Let f(x,y) = x²y + 3y². What is ∇f at the point (1, 2)?
A⟨4, 13⟩
B⟨4, 1⟩
C⟨2, 13⟩
D⟨2y, x² + 6y⟩
∂f/∂x = 2xy and ∂f/∂y = x² + 6y. At (1, 2): ∂f/∂x = 2(1)(2) = 4 and ∂f/∂y = (1)² + 6(2) = 1 + 12 = 13. So ∇f(1,2) = ⟨4, 13⟩. Option D gives the symbolic formula (correct in general) but not the evaluated vector at (1,2).
Question 2 True / False
The gradient vector ∇f at a point is perpendicular to the level curve of f passing through that point.
TTrue
FFalse
Answer: True
This is a fundamental geometric property. A level curve is defined by f(x,y) = c, so moving along it leaves f unchanged — the directional derivative in that direction is zero. Since the directional derivative equals ∇f · u, this means ∇f must be orthogonal to every tangent direction of the level curve, i.e., perpendicular to it.
Question 3 Short Answer
A student claims that the gradient ∇f at a point gives the direction of steepest *descent*. What is wrong with this claim, and what direction does ∇f actually indicate?
Think about your answer, then reveal below.
Model answer: The gradient points in the direction of steepest *ascent* (increase), not descent. The steepest descent direction is −∇f.
The directional derivative Dᵤf = ∇f · u = |∇f| cos(θ). This is maximized (greatest increase) when θ = 0, meaning u points in the same direction as ∇f. It is minimized (greatest decrease) when θ = π, meaning u = −∇f/|∇f|. So ∇f is steepest ascent, and −∇f is steepest descent.