XII-MATHEMATICS
MATHEMATICS - APPLICATION OF DERIVATIVE (RATE OF CHARGE)
1. The volume of a cube is increasing at the 9. A particle moves along the curve 6y = x3 +
rate of 8 cm3/s. How fast is the surface area 2. Find the points on the curve at which the
increasing when the length of an edge is y-coordinate is changing 8 times as fast as
12 cm ? the x-coordinate.
2. The radius of a circle is increasing 10. The radius of an air bubble is increasing at
uniformly at the rate of 3 cm/s. Find the 1
rate at which the area of the circle is the rate of cm/s. At what rate is the
2
increasing when the radius is 10 cm. volume of the bubble increasing when the
radius is 1 cm ?
3. An edge of a variable cube is increasing at
the rate of 3 cm/s. How fast is the volume 11. A balloon, which always remains
of the cube increasing when the edge is 10 spherical, has a variable diameter
cm long ? 3
(2 x 1) . Find the rate of change of its
2
4. A stone is dropped into a quiet lake and
volume with respect to x.
waves move in circles at the speed of 5
cm/s. At the instant when the radius of the
12. The total cost C(x) in Rupees associated
circular wave is 8 cm, how fast is the
with the production of x units of an item is
enclosed area increasing ?
given by
C (x) = 0.007x3 – 0.003x2 + 15x + 4000
5. The radius of a circle is increasing at the
Find the marginal cost when 17 units are
rate of 0.7 cm/s. What is the rate of
produced.
increase of its circumference ?
13. The total revenue in Rupees received from
6. A balloon, which always remains spherical
the sale of x units of a product is given by
on inflation, is being inflated by pumping
R (x) = 13x2 + 26x + 15
in 900 cubic centrimetres of gas per
Find the marginal revenue when x = 7
second. Find the rate at which the radius of
the balloon increases when the radius is 15
14. The surface area of a spherical bubble is
cm.
increasing at the rate of 2 cm2/sec. Find the
rate at which the volume of the bubble is
7. The length x of a rectangle is decreasing at
increasing at the instant radius 6 cm.
the rate of 5 cm/minute and the width y is
increasing at the rate of 4 cm/minute.
15. The radius of a circle is increasing
When x = 8 cm and y = 6 cm, find the rates
uniformly at the rate of 4 cm/sec. Find the
of change of (a) the perimeter, and (b) the
rate at which the area of the circle is
area of the rectangle.
increasing when the radius is 8 cm.
8. A balloon, which always remains spherical
16. The radius of a balloon is increasing at the
has a variable radius. Find the rate at which
rate of 10 cm/sec. At what rate is the
its volume is increasing with the radius
surface of the balloon increasing when the
when the later is 10 cm.
radius is 15 cm ?
1
Meta Phase Institute| Delanipur Main road, Near Uma Bakery, Opposite Muthoot finance|
Contact No. 9679585359, 7585049444| Port Blair 744102
MATHEMATICS - APPLICATION OF DERIVATIVE (RATE OF CHARGE)
1. The volume of a cube is increasing at the 9. A particle moves along the curve 6y = x3 +
rate of 8 cm3/s. How fast is the surface area 2. Find the points on the curve at which the
increasing when the length of an edge is y-coordinate is changing 8 times as fast as
12 cm ? the x-coordinate.
2. The radius of a circle is increasing 10. The radius of an air bubble is increasing at
uniformly at the rate of 3 cm/s. Find the 1
rate at which the area of the circle is the rate of cm/s. At what rate is the
2
increasing when the radius is 10 cm. volume of the bubble increasing when the
radius is 1 cm ?
3. An edge of a variable cube is increasing at
the rate of 3 cm/s. How fast is the volume 11. A balloon, which always remains
of the cube increasing when the edge is 10 spherical, has a variable diameter
cm long ? 3
(2 x 1) . Find the rate of change of its
2
4. A stone is dropped into a quiet lake and
volume with respect to x.
waves move in circles at the speed of 5
cm/s. At the instant when the radius of the
12. The total cost C(x) in Rupees associated
circular wave is 8 cm, how fast is the
with the production of x units of an item is
enclosed area increasing ?
given by
C (x) = 0.007x3 – 0.003x2 + 15x + 4000
5. The radius of a circle is increasing at the
Find the marginal cost when 17 units are
rate of 0.7 cm/s. What is the rate of
produced.
increase of its circumference ?
13. The total revenue in Rupees received from
6. A balloon, which always remains spherical
the sale of x units of a product is given by
on inflation, is being inflated by pumping
R (x) = 13x2 + 26x + 15
in 900 cubic centrimetres of gas per
Find the marginal revenue when x = 7
second. Find the rate at which the radius of
the balloon increases when the radius is 15
14. The surface area of a spherical bubble is
cm.
increasing at the rate of 2 cm2/sec. Find the
rate at which the volume of the bubble is
7. The length x of a rectangle is decreasing at
increasing at the instant radius 6 cm.
the rate of 5 cm/minute and the width y is
increasing at the rate of 4 cm/minute.
15. The radius of a circle is increasing
When x = 8 cm and y = 6 cm, find the rates
uniformly at the rate of 4 cm/sec. Find the
of change of (a) the perimeter, and (b) the
rate at which the area of the circle is
area of the rectangle.
increasing when the radius is 8 cm.
8. A balloon, which always remains spherical
16. The radius of a balloon is increasing at the
has a variable radius. Find the rate at which
rate of 10 cm/sec. At what rate is the
its volume is increasing with the radius
surface of the balloon increasing when the
when the later is 10 cm.
radius is 15 cm ?
1
Meta Phase Institute| Delanipur Main road, Near Uma Bakery, Opposite Muthoot finance|
Contact No. 9679585359, 7585049444| Port Blair 744102
