(4A-2) The full function is z = x2 + 2y2
In this instance, y is constrained to be equal to -0.5. Therefore the function simplifies to:
z = x2 + 0.5.
The gradient of the tangent is given by the derivative dz/dx = 2x.
When x = -0.5, the gradient is therefore equal to -1.
When, as in this case, the value of y is held constant the partial derivative of the function z = x2 + 2y2, ∂z/∂x = 2x. This is, as it must be, the same as the derivative of the function formed by the intersection of the z = x2 + 0.5 surface with the plane y = -0.5 (ie y is constant).