A tangent line was the best straight approximation to a curve. A tangent plane is the best flat approximation to a surface.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The equation
At (a, b) the tangent plane is z = f(a, b) + fₓ(a, b)(x − a) + fᵧ(a, b)(y − b). It is a starting height plus a slope in each direction.
How to read it
Move a little in x and gain fₓ times that step. Move a little in y and gain fᵧ times that step. Add both.
Linear approximation
Near the point, the plane and the surface are close. Substituting nearby values into the plane estimates the function cheaply.
The differential
dz = fₓ dx + fᵧ dy estimates how much the output changes when both inputs shift slightly.
Where it breaks
The approximation degrades as you move away from the point, and it degrades faster where the surface curves sharply.
Zoom in and the surface flattens
A differentiable surface looks like a plane close enough to any point. That plane is the tangent plane, and it is the two-variable analogue of the tangent line.
Built from the partials
The tangent plane at a point uses both partial derivatives as its slopes in the two axis directions. Together they determine the plane completely.
Linear approximation in two variables
Using the tangent plane in place of the surface estimates nearby values. It is the same trade as in one variable — accuracy near the point in exchange for a far simpler function.
Partials existing is not enough
A function can have both partial derivatives at a point and still fail to be differentiable there. Differentiability requires the tangent plane to approximate well in every direction, not just along the axes.
Step 2: Try It Yourself
Tap and try it out.
- The surfacez = a·x + b·y
- Height at the origin0
Each line holds one coordinate fixed and varies the other, so every line is a cross-section of the surface.
Step 3: Watch an Example
One step at a time.
Watch Lena Approximate a Value
Lena has f(x, y) = x² + y², with f(3, 4) = 25, fₓ(3, 4) = 6 and fᵧ(3, 4) = 8. She estimates f(3.1, 4.1).
- Step 1
She notes the x-step is 0.1 and the y-step is 0.1.
Step 4: Your Turn
Practice makes it stick.
The Small Step
Problem 1 of 2
f(2, 5) = 40, fₓ = 3 and fᵧ = −1. Estimate f(2.2, 5.4).
The Differential
Problem 2 of 2
fₓ = 5 and fᵧ = 2. Both inputs rise by 0.1. What is dz?
Flatten the Surface
1 of 8
f(1, 1) = 10, fₓ = 4, fᵧ = 0. Estimate f(1.5, 1).
2 of 8
f(1, 1) = 10, fₓ = 4, fᵧ = 6. Estimate f(1, 1.5).
3 of 8
fₓ = 2 and fᵧ = 3, with dx = 1 and dy = 1. What is dz?
4 of 8
A tangent plane at a point where fₓ = 0 and fᵧ = 0. Is it horizontal? 1 yes, 0 no.
5 of 8
f(0, 0) = 7, fₓ = −2, fᵧ = 5. Estimate f(1, 1).
6 of 8
Does linear approximation get better or worse further from the point? Enter 1 for better, 2 for worse.
7 of 8
Order the steps for building a tangent plane.
- 1Compute both partial derivatives
- 2Evaluate both partials at the point
- 3Assemble the plane equation
- 4Evaluate f at the point
8 of 8
f(4, 2) = 30 and both partials are 0. Estimate f(4.1, 2.1).
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(2, 3) = 20, fₓ = 5, fᵧ = 1. Estimate f(2.2, 3.0).
Question 2 of 2
What is a tangent plane?
What You Learned
- A tangent plane is the two-variable version of a tangent line.
- z = f(a, b) + fₓ(x − a) + fᵧ(y − b): a starting height plus a gain in each direction.
- The approximation is good nearby and degrades as you move away.