Tangent Plane Graph Calculator

Tangent Plane Equation:

\[ z - z_0 = f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y - y_0) \]

Partial Derivative f_x(x₀,y₀):

Partial Derivative f_y(x₀,y₀):

x₀:

y₀:

z₀:

Tangent Plane Equation:

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1. What is the Tangent Plane Equation?

The tangent plane equation represents the plane that best approximates a surface z = f(x,y) at a given point (x₀,y₀,z₀). It provides a linear approximation of the surface near that point.

2. How Does the Calculator Work?

The calculator uses the tangent plane equation:

\[ z - z_0 = f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y - y_0) \]

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 represents the linear approximation of the surface at the given point, where the partial derivatives determine the slope in each direction.

3. Importance of Tangent Plane Calculation

Details: Calculating tangent planes is essential in multivariable calculus for approximating surfaces, optimizing functions, and understanding local behavior of surfaces in 3D space.

4. Using the Calculator

Tips: Enter the partial derivatives fₓ and fᵧ at the point (x₀,y₀), and the coordinates of the point (x₀,y₀,z₀). All values must be valid real numbers.

5. Frequently Asked Questions (FAQ)

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

Q2: When is the tangent plane a good approximation?
A: The tangent plane provides a good approximation when the surface is differentiable at the point and when considering points close to (x₀,y₀).

Q3: Can this calculator handle any surface?
A: This calculator works for any differentiable surface, provided you input the correct partial derivatives at the point of interest.

Q4: What if the partial derivatives are zero?
A: If both partial derivatives are zero, the tangent plane is horizontal (parallel to the xy-plane) at that point.

Q5: How is this related to linear approximation?
A: The tangent plane equation is essentially the multivariable version of linear approximation, providing the best linear approximation of the surface at the given point.