Given that, dy/dx = 6x^2 - 3x + 4, and y = 14 when x = 2, express y in terms of x.
dy/dx = 6x2 - 3x + 4
To retrieve the original function y from dy/dx you have to integrate the derivative with respect to x.
y = ∫(dy/dx)dxy = ∫(6x2 - 3x + 4)dx
To integrate, the power is raised by one and the whole term is then divided by the new power on x. The constant of integration is to be included since this is indefinite integration.y = 6x3/3 - 3x2/2 + 4x + cy = 2x3 - 3x2/2 + 4x + c
Since values for y and x are given, they can be substituted into the function to solve for c.y = 14 and x = 214 = 2(2)3 - 3(2)2/2 + 4(2) + c14 = 2(8) - 3(4)/2 + 8 + c14 = 16 - 6 + 8 + c14 = 18 + cc = -4Evaluating the function yields a value of c = -4.This value of c = -4 is written back into the function of y to give the final answer:y = 2x3 - 3x2/2 + 4x - 4
CB
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