Question 373641: Find the cubic polynomial which takes the following values y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10 , hence obtain the value of y(0.5)?
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! Find the cubic polynomial which takes the following values y(0) = 1, y(1) = 0, y(2) = 1 and y(3) = 10 , hence obtain the value of y(0.5)?
y = ax^3 + bx^2 + cx + d
y(0) = 1 = 0 + 0 + 0 + d, d = 1
y(1) = 0 = a + b + c + 1, a + b + c = -1
y(2) = 1 = 8a + 4b + 2c + 1, 8a + 4b + 2c = 0
y(3) = 10 = 27a + 9b + 3c + 1, 27a + 9b + 3c = 9 --> 9a + 3b + c = 3
a + b + c = -1
8a + 4b + 2c = 0
9a + 3b + c = 3
eq. 1 - eq. 3: a - 9a + b - 3b + c - c = -1 - 3
-8a - 2b = -4
-4a - b = -2
4a + b = 2
eq. 2 - 2*eq.3: 8a - 18a + 4b - 6b + 2c - 2c = 0 - 6
-10a - 2b = -6
-5a - b = -3
5a + b = 3
subtract 5a + b = 3 from 4a + b = 2:
4a - 5a + b - b = 2 - 3
-a = -1
a = 1
substitute a = 1 into 4a + b = 2:
4 + b = 2 --> b = -2
subsitute a = 1 and b = -2 into a + b + c = -1:
1 - 2 + c = -1
-1 + c = -1
c = 0
cubic polynomial equation is:
y = x^3 - 2x^2 + 1
y(0) = 1, yes
y(1) = 1 - 2 + 1 = 0, yes
y(2) = 8 - 8 + 1 = 1, yes
y(3) = 27 - 18 + 1 = 9 + 1 = 10, yes
y(0.5) = (0.5)^3 - 2(0.5)^2 + 1
y(0.5) = (1/2)^3 - 2(1/2)^2 + 1
y(0.5) = 1/8 - 2 * 1/4 + 1
y(0.5) = 1/8 - 2/4 + 1
y(0.5) = 1/8 - 4/8 + 8/8
y(0.5) = -3/8 + 8/8
y(0.5) = 5/8
y(0.5) = 0.625
|
|
|
