Step by step
The question
Question: find the tangent plane to z = x² + y² at (2, −1).
1 Put the point in
First the height of the surface at the point: put x = 2 and y = −1 in.
Work out each term.
So the surface goes through (2, −1, 5).
2 The partial derivatives
The plane's two slopes are the partial derivatives at the point.
Term by term.
x²: the power 2 comes down.
The power goes down by 1.
So 2x.
y² has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 2x.
Term by term.
x² has no y: it is a fixed number, so it gives 0.
y²: the power 2 comes down.
The power goes down by 1.
So 2y.
Drop the zeros: ∂f/∂y = 2y.
At the point: f_x(2, −1) = 4.
At the point: f_y(2, −1) = −2.
3 Write the tangent plane
The rule: z minus z₀ equals the x slope times x minus x₀, plus the y slope times y minus y₀.
Put in the point and the two slopes: 4 for x and −2 for y.
Careful!
Use the slopes at the point, the numbers 4 and −2.
Not the formulas with x and y in them: the plane must be flat.
What is in front of a bracket multiplies every part inside.
Add 5 on both sides.
So the tangent plane is z = 4x − 2y − 5.
4 Check
Let's check with a picture: the surface near (2, −1, 5).
The orange plane rests on it and touches it at the point.
At the point, the plane gives 5, the height of the surface.
Right!
Close by, at (2.1, −1), the surface is 5.41 and the plane is 5.4.
Almost the same.
Right!
Let's look back.
First: put the point in.
Then: the partial derivatives.
Then: write the tangent plane.
Last: check.