The Gradient Vector

Both partial derivatives, packed into one arrow.

12 problems · about 23 minutes · N-VM.A.1, F-IF.B.6

What this lesson teaches

The student computes the gradient and interprets it as the direction of steepest ascent.

The gradient ∇f = ⟨fₓ, fᵧ⟩ packs both partial derivatives into one vector.
It points in the direction of steepest ascent, and its magnitude is how steep that is.
A zero gradient marks a critical point, where no direction goes up.

Straightforward practice. Get the method working first.

5 problems

f = x² + 3y. What is the x-component of ∇f at (5, 1)?
Answer 10
Why 10.
Which way does the gradient point?
Answer In the direction of steepest increase.
Why Steepest ascent.
f = 3x + 5y. What is the x-component of ∇f?
Answer 3
Why fₓ = 3 everywhere.
f = x² + y². What is the y-component of ∇f at (1, 4)?
Answer 8
Why fᵧ = 2y.
∇f = ⟨5, 12⟩. What is the steepest rate of increase?
Answer 13
Why 25 + 144 = 169.

The same ideas with more to keep track of.

3 problems

∇f = ⟨0, 0⟩ at a point. Is it a critical point? 1 yes, 0 no.
Answer 1
Why No uphill direction exists.
f = xy. What is the x-component of ∇f at (2, 7)?
Answer 7
Why fₓ = y.
f = 10 for every point. What is the magnitude of ∇f?
Answer 0
Why A flat surface has no uphill.

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

4 problems

Match each gradient fact to its meaning.
Answer The direction of ∇f → Steepest ascent; The magnitude of ∇f → How steep that ascent is; The direction of −∇f → Steepest descent
Why Direction and length carry different information.
f = 4x. What is the y-component of ∇f?
Answer 0
Why No y appears.
The Hillside: A hill has ∇h = ⟨3, 4⟩ at your feet. How steep is the steepest climb from there?
Answer 5
Why 5.
The Descent: ∇f = ⟨6, −8⟩. What is the x-component of the steepest descent direction?
Answer -6
Why −6.