#### Examples : Activity 1

- Find the rate of change of the area of a circle with respect to its radius r when

a) r = 3 cm

b) r= 4 cm

- The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

- An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

- The radius of a circle is increasing at the rate of 7 cm/s. What is the rate of increase of its circumference?

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

- A balloon, which always remains spherical, has a variable diameter \(\frac{3}{2}(2x+1)\). Find the rate of change of its volume with respect to x.

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

- Air is being pumped into a spherical balloon such that its radius increases at a rate of 0.75 in/min. Find the rate of change of its volume when the radius is 5 inches.

#### Examples : Activity 2

- A particle moves along the curve \(6y = x^3 +2\). Find the points on the curve at which the y-coordinate changes 8 times as fast as the x-coordinate.

- 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 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall ?

- The volume of a cube is increasing at the rate of \(8 cm^3/s\). How fast is the surface area increasing when the length of an edge is 12 cm?

- The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y= 6cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

- 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?

- The position of a particle on a line is given by \(s(t) = t 3 − 3 t 2 − 6 t + 5\) , where t is measured in seconds and s is measured in feet. Find

a. The velocity of the particle at the end of 2 seconds.

b. The acceleration of the particle at the end of 2 seconds.

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