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Math101Tangent Planes
A rigorous, example-driven guide to tangent planes, including hypotheses, method choice, verification, and practice.
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
- Verify the point lies on the surface.
- Compute needed partial derivatives and evaluate at the point.
- Choose graph form or level-surface gradient form.
- Write point-normal or tangent-plane form before simplifying.
- Check the plane contains the point and has the expected normal.
