Tangent Planes and Linear Approximation

Zoom in far enough and every surface is flat.

12 problems · about 24 minutes · F-IF.B.6, G-GPE.A.1

What this lesson teaches

The student writes the tangent plane to a surface and uses it to approximate values.

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.

Straightforward practice. Get the method working first.

5 problems

f(2, 3) = 20, fₓ = 5, fᵧ = 1. Estimate f(2.2, 3.0).
Answer 21
Why 21.
What is a tangent plane?
Answer The best flat approximation to the surface at a point.
Why The flat surface that best matches locally.
f(1, 1) = 10, fₓ = 4, fᵧ = 0. Estimate f(1.5, 1).
Answer 12
Why 10 + 4(0.5).
f(1, 1) = 10, fₓ = 4, fᵧ = 6. Estimate f(1, 1.5).
Answer 13
Why Only y moved.
fₓ = 2 and fᵧ = 3, with dx = 1 and dy = 1. What is dz?
Answer 5
Why Add both contributions.

The same ideas with more to keep track of.

3 problems

A tangent plane at a point where fₓ = 0 and fᵧ = 0. Is it horizontal? 1 yes, 0 no.
Answer 1
Why No slope in either direction.
f(0, 0) = 7, fₓ = −2, fᵧ = 5. Estimate f(1, 1).
Answer 10
Why 7 − 2 + 5.
Does linear approximation get better or worse further from the point? Enter 1 for better, 2 for worse.
Answer 2
Why The surface curves away from the plane.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Order the steps for building a tangent plane.
Answer 1. Evaluate f at the point 2. Compute both partial derivatives 3. Evaluate both partials at the point 4. Assemble the plane equation
Why You need the numbers before you can assemble them.
f(4, 2) = 30 and both partials are 0. Estimate f(4.1, 2.1).
Answer 30
Why A flat plane predicts no change.
The Small Step: f(2, 5) = 40, fₓ = 3 and fᵧ = −1. Estimate f(2.2, 5.4).
Answer 40.2
Why 40 + 0.6 − 0.4 = 40.2.
The Differential: fₓ = 5 and fᵧ = 2. Both inputs rise by 0.1. What is dz?
Answer 0.7
Why 0.7.