Directional Derivatives

The rate of change in any direction you choose.

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

NameDate

Figures — use these to answer the problems

One arrow is the gradient and the other is your walking direction. Turn them perpendicular and the rate of change dies to zero.

gradient, x-component3

gradient, y-component4

direction, x-component1

direction, y-component0

Figure 1

The gradient at any point crosses these rings at a right angle. Along a ring, nothing changes.

Figure 2

Straightforward practice. Get the method working first.

5 problems

∇f = ⟨8, 6⟩. What is the largest possible directional derivative?
Answer
2
How is the gradient related to a level curve?
- It is perpendicular to the level curve at every point.
- It runs along the level curve.
∇f = ⟨2, 7⟩ and u = ⟨0, 1⟩. What is the directional derivative?
Answer
∇f = ⟨3, 4⟩. What is the largest possible directional derivative?
Answer
∇f = ⟨3, 4⟩. What is the smallest possible directional derivative?
Answer

The same ideas with more to keep track of.

3 problems

Walking along a level curve. What is the directional derivative?
Answer
A direction vector ⟨6, 8⟩. What is its magnitude, needed before using it?
Answer
∇f = ⟨1, 1⟩ and u = ⟨0.6, 0.8⟩. What is the directional derivative, as a decimal?
Answer

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

4 problems

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