Question 4

a. Expand $\tan^{-1}\dfrac{y}{x}$ in the neighborhood of $(1,1)$ by Taylor's theorem. Hence compute $f(1.1, 0.9)$. (2 marks)

b. If $u = \tan^{-1}\dfrac{x^3+y^3}{x+y}$, then show that $x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} = \sin 2u$ and$$x^2\frac{\partial^2 u}{\partial x^2} + 2xy\frac{\partial^2 u}{\partial x\,\partial y} + y^2\frac{\partial^2 u}{\partial y^2} = \sin 4u - \sin 2u.$$ (3 marks)