Question

Найти модуль градиента функции z=sin(y/x) в точке (1;0)

Answer 1

$z = sinfrac{y}{x}\z'_x = cos(frac{y}{x}) * (frac{y}{x})'_x = - frac{y* cos(frac{y}{x})}{x^2}\z'_y = cos(frac{y}{x}) * (frac{y}{x})'_y = frac{1}{x}*cos(frac{y}{x})\grad(z) = (- frac{y* cos(frac{y}{x})}{x^2}; frac{1}{x}*cos(frac{y}{x}))\ |grad(z)| = sqrt{frac{y^2*cos^2(frac{y}{x})}{x^4} + frac{cos^2(frac{y}{x}) }{x^2}} = sqrt{frac{cos^2(frac{y}{x})(x^2+y^2)}{x^4}} = frac{|cos(frac{y}{x})|}{x^2}sqrt{x^2+y^2}$

$x = 1\y = 0\|grad(z)| = frac{|cos(frac{0}{1})|}{1^2}sqrt{1^2+0^2} = cos0 = 1\Answer: 1$