MATHS.maths.maths.maths.maths.maths.PAS.pptx

Second degree Parabola and more general curve

1. Second Degree Parabola And More General Curve

2. What is a Curve? A curve is a smooth, continuous line with no sharp corners. Why Study Curves? They model real-world phenomena (e.g., bridges, ball trajectories). Help solve problems in physics, engineering, and economics. Example:A circle, a straight line, or a wavy road.

3. Second Degree Parabola Let ( x i , y i ) , i = 1, 2 ,...,n be the set of n values and let the relation between x and y be y = a + bx + cx 2 The constants a, b, and care selected such that the parabola is the best fit to the data. The residual at x = x i is

4. Second Degree Parabola

5. These equations are known as normal equations . They can be solved simultaneously to determine the best values of a , b , and c . Once these values are obtained, the best-fitting parabola is given by the equation: y= a+bx+c x 2 Substituting the calculated values of a , b , and c into this equation provides the required parabolic curve that best fits the given data.

6. Example Find the least squares quadratic curve y=a+bx+cx2 for the data: x 1 2 3 4 y 1 1.8 1.3 2.5 6.3 ∑y = na + b ∑x + c ∑ x ^ 2 ∑ xy = a ∑x + b ∑ x ^ 2 + c ∑x ^ 3 ∑x ^ 2 y = a x ^ 2 + b ∑x ^ 3 + c ∑x ^ 4 Solution .

7. ANALYSIS X Y x 2 x 3 X 4 xy x 2 y 1 1 1.8 1 1 1 1.8 1.8 2 1.3 4 8 16 2.6 5.2 3 2.5 9 27 81 7.5 22.5 4 6.3 16 64 256 25.2 100.8 ∑x=10   ∑y=12.9   ∑x 2 =30  ∑x 3 =100  ∑x 4 =354   ∑ xy =37.1   ∑x 2 y=130.3

8. Normal Equations: 12.9=5a+10b+30c 37.1=10a+30b+100c 130.3=30a+100b+354c Equation: y=1.42−1.07x+0.55x 2 Solution: a=1.42 a =1.42 b=−1.07 b =−1.07 c=0.55 c =0.55

9. Let (x, y), i = 1, 2, ..., n be the set of n values and let the relation between x and y be y = a b x . Taking logarithm on both the sides of the equation y = a b x . log e y= log e a+ x log e b Putting log e y = Y, log e a = A, x = X, and log e = B Y = A + BX This is a linear equation in X and Y. The normal equations are ΣΥ =  ∑ Α+ΒΣ x ΣΧΥ = ΑΣΧ+ΒΣ x 2 Solving these equations, A and B, and, hence, a and b can be found. The best fitting exponential curve is obtained by substituting the values of a and b in the equation y = a b x . Similarly, the best fitting exponential curves for the relation y= ax b and y= ae bx can be obtained.

10. y = ax b Taking logarithm on both the sides log e y = log e a+ b log e x Putting log e y=Y, log e a= A,b =B and log e x=X , Y=A+BX The normal equations are ΣΥ = n  A +ΒΣ x ΣΧΥ = ΑΣΧ+ΒΣ x 2

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