amaths

1.It is given that at every point on a curve,dy/dx=3x^2 -6x +1.If the curve passes through the point(2,11),find the equation of the curve.
dy/dx = 3x^2 - 6x + 1
y = x^3 - 3x^2 + x + C, C is constant
11 = 2^3 - 3(2)^2 + 2 + C, C = 13
y = x^3 - 3x^2 + x + 13
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2.It is given that at every point on a curve,d^2y/dx^2 =6.If the slope of the curve at the point (0,4) is 2,find the equation of the curve
d^2y/dx^2 = 6
dy/dx = 6x + m, m is constant
slope = 6(0) + m = 2, m = 2
dy/dx = 6x + 2
y = 3x^2 + 2x + n, n is constant
4 = 3(0)^2 + 2(0) + n, n = 4
y = 3x^2 + 2x + 4

1)
dy/dx = 3x^2 - 6x + 1
y = x^3 - 3x^2 + x + C where C is a constant
As (2, 11) passes through this curve,
Therefore, 11 = 2^3 - 3(2)^2 + 2 + C
C = 13

Therefore, y = x^3 - 3x^2 + x + 13

2)
d^2y/dx^2 = 6
dy/dx = 6x + C where C is a constant
2 = 6(0) + C
C = 2
Therefore, dy/dx = 6x + 2
y = 3x^2 + 2x + B where B is another constant
4 = 3(0)^2 + 2(0) + B
B = 4
Therefore, y = 3x^2 + 2x + 4