Exercise · ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-009 · ★★★ Advanced
Let \(f(x,y)=\dfrac{x^2y}{x^2+y^2}\) with \(f(0,0)=0\). Which statement is correct?
Choose one option.
Show answer
- C. Its partials exist at the origin, but it is not differentiable
Explanation
Since \(f(h,0)=f(0,k)=0\), both partials equal \(0\) at the origin. But along \(h=k\) the error quotient equals \(\sqrt2/4\neq0\), so it is not differentiable. It is continuous: \(|f|\le|y|\).
Review the concept: Differentiability in several variables
A function \(f(x,y)\) is differentiable at a point \((a,b)\) if its graph has a tangent plane there: the error of the linear approximation is negligible compared with the distance to the point.
\[\lim_{(h,k)\to(0,0)}\frac{f(a+h,b+k)-f(a,b)-f_x(a,b)\,h-f_y(a,b)\,k}{\sqrt{h^2+k^2}}=0\]
More practice problems: differentiability in several variables
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-001 Let f(x,y)=x^2y. Use the tangent plane at (1,2) to approximate f(1.1, 1.9).
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-002 For f(x,y)=x^2+3xy at (1,1), compute the differential df=f_x dx+f_y dy with dx=0.1, dy=-0.2.
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-003 Let f(x,y)=xe^y. At (2,0), with dx=0.05 and dy=0.1, what is df?
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-004 Find the linearization of f(x,y)=sin (xy) at the point (1,0).
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-005 Which condition guarantees that f is differentiable at (a,b)?
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-006 If f is differentiable at P, which statement is always true?
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-007 Let f(x,y)=√x^2+y^2. Why is it not differentiable at the origin?
- ZW-DIFFERENTIABILITY-SEVERAL-VARIABLES-008 Which of these functions is differentiable on all of ℝ^2?
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