Tangent Planes and Normal Lines - MTH 254 Module 5 Tech Lab

① This example demonstrates the geometric construction of the tangent plane and normal line to a surface at a given point, visualizing how the tangent plane touches the surface and the normal line is perpendicular to it. ② Core mathematical concepts include partial derivatives, gradient vectors, tangent plane equations, normal line equations, and parametric curves in 3D space. ③ Construction details: using GeoGebra commands such as Surface to draw the surface S, Function to define f, Vector to create vectors, and CurveCartesian for curves; key objects include point P (tangency point), normal vector N, tangent plane T, and normal line L, with sliders or dynamic points to adjust the tangency point and update the visualization in real time. ④ Suitable for university-level calculus or multivariable calculus courses (e.g., MTH 254), helping students grasp the geometric meaning of tangent planes and normal lines, and useful for classroom demonstrations or self-paced exploration.