Questions: Directional Derivatives and the Gradient

Question 1

At a point P, the gradient is ∇f = ⟨3, 4⟩. What is the maximum possible value of the directional derivative at P?

• A. 3.5 (the average of the two components)
• B. 7 (the sum of the components)
• C. 5 (the magnitude of the gradient vector)
• D. 4 (the larger component)

Answer: (C)

The directional derivative in direction u is D_u f = ∇f · u = ‖∇f‖ cos θ. This is maximized when cos θ = 1 (u points in the exact direction of ∇f), giving the maximum value ‖∇f‖. Here ‖⟨3,4⟩‖ = √(9+16) = 5. The other options reflect common errors: summing components (ignoring the unit-vector requirement) or averaging them.

Question 2

At point P, ∇f = ⟨4, 3⟩. In which direction u should you travel to achieve the maximum rate of decrease of f?

• A. u = ⟨4, 3⟩/5 — the direction of the gradient
• B. u = ⟨-4, -3⟩/5 — the direction opposite to the gradient
• C. u = ⟨3, -4⟩/5 — a direction perpendicular to the gradient
• D. u = ⟨0, -1⟩ — directly downward, regardless of gradient direction

Answer: (B)

D_u f = ‖∇f‖ cos θ is most negative when cos θ = −1, meaning u points exactly opposite to ∇f. The direction of steepest descent is −∇f/‖∇f‖ = ⟨-4,-3⟩/5. Option C (perpendicular to gradient) gives D_u f = 0 — you would stay on a level curve, not descend. This is the foundation of gradient descent algorithms in machine learning and optimization.

Question 3

If you travel in a direction perpendicular to the gradient ∇f at a point, the value of f stays constant — you are moving along a level curve.

• T. True
• F. False

Answer: True

D_u f = ∇f · u = ‖∇f‖ cos θ. When u is perpendicular to ∇f, θ = 90° and cos 90° = 0, so D_u f = 0. No change in f means you are tracing a path where f is constant — a level curve. Equivalently, the gradient is always perpendicular to the level curves of f.

Question 4

The gradient vector ∇f at a point points in the direction of steepest descent of f.

• T. True
• F. False

Answer: False

The gradient points in the direction of steepest ASCENT — the direction in which f increases most rapidly. The direction of steepest descent is −∇f. This distinction matters greatly in practice: gradient descent algorithms step in the −∇f direction precisely because ∇f itself points uphill.

Question 5

Explain why the gradient is always perpendicular to the level curves of f, using the directional derivative formula.

Model answer: On a level curve, f is constant, so the rate of change of f in any direction tangent to the curve is zero. Since D_u f = ∇f · u = 0 for every direction u tangent to the level curve, the gradient must be orthogonal to all such tangent directions — meaning ∇f is perpendicular to the level curve at every point.

This is the geometric heart of the gradient. The dot product formula D_u f = ∇f · u = 0 forces ∇f to be perpendicular to any direction with zero rate of change. Since level curves are exactly the paths along which f changes at rate zero, ∇f must be normal to them everywhere. This perpendicularity relationship is why contour maps (level curves) and gradient arrows always meet at right angles.