16 Find the partial derivative of the dependent variable with respect to each of the independent variables. \[ \frac{\partial z}{\partial x}=\left(5 x^{2}+x y^{3}\right)^{7} \]

Find the exact area of the resulting surface obtained by rotating about the y-axis the curve: \( y=\frac{1}{3} x^{\frac{3}{2}}, 0 \leq x \leq 12 \).

- Determine os limites infinitos, quando existirem: \( \begin{array}{ll}\text { a) } \lim _{x \rightarrow 2} \frac{-1}{2-x} & \text { d) } \lim _{x \rightarrow 2} \frac{2 x-1}{x^{2}-2 x} \\ \text { b) } \lim _{x \rightarrow 0} \frac{1}{x^{6}} & \text { e) } \lim _{x \rightarrow 0} \frac{1-x}{-x^{2}} \\ \text { c) } \lim _{x \rightarrow 3} \frac{x-1}{x-3} & \text { f) } \lim _{x \rightarrow-2} \frac{1-x^{2}}{x+2}\end{array} \)

Find \( \frac{d^{2} y}{d x^{2}} \) given that \( x^{3}-3 x y+y^{3}=1 \)

(b) Prove that between any two real roots of \( e^{x} \cos 2 x=2 \), there is at least one real root of \( e^{x} \sin 2 x=1 \).

\( \int ^ { 3 A 3 } ( 3 ^ { x } - 3 ^ { 2 } ) : 3 + 3 ^ { 3 } : 3 ^ { 2 } = \)