Applied Mathematics for Electrical Engineering (3130908)

Question-4b-OR

BE | Semester-3 Winter-2019 | 26-11-2019

Q4) (b)

The normal equations for the second degree polynomial

y = A + Bx + {Cx}^{2}

are,

\sum_{}^{} y = nA + B \sum_{}^{} x + C \sum_{}^{} x^{2}

{\sum x}_{}^{} y = A \sum_{}^{} x + B \sum_{}^{} x^{2} + C \sum_{}^{} x^{3}

{\sum x^{2}}_{}^{} y = A \sum_{}^{} x^{2} + B \sum_{}^{} x^{3} + C \sum_{}^{} x^{4}

So, we need to find the value of

\sum_{}^{} x, \sum_{}^{} y, \sum_{}^{} x^{2}, \sum_{}^{} x^{3}, \sum_{}^{} x^{4}, \sum_{}^{} xy and, \sum_{}^{} x^{2} y .

To find the value of A, B and C.

$x$	$y$	$x y$	$x^{2}$	$x^{2} y$	$x^{3}$	$x^{4}$
$1$	$6$	$6$	$1$	$6$	$1$	$1$
$2$	$11$	$22$	$4$	$44$	$8$	$16$
$3$	$18$	$54$	$9$	$162$	$27$	$81$
$4$	$27$	$108$	$16$	$432$	$64$	$256$
$\sum_{} x = 10$	$\sum_{} y = 62$	$\sum_{} x y = 190$	$\sum_{} x^{2} = 30$	$\sum_{} x^{2} y = 644$	$\sum_{} x^{3} = 100$	$\sum_{} x^{4} = 354$

The normal equations for the second degree polynomial are,

\sum_{} y = n A + B \sum_{} x + C \sum_{} x^{2} \Rightarrow 62 = 4 A + 10 B + 30 C . . . . . . . . . (1)

\underset{}{\sum x} y = A \underset{}{\sum x} + B \sum_{} x^{2} + C \sum_{} x^{3} \Rightarrow 190 = 10 A + 30 B + 100 C . . . . . . . . . (2)

\underset{}{\sum x^{2}} y = A \underset{}{\sum x^{2}} + B \sum_{} x^{3} + C \sum_{} x^{4} \Rightarrow 644 = 30 A + 100 B + 354 C . . . . . . . . . (3)

By equation ( 1 ), ( 2 ) and ( 3 ), we have

A = 3, B = 2

and

C = 1

Therefore, the best fit line is

y = A + Bx + {Cx}^{2}

Therefore, the best fit li

\Rightarrow y = 3 + 2 x + x^{2}