Exercise · ZW-GRADIENT-006 · ★★☆ Intermediate
In which unit vector direction does \(f(x,y)=xy\) increase fastest from \((2,3)\)?
Choose one option.
Show answer
- A. \(\dfrac{\langle3,2\rangle}{\sqrt{13}}\)
Explanation
\(\nabla f=\langle y,x\rangle=\langle3,2\rangle\), of norm \(\sqrt{13}\). The direction of fastest increase is the normalized gradient; the unnormalized vector is not a unit vector.
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-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?
- ZW-GRADIENT-009 Find the tangent line to the curve x^2+4y^2=8 at the point (2,1).