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Calculus IIIChapter 2: Partial DerivativesSection 2-7

Directional Derivatives — Question 5

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Question 5

Let ff be continuously differentiable near PP. At PP, its level curve is smooth and ∇f(P)≠0\nabla f(P)\ne 0. A unit direction uu makes angle α\alpha with ∇f(P)\nabla f(P).

Tasks

  1. Derive Duf(P)D_{u}f(P) in terms of α\alpha.

  2. Explain why tangent motion along the level curve gives zero rate.

  3. Determine the sign of the rate for acute, right, and obtuse α\alpha.

Original worksheet page 1: question and worked solution for 2-7-005
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Question 5 – Solution

Strategy. Combine the dot-product formula with the gradient’s normal relationship to level curves.

See the diagram in the original worksheet below.

Step 1: Angular formula Duf(P)=∇f(P)⋅u=∥∇f(P)∥cos⁡α.\boxed{D_{u}f(P)=\nabla f(P)\cdot u =\|\nabla f(P)\|\cos\alpha}.

Step 2: Tangent direction Along the level curve, ff is constant, so its rate is zero. Geometrically the tangent is perpendicular to ∇f\nabla f, giving α=π/2\alpha=\pi/2 and cos⁡α=0\cos\alpha=0.

Step 3: Signs The rate is positive for acute α\alpha, zero for a right angle, and negative for obtuse α\alpha. Its magnitude is greatest when uu is parallel or antiparallel to the gradient.

Original worksheet page 2: question and worked solution for 2-7-005

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