The radius of a cylinder is increasing at a rate of $3 \text{ m/s}$ and its height is decreasing at a rate of $4 \text{ m/s}$. Find the rate of change of its volume when the radius is $4 \text{ m}$ and the height is $6 \text{ m}$.

$A$ particle moves along the curve $y = x^{3/2}$ in the first quadrant in such a way that its distance from the origin increases at the rate of $11$ units per second. The value of $\frac{dx}{dt}$ when $x = 3$ is

The length and breadth of a rectangle are $x \text{ cm}$ and $y \text{ cm}$ respectively. If the length decreases at the rate of $5 \text{ cm/min}$ and the breadth increases at the rate of $3 \text{ cm/min}$,then the rates of change of the perimeter and area respectively when $x = 5 \text{ cm}$ and $y = 2 \text{ cm}$ are:

For what values of $x$ does the rate of change of $x^3 - 5x^2 + 5x + 8$ become twice the rate of change of $x$?

The equation of motion of a particle is given by $s = 2t^3 - 9t^2 + 12t + 1$,where $s$ and $t$ are measured in $cm$ and $sec$. The time when the particle stops momentarily is

$A$ ladder,$5 \ m$ long,rests against a vertical wall. If its top slides downwards at the rate of $10 \ cm/s$,then the angle between the ladder and the floor is decreasing at the rate of how many radians/second when its lower end is $4 \ m$ away from the wall?