fitting of second degree curveProb.-

Year(X) 1 2 3 4 5

Profit(Y) 24 27 32 38 45

Fit the second degree curve for the above data.

Solution-

Let the second degree curve be y=a+bx+cx2 ---------------------(1) By changing variable x to u=x-x , eq. (1) becomes y=a'+b'u+c'u2 ---------------------(2)

Claim-To find a' ,b' & c'.The normal eq.s to find a' ,b' & c' are

∑y=na'+b'∑u+c'∑u2 -------------(3) ∑uy= a'∑u+b'∑u2+c'∑u3 -----------(4) ∑u2y= a'∑u2+b'∑u3+c'∑u4 ----------(5)

Obtain the values of ∑y, ∑uy, ∑u2y ,∑u , ∑u2 , ∑u3 & ∑u4 . Use these values in eq. (3),(4) & (5) to get a' ,b' & c'.Put the values of u, a' ,b' & c‘ in eq. (2) , we get the required second degree curve.

Here x = 3Let u=x-x =x-3

x y u=x-3 u2 u3 u4 uy u2y

∑y= ∑u= ∑u2= ∑u3= ∑u4= ∑uy= ∑u2y=

x y u=x-3 u2 u3 u4 uy u2y

1 24 -2 4 -8 16 -48 96

2 27 -1 1 -1 1 -27 27

3 32 0 0 0 0 0 0

4 38 1 1 1 1 38 38

5 45 2 4 8 16 90 180

∑y=166

∑u=0 ∑u2=1o

∑u3=0

∑u4=34

∑uy=53

∑u2y=341

a‘=31.9143 ,b' =5.3, c' =0.6429

Prob.-Fit second degree curve.

Year(X)

1989 1990 1991 1992 1993 1994 1995

Profit(Y)

10 12 18 20 15 13 16

2) Fitting of exponential curve y=a bX Prob.-

Year(X) 1951 1961 1971 1981 1991

Profit(Y) 140 170 200 250 300

Fit the exponential curve y=a bX for the above data.

Solution-

Let the exponential curve be y=a bX ---------------------(1) Taking log (to the base 10) of both sides. log (y)=log(a)+xlog(b) V=A+BX ,Where log (y)=v, log(a)=A, log(b)=BBy changing variable x to u=(x-x)/10 we get v=A +Bu-----------------------(2)Then eq.(1) becomes y=a bu ---------------------(3) The normal eq.s to find A & B are

∑v=n A+ B ∑u -------------(4) ∑uv= A∑u+B∑u2-----------(5)

Obtain the values of ∑v, ∑uv, ∑u , ∑u2 .

Use these values in eq. (4),(5) to get A & B.

Taking antilog of A & B we get a & b.

putting u, a & b values in eq. (3) we get required eq.

x y u=(x-1971)/10

V=logy u2 uv

1951 140

1961 170

1971 200

1981 250

1991 300

∑u= ∑v= ∑u2

=∑uv=

Here x = 1971Let u=(x-x )/10=(x-1971)/10

x y u=(x-1971)/10

V=logy u2 uv

1951 140 -2 2.1461 4 -4.2922

1961 170 -1 2.2304 1 -2.2304

1971 200 0 2.3010 0 0

1981 250 1 2.3979 1 2.3979

1991 300 2 2.4771 4 4.9542

∑u=0 ∑v=11.5525

∑u2

=10∑uv=0.8295

A=2.3105,B=0.08295a=204.41,b=1.2105

Prob.-Fit the curve y=a bX

Year(X)

1 2 3 4 5 6 7

Sale(Y)

32 47 65 92 132 190 275

3) Fitting of exponential curve y=a ebx Prob.-

X 1 2 3 4 5 6

Y 1.6 4.5 13.8 40.2 125 300

Fit the exponential curve y=a ebx for the above data.

Solution-

Let the exponential curve be y=a ebx ---------------------(1) Taking log (to the base 10) of both sides. log (y)=log(a)+bxlog(e) u=A+BX --------------------------(2) Where log (y)=u, log(a)=A, blog(e)=b x 0.4343=BThe normal eq.s to find A & B are

∑u=n A+ B ∑x -------------(3) ∑ux= A∑x+B∑x2-----------(4)

x y u=log(y) x2 ux

1 1.6

2 4.5

3 13.8

4 40.2

5 125

6 300

∑x= ∑u= ∑x2= ∑ux=

x y u=log(y) x2 ux

1 1.6 0.2410 1 0.2041

2 4.5 0.6532 4 1.3064

3 13.8 1.1399 9 3.4197

4 40.2 1.6042 16 6.4168

5 125 2.0969 25 10.4845

6 300 2.4771 36 14.8626

∑x=21 ∑u=8.1754 ∑x2=91 ∑ux=36.6941

Use eq. 3 & 4 ,find the values of A & B .

Then

a= antilog (A), b=B/0.4343

Put these values of a & b in eq. 1 we get required curve.

A= -0.25347, B= 0.4617a= 0.5578, b= 1.063

Prob.-Fit the curve y=a ebx

Year(X)

1989 1990 1991 1992 1993 1994 1995

Profit(Y)

10 12 18 20 15 13 16

Year(X)

1 2 3 4 5 6 7

Sale(Y)

32 47 65 92 132 190 275

Prob.-Fit second degree & the curve y=a ebx ,y=a bX

Prob.-Fit second degree & the curve y=a bX

Year(X)

1989 1990 1991 1992 1993 1994 1995

Profit(Y)

10 12 18 20 15 13 16

Year(X)

1 2 3 4 5 6 7

Index(Y)

10 12 20 24 25 26 30

Prob.-Fit second degree curve & curve y=a ebx ,y=a bX