Equation Of The Tangent Plane Calculator Desmos

Equation Of The Tangent Plane:

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

Partial Derivative f_x(x₀,y₀):

Partial Derivative f_y(x₀,y₀):

x-coordinate (x₀):

y-coordinate (y₀):

z-coordinate (z₀):

Tangent Plane Equation:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Is The Equation Of The Tangent Plane?

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

3. Importance Of Tangent Plane Calculation

Details: Calculating the tangent plane 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 f_x and f_y at the point, along with the coordinates (x₀, y₀, z₀) of the point of tangency. The calculator will generate the equation of the tangent plane.

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 horizontal?
A: The tangent plane is horizontal when both partial derivatives f_x and f_y are zero at the point.

Q3: Can this be used for surfaces defined implicitly?
A: For implicitly defined surfaces F(x,y,z)=0, the tangent plane equation uses gradient vectors instead of partial derivatives.

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 further away.

Q5: What's the relationship with directional derivatives?
A: The tangent plane contains all possible tangent directions, and directional derivatives can be calculated from the partial derivatives.