A hollow cone with base radius a cm and, height b em is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is \(\frac { 4 }{ 9 } \) times volume of the cone.

Write the Maclaurin series expansion of the following function
tan^-1(x); -1 ≤ x ≤ 1

Find the asymptotes of the following curve \(f(x)=\frac { { x }^{ 2 } }{ { x }^{ 2 }-1 } \)

Evaluate the following limit, if necessary use l ’Hôpital Rule 
\(\underset { x\rightarrow { \frac { \pi }{ 2 } } }{ lim } { \left( sinx \right) }^{ tanx }\)

A steel plant is capable of producing x tonnes per day of a low-grade steel and y tonnes per day of a high-grade steel, where \(y=\frac { 40-5x }{ 10-x } \). If the fixed market price of low-grade steel is half that of high-grade steel, then what should be optimal productions in low-grade steel and high-grade steel in order to have maximum receipts.