(20p)Evaluate \( \left|\begin{array}{rrcrr}0 & 3 & 1 & -2 & 4 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 & 0 \\ 3 & 2 & 0 & 10 & 0 \\ 4 & 0 & 0 & 5 & 1\end{array}\right| \)

\( \frac { \sqrt[ 3 ] { ( ( x _ { 0 } + \Delta x ) - 1 ) ^ { 2 } + \sqrt[ 3 ] { ( x _ { 0 } - 1 ) ^ { 2 } } } } { \Delta x \cdot ( \sqrt[ 3 ] { x _ { 0 } - 1 } + \sqrt[ 3 ] { x _ { 0 } - 1 } ) } \)

\( \lim _ { x \rightarrow 9 } ^ { 9 : 03 } ( \frac { \sqrt { x - 9 } } { \sqrt { x } - 3 } ) \)

1. If \( y=3 t^{4}-\operatorname{tand} x=e^{-t^{2}} \), compute \( \left.\frac{d y}{d x}\right|_{t=0} \) 2. If \( y=3 t^{-2} \) and \( x=e^{-3 t} \), compute \( \left.\frac{d y}{d x}\right|_{t=0} \)

Розв'язати ДР з розподіленими змінними \( \mathrm{x}^{2} y^{\prime}=\frac{1}{\cos y} \).

Determine the volume generated by revolving the area between \( y = \sqrt{x} \) and the x-axis from \( x = 0 \) to \( x = 4 \) about the y-axis with the washer method.