切平面水平条件

解析函数

我在理解函数z = x³的三维表面，寻找切平面水平的点。這個進程讓我注意到，切平面水平的點需要特定的x值。

确定槁杠的水平

嘿！我在找槁杠的平面是水平的，担心它的法向量垂直。然后我发现，它在 x = 0 时是水平的。

To find all points $(x, y)$ where the tangent plane to the surface $z = x^3$ is horizontal, we need to determine where the partial derivatives of $z$ with respect to $x$ and $y$ indicate a horizontal tangent plane.

The tangent plane to a surface $z = f(x, y)$ is horizontal when its normal vector is vertical. This occurs when the gradient vector in the $x$ and $y$ directions is zero, meaning both partial derivatives $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ are zero.

Given: $z = x^3$

Compute the partial derivatives:

$$\frac{\partial z}{\partial x} = 3x^2 \\ \frac{\partial z}{\partial y} = 0$$

Set the partial derivatives equal to zero:

$$3x^2 = 0 \implies x = 0 \\ \frac{\partial z}{\partial y} = 0 \text{ (always true)}$$

The partial derivative with respect to $y$ is always zero, so the condition for a horizontal tangent plane reduces to $x = 0$.

Conclusion: The tangent plane to $z = x^3$ is horizontal at all points where $x = 0$, regardless of the value of $y$.

**Answer: All points where x = 0; that is, all (0, y) in the domain have horizontal tangent planes.