Tangent Planes

The central idea

If $f:\mathbb R^2\to\mathbb R$ is differentiable at $(a,b)$, the tangent plane to $z=f(x,y)$ at $(a,b,f(a,b))$ is $z-f(a,b)=f_x(a,b)(x-a)+f_y(a,b)(y-b)$. For a regular level surface $F(x,y,z)=c$, the plane has normal $\nabla F$ at the point.

Definitions, hypotheses, and notation

Continuous first partial derivatives near a point are a common sufficient condition for differentiability and a valid tangent plane. Merely finding $f_x$ and $f_y$ at one point does not guarantee the remainder is small in all directions. For implicit surfaces, the nonzero gradient condition allows a local tangent plane and often a graph representation.

A vertical tangent plane cannot be written as $z=L(x,y)$, but level-surface normal form handles it naturally. Choosing which coordinate to solve for depends on which gradient component is nonzero, linking tangent geometry to the implicit function theorem.

Conceptual meaning

The tangent plane is the graph of the best linear approximation. Its slopes in coordinate directions match the surface's partial derivatives, and its normal packages those slopes as $\langle f_x,f_y,-1\rangle$ or the opposite.

A dependable method and decision rule

1. Verify the point lies on the surface.
2. Compute needed partial derivatives and evaluate at the point.
3. Choose graph form or level-surface gradient form.
4. Write point-normal or tangent-plane form before simplifying.
5. Check the plane contains the point and has the expected normal.

Fully worked example

Graphical or geometric meaning

Coordinate trace tangents lie within the plane, one in the $x$ direction and one in the $y$ direction. Their span forms the local flat patch approximating the surface.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute the tangent point into the plane equation.
• Dot the normal with two independent tangent directions.
• Compare with the multivariable linearization.

Regular level surfaces supply normals

For a graph $z=f(x,y)$, the tangent plane at $(a,b,f(a,b))$ comes from linearization. For an implicit surface $F(x,y,z)=k$, the gradient $\nabla F$ is normal when it is nonzero. First verify that the point lies on the surface and that the chosen normal is not zero. If the gradient vanishes, the standard tangent-plane formula is inconclusive; a plane may fail to be unique or may require higher-order analysis. Two independent tangent vectors offer an alternative normal via their cross product. Substitute the base point into the final plane equation and check orthogonality with known tangent directions. This catches evaluation and sign errors without redoing the derivation.

Practice

1. Find the tangent plane to $z=x^2+y^2$ at $(1,1,2)$.
2. Find a normal to $x^2+y^2+z^2=9$ at $(1,2,2)$.
3. What hypothesis supports graph tangent-plane approximation?

Connections and next steps

• Linearization in Several Variables
• Gradient
• Planes