1. What is the Tangent Plane Equation?

The tangent plane equation represents the plane that just touches a surface at a given point. For a surface z = f(x,y), the tangent plane at point (x₀,y₀,z₀) is given by the linear approximation using partial derivatives.

2. How Does the Calculator Work?

The calculator uses the tangent plane equation:

Where:

• \( f_x(x_0,y_0) \) — Partial derivative with respect to x at point (x₀,y₀)
• \( f_y(x_0,y_0) \) — Partial derivative with respect to y at point (x₀,y₀)
• \( x_0, y_0, z_0 \) — Coordinates of the point of tangency

Explanation: The equation provides the best linear approximation to the surface near the point of tangency.

3. Importance of Tangent Plane Calculation

Details: Tangent planes are fundamental in multivariable calculus for approximating functions, optimization problems, and understanding surface behavior locally.

4. Using the Calculator

Tips: Enter the partial derivatives f_x and f_y at the point (x₀,y₀), along with the coordinates of the point of tangency (x₀,y₀,z₀).

5. Frequently Asked Questions (FAQ)

Q1: What are partial derivatives?
A: Partial derivatives measure how a function changes as one variable changes while keeping other variables constant.

Q2: When does a tangent plane not exist?
A: A tangent plane may not exist if the function is not differentiable at the point, or if the partial derivatives are discontinuous.

Q3: How is this related to the gradient?
A: The normal vector to the tangent plane is given by the gradient vector (f_x, f_y, -1) at the point.

Q4: Can this be extended to higher dimensions?
A: Yes, the concept generalizes to tangent hyperplanes for functions of more than two variables.

Q5: What's the difference between tangent plane and linear approximation?
A: The tangent plane equation gives the linear approximation of the function near the point (x₀,y₀).