Exercise · ZW-DIRECTIONAL-DERIVATIVE-009 · ★★★ Advanced
If \(|\nabla f(p)|=5\) and for a unit vector \(\mathbf{u}\) we have \(D_{\mathbf{u}}f(p)=5\), what follows about \(\mathbf{u}\)?
Choose one option.
A It is perpendicular to \(\nabla f(p)\) B It points opposite to \(\nabla f(p)\) C It points in the same direction as \(\nabla f(p)\) D It cannot be determined Show answer C. It points in the same direction as \(\nabla f(p)\)← Back to the question
Explanation Since \(D_{\mathbf{u}}f=|\nabla f|\cos\theta\), the value \(5=5\cos\theta\) forces \(\cos\theta=1\), that is, \(\theta=0\): \(\mathbf{u}\) points along the gradient .
The directional derivative \(D_{\mathbf{u}}f\) of \(f\) at a point \(\mathbf{x}_0\), along a unit vector \(\mathbf{u}\), is the rate of change of \(f\) moving from \(\mathbf{x}_0\) in the direction of \(\mathbf{u}\).
\[D_{\mathbf{u}}f(\mathbf{x}_0)=\lim_{h\to0}\frac{f(\mathbf{x}_0+h\mathbf{u})-f(\mathbf{x}_0)}{h}\]
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