1. The problem is to understand what a tangent plane is. 2. A tangent plane to a surface at a given point is a plane that just touches the surface at that point and is "flat" relative to the surface there. 3. For a surface defined by a function $z=f(x,y)$, the tangent plane at point $(x_0,y_0,z_0)$ is given by the formula: $$z = f(x_0,y_0) + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y - y_0)$$ where $f_x$ and $f_y$ are the partial derivatives of $f$ with respect to $x$ and $y$. 4. This plane approximates the surface near the point $(x_0,y_0,z_0)$ and is useful in calculus and geometry to study the surface's behavior. 5. In simple terms, the tangent plane is like a flat sheet that just touches a curved surface at one point without cutting through it nearby.