Applications of Derivatives

Q. Of all rectangles of area 100, which has the smallest perimeter?

Q. Find the rate of change of the area of a circle with respect to its radius $r$ when $r = 4$ cm

Q. A stone is dropped into a quiet lake and waves move in circles at the rate of 5 cm/sec. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

Q. A balloon which always remains spherical has variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.

Q. A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall at the rate of 2 cm/sec. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Q. A particle moves along the curve $6y = x^3 + 2$. Find the points on the curve at which the y-coordinate is changing 8 times as fast as x-coordinate.

Q. The total cost C(x) in rupees associated with the production of x units of an item given by $C(x) = 0.007x^3$ - $ 0.003x^2$ + $15x$ + $4000$
Find the marginal cost when 17 units are produced.

Q. The total revenue in rupees received from sale of x units of a product is given by $R(x) = 13x^2 + 26x + 15$
Find the marginal revenue when x = 7.

Q. If $s=4t^3-2t^2+3t+7$, find velocity and acceleration at $t=2s$

Q. Sand is pouring from a pipe at the rate of 12 $cm^3/s$. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?