Directional Derivatives

The rate of change in any direction you choose.

12 problems · about 23 minutes · N-VM.B.4, F-IF.B.6

What this lesson teaches

The student computes directional derivatives and relates them to the gradient.

A directional derivative is ∇f dotted with a unit vector in the chosen direction.
It is largest along the gradient and zero perpendicular to it.
Because zero happens along the level curve, the gradient is perpendicular to level curves.

Straightforward practice. Get the method working first.

5 problems

∇f = ⟨8, 6⟩. What is the largest possible directional derivative?
Answer 10
Why 10.
How is the gradient related to a level curve?
Answer It is perpendicular to the level curve at every point.
Why Perpendicular.
∇f = ⟨2, 7⟩ and u = ⟨0, 1⟩. What is the directional derivative?
Answer 7
Why The dot product keeps the y-part.
∇f = ⟨3, 4⟩. What is the largest possible directional derivative?
Answer 5
Why The magnitude of the gradient.
∇f = ⟨3, 4⟩. What is the smallest possible directional derivative?
Answer -5
Why Walking straight downhill.

The same ideas with more to keep track of.

3 problems

Walking along a level curve. What is the directional derivative?
Answer 0
Why The height is unchanged.
A direction vector ⟨6, 8⟩. What is its magnitude, needed before using it?
Answer 10
Why 36 + 64 = 100.
∇f = ⟨1, 1⟩ and u = ⟨0.6, 0.8⟩. What is the directional derivative, as a decimal?
Answer 1.4
Why 0.6 + 0.8.

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

4 problems

Sort each direction by the directional derivative it produces.
Answer Positive: Along the gradient, Slightly off the gradient, less than 90 degrees away · Zero: Perpendicular to the gradient, Along the level curve
Why Perpendicular and along the level curve are the same direction.
∇f = ⟨0, 0⟩. What is the directional derivative in any direction?
Answer 0
Why Dotting with the zero vector.
The Due East Walk: ∇T = ⟨6, −2⟩. What is the rate of change walking in the unit direction ⟨1, 0⟩?
Answer 6
Why 6.
The Level Walk: ∇f = ⟨0, 5⟩. What is the rate of change walking in the unit direction ⟨1, 0⟩?
Answer 0
Why 0.