What is a tangent plane?
The unique plane that best approximates a smooth surface at a point.
Tangent Plane Calculator
Evaluate f and its first partials at (x₀,y₀) to write the tangent-plane equation.
Enter f and the point
Provide a, b, c and the point (x₀, y₀).
About this calculator
How to use Tangent Plane Calculator, plus the formulas and logic behind the results.
Tool description
The tangent plane is the best linear approximation to z = f(x,y) near a point on the surface.
How to use
- Enter coefficients of f(x,y)=ax²+by²+cxy.
- Enter the point (x₀, y₀).
- Select Calculate to view z₀, partials, and the plane equation.
- Use the plane to approximate f near the point.
Formulas & reference
z = z₀ + fₓ(x₀,y₀)(x − x₀) + fᵧ(x₀,y₀)(y − y₀)fₓ = 2ax + cy, fᵧ = 2by + cx
Tangent plane tips
- Analog of the tangent line in one variable
- Requires differentiability (true for polynomials)
- Normal vector can be taken as (−fₓ, −fᵧ, 1)
- Good for local linear approximations
Calculation logic
Computes z₀ = f(x₀,y₀) and the gradient, then assembles the plane equation.
No constant or linear terms in f for this simplified surface model.
Tangent Plane Calculator FAQ
Common questions about this calculator and how to use the results.
What f is used?
f(x,y)=ax²+by²+cxy.
What is z₀?
The surface height f(x₀,y₀).
How do I approximate f(x,y)?
Plug nearby (x,y) into the plane equation.
Is the normal included?
Partials define the plane; you can form a normal from them.
Is this free?
Yes.