Exercise · ZW-DIRECTIONAL-DERIVATIVE-007 · ★★☆ Intermediate
For \(f(x,y)=x^2+y^2\) at \((1,2)\), in which unit direction \(\mathbf{u}\) is the directional derivative \(0\)?
Choose one option.
Show answer
- D. \(\tfrac{1}{\sqrt5}\langle2,-1\rangle\)
Explanation
\(\nabla f(1,2)=\langle2,4\rangle\), and \(D_{\mathbf{u}}f=0\) requires \(\mathbf{u}\perp\nabla f\). Only \(\langle2,-1\rangle\) works, since \(2\cdot2+4\cdot(-1)=0\); the other options give a nonzero dot product.
Review the concept: Directional derivative
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 \(\mathbf{u}\).
\[D_{\mathbf{u}}f(\mathbf{x}_0)=\lim_{h\to0}\frac{f(\mathbf{x}_0+h\mathbf{u})-f(\mathbf{x}_0)}{h}\]
More practice problems: directional derivative
- ZW-DIRECTIONAL-DERIVATIVE-001 Compute D_uf(1,2) for f(x,y)=x^2y and u=langletfrac35,tfrac45rangle.
- ZW-DIRECTIONAL-DERIVATIVE-002 Compute D_uf(2,3) for f(x,y)=xy and u=i=langle1,0rangle.
- ZW-DIRECTIONAL-DERIVATIVE-003 Compute D_uf(1,1) for f(x,y)=x^2+y^2 and u=langletfrac1√2,tfrac1√2rangle.
- ZW-DIRECTIONAL-DERIVATIVE-004 Compute the derivative of f(x,y)=e^xy at (0,1) in the direction of the vector v=langle3,4rangle.
- ZW-DIRECTIONAL-DERIVATIVE-005 A point at (1,1) moves toward (4,5). Compute the rate of change of f(x,y)=x^2y+y in that direction.
- ZW-DIRECTIONAL-DERIVATIVE-006 What is the maximum rate of change of f(x,y)=x^2+3y^2 at (1,1)?
- ZW-DIRECTIONAL-DERIVATIVE-008 Compute D_uf(1,2,3) for f(x,y,z)=xyz and u=langletfrac23,tfrac13,tfrac23rangle.
- ZW-DIRECTIONAL-DERIVATIVE-009 If |∇ f(p)|=5 and for a unit vector u we have D_uf(p)=5, what follows about u?
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