) A rod in a mechanism rotates according to the following constraint equation, 5 \cos \theta+4 \cos 3 t=\cos \left(\theta^{2}\right) where theta(t) is the angle of the rod at time

t. Derive an expression for the angular velocity, theta(t). b) An equation of state for a non-ideal gas gives the pressure, p, as a function of volume, V, and temperature, T, according to the relationship p(V, T)=\frac{8 T}{V-3}-\frac{4}{V^{2} \sqrt{T}} . \text { Find } \frac{\partial p}{\partial T} \text { and } \frac{\partial p}{\partial V} (c) The temperature T(x) along a handle is given by the solution of the equation \frac{d T}{d x}+T=x e^{-2 x} . \text { If } T=50^{\circ} \mathrm{C} \text { at } x=0, \text { find the function } T(x) \text { Given that } \frac{\partial f}{\partial x}=2 x-y \sin (x y)+3 x^{2} y^{2} \text { and } \frac{\partial f}{\partial y}=-x \sin (x y)+2 x^{3} y-1 \text {, find } f(x, y)