Exercise · ZW-GRADIENT-010 · ★★★ Advanced
The temperature is \(T(x,y)=x^2y\). From \((1,1)\), toward which unit vector does it decrease fastest?
Choose one option.
Show answer
- C. \(\dfrac{\langle-2,-1\rangle}{\sqrt5}\)
Explanation
\(\nabla T=\langle2xy,x^2\rangle=\langle2,1\rangle\) at \((1,1)\). The fastest decrease is along \(-\nabla T\) normalized: \(\langle-2,-1\rangle/\sqrt5\).
Review the concept: Gradient
The gradient \(\nabla f\) of a differentiable function \(f\) is the vector that collects its partial derivatives at a point; it points where \(f\) increases fastest.
\[\nabla f=\langle f_x,\,f_y\rangle=f_x\,\mathbf{i}+f_y\,\mathbf{j}\]
More practice problems: gradient
- ZW-GRADIENT-001 Compute ∇ f(1,2) for f(x,y)=x^2y.
- ZW-GRADIENT-002 Compute ∇ f(2,0) for f(x,y)=xe^y.
- ZW-GRADIENT-003 Compute ∇ f(1,2,3) for f(x,y,z)=xyz.
- ZW-GRADIENT-004 Compute ∇ f(1,1) for f(x,y)=ln (x^2+y^2).
- ZW-GRADIENT-005 For f(x,y)=x^2+y^2 at (3,4), what is the maximum rate of change of f?
- ZW-GRADIENT-006 In which unit vector direction does f(x,y)=xy increase fastest from (2,3)?
- ZW-GRADIENT-007 The level curve of f(x,y)=x^2+y^2 through (1,1) has a tangent vector there. Which one?
- ZW-GRADIENT-008 If ∇ f(P)=0 and f is differentiable, what can be said at P?