Answer: -75.96°
(i) To find the gradient of the normal to the curve at the point where the curve intersects the y-axis, we need to find the derivative of the curve equation and negate it. The derivative of y = -x³ + 3x² - 4x + 2 is given by:
dy/dx = -3x² + 6x - 4
At the point where the curve intersects the y-axis, x = 0. So, the gradient of the normal to the curve at this point is:
-3 * 0² + 6 * 0 - 4 = -4
(ii) To find the angle that this normal to the curve makes with the x-axis, we can use the tangent function. The tangent of an angle is equal to the gradient of a line, so we have:
tan(θ) = gradient of the normal = -4
The inverse tangent function (arctan) gives us the angle θ in radians, which we can then convert to degrees:
θ = arctan(-4) = -75.96°
So, the angle that the normal to the curve makes with the x-axis is approximately -75.96°.
Explanation: