1. What is the Tangent Plane Equation?

The tangent plane to a surface z = f(x,y) at a point (x₀,y₀,z₀) is the plane that best approximates the surface near that point. It's defined by the equation z - z₀ = fₓ(x₀,y₀)(x - x₀) + fᵧ(x₀,y₀)(y - y₀), where fₓ and fᵧ are the partial derivatives of f with respect to x and y.

2. How Does the Calculator Work?

The calculator uses the tangent plane equation:

Where:

• \( f_x \) — Partial derivative with respect to x
• \( f_y \) — Partial derivative with respect to y
• \( x_0, y_0, z_0 \) — Coordinates of the point of tangency
• \( x, y, z \) — Variables in the tangent plane equation

Explanation: The equation represents a linear approximation of the surface at the specified point, using the partial derivatives as slopes in the x and y directions.

3. Importance of Tangent Plane Calculation

Details: Calculating tangent planes is fundamental in multivariable calculus, optimization problems, and understanding local behavior of surfaces. It's used in physics, engineering, and computer graphics for surface approximations.

4. Using the Calculator

Tips: Enter the partial derivatives fₓ and fᵧ as mathematical expressions, and the coordinates (x₀, y₀, z₀) as numerical values. The calculator will generate the complete tangent plane equation.

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 doesn't exist at points where the function is not differentiable or where partial derivatives don't exist.

Q3: Can I use this for functions of more than two variables?
A: This calculator is specifically designed for functions of two variables z = f(x,y). For higher dimensions, the concept extends to tangent hyperplanes.

Q4: How accurate is the tangent plane approximation?
A: The approximation is most accurate very close to the point of tangency and becomes less accurate as you move farther away.

Q5: What's the relationship between tangent planes and differentiability?
A: A function is differentiable at a point if it has a well-defined tangent plane at that point.