Directional Derivatives — Question 5

Question 5

Let $f$ be continuously differentiable near $P$. At $P$, its level curve is smooth and $\nabla f(P)\ne 0$. A unit direction $u$ makes angle $\alpha$ with $\nabla f(P)$.

Tasks

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

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

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

Question 5 – Solution

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

Step 1: Angular formula $$\boxed{D_{u}f(P)=\nabla f(P)\cdot u =\|\nabla f(P)\|\cos\alpha}.$$

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

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