Partial Derivatives, Tangent Plane, and Gradient

This applet demonstrates the geometric meaning of partial derivatives, tangent planes, and gradient vectors for multivariable functions. Through interactive controls, users can adjust the position of a point on a surface and observe how the tangent plane changes at that location. The applet visualizes the two partial derivatives as slopes along the coordinate directions and constructs the tangent plane from them. It also displays the gradient vector at the selected point, allowing users to explore its relationship with contour lines on the surface. The gradient is shown to be perpendicular to the level curves, pointing in the direction of steepest ascent, with its magnitude representing the maximum rate of change. Students can manipulate sliders and drag points to investigate how these concepts behave across different surfaces and locations. This interactive exploration helps learners connect the algebraic definitions of partial derivatives and gradients with their geometric interpretations, reinforcing key ideas in multivariable calculus. The applet is particularly useful for building intuition about how a smooth surface can be approximated locally by its tangent plane and how the gradient encodes directional information about the function's behavior.