The distance travelled $s$ (in $cm$) by a particle in $t$ seconds is given by $s = t^3 + 2t^2 + t$. The speed of the particle after $1$ second will be ......... $cm/sec$.

$A$ particle starts moving from rest from a fixed point in a fixed direction. The distance $s$ from the fixed point at a time $t$ is given by $s = t^{2} + at - b + 17$, where $a$ and $b$ are real numbers. If the particle comes to rest after $5 \ s$ at a distance of $s = 25$ units from the fixed point, then the values of $a$ and $b$ are respectively:

The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x)=3 x^{2}+36 x+5$. The marginal revenue,when $x=15$ is

$A$ particle is moving along a line according to the law $S = t^3 - 3t^2 + 4t - 2$,where $S$ is measured in meters and $t$ is measured in seconds. Then the velocity (in $m/s$) of the particle when its acceleration is zero is:

The side of an equilateral triangle is increasing at the rate of $2 \text{ cm/s}$. Find the rate at which the area is increasing when the side of the triangle is $20 \text{ cm}$.

The rate of change of the radius of a sphere is $0.1 \ cm/s$. Find the rate of change of its surface area when its radius is $200 \ cm$.