If $y = \sqrt{(1 - x)(1 + x)}$,then

Let $f$ be a differentiable function such that $(x - y) f(x + y) - (x + y) f(x - y) = 4xy(x^2 - y^2)$ and $f(1) = 2$. Then the area enclosed by $\frac{|f(x) - x|^{1/3}}{17} + \frac{|f(y) - y|^{1/3}}{2} \le \frac{1}{4}$ is

The focal length of a mirror is given by $\frac{2}{f} = \frac{1}{v} - \frac{1}{u}$. In finding the values of $u$ and $v$,the errors are equal to $p$. Then,the relative error in $f$ is

Let $f(x)$ and $g(x)$ be twice differentiable functions such that $f(x) = x^2 + g'(1)x + g''(2)$ and $g(x) = f(1)x^2 + xf'(x) + f''(x)$. Then $f(x) - g(x) =$

If $x^2+y^2=4$,then $y \frac{dy}{dx}+x=$

If $(x e)^{y}=e^{x}$,then $\frac{d y}{d x}$ is