1. What is the Equation of Tangent Plane?

The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. It provides the best linear approximation to the surface near that point.

2. How Does the Calculator Work?

The calculator uses the tangent plane equation:

Where:

• \( f(x_0,y_0) \) — Value of the function at the point (x₀,y₀)
• \( f_x \) — Partial derivative of f with respect to x
• \( f_y \) — Partial derivative of f with respect to y
• \( x_0 \) — x-coordinate of the point
• \( y_0 \) — y-coordinate of the point

Explanation: The equation represents a plane that approximates the surface z = f(x,y) near the point (x₀,y₀).

3. Importance of Tangent Plane

Details: Tangent planes are fundamental in multivariable calculus, used for linear approximations, optimization problems, and understanding surface behavior at specific points.

4. Using the Calculator

Tips: Enter all required values - the function value at the point, both partial derivatives, and the point coordinates. The calculator will provide the equation in the form z = Ax + By + C.

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of the tangent plane?
A: It's the plane that best approximates the surface near the given point, touching the surface at that point and having the same slope as the surface in all directions.

Q2: How is this different from a tangent line?
A: A tangent line approximates a curve in 2D, while a tangent plane approximates a surface in 3D.

Q3: When does a tangent plane not exist?
A: When the function is not differentiable at the point (the surface has a sharp corner or edge there).

Q4: Can this be extended to higher dimensions?
A: Yes, the concept generalizes to tangent hyperplanes in n-dimensional spaces.

Q5: How accurate is the tangent plane approximation?
A: It's very accurate very close to the point (x₀,y₀), but the approximation error grows as you move further away.