Find the rate of change of the area of a circle with respect to its radius $r$ when $r = 3 \text{ cm}$. (in $\pi \text{ cm}$)

The rate of change of the area of a sphere with respect to its radius, when the radius is $6 \text{ cm}$, is . . . . . . . (in $\pi$)

The interval in which the function $f(x) = x^3$ increases less rapidly than $g(x) = 6x^2 + 15x + 5$ is:

The diameter of one of the bases of a truncated cone is $100 \, mm$. If the diameter of this base is increased by $21 \%$ such that it still remains a truncated cone with the height and the other base unchanged,the volume also increases by $21 \%$. The radius of the other base (in $mm$) is

$A$ particle is moving in a straight line. At time $t$, the distance of the particle from its starting point is given by $x = t^3 - 6t^2 + t$. Its acceleration will be zero at

If $y = x - x^2$,then the rate of change of $y^2$ with respect to $x^2$ at $x = 2$ is