Find the equation of the normal to the curve y=2x^3 at the point on the curve where x=2. Write in the form of ax+by=c.

For x = 2, y = 16. Calculate the gradient of the curve at y = 2, dy/dx = 6x^2, dy/dx = 24. This is also the gradient of the tangent to the curve at x = 2. It is a rule that the products of the gradients of two lines that are perpendicular to each other must equal -1 (m1m2 = -1). Using this, you can calculate the gradient of the normal to the curve, m2 = -1/24. You can now find the y intercept of the normal by substituting values into the equation y = mx + c. 16 = (-1/24)(2) + c, from rearranging c = 193/12. To get the final answer substitute values into the form ax + by = c which is rearranged from by = ax + c. y = (-1/24)x + 193/12, rearrange this to get (1/24)x + y + 193/12. To make this a lot nicer to read by having whole numbers and no fractions, multiply everything by 24 to get x + 24 y = 386, which is your final answer.