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An oil tank is being filled. The oil volume $V$ in litres after $t$ minutes is given by: $V = 3 t + t^{2}$ .How fast is the volume increasing when there is $10 l i t r e s$ of oil in the tank?

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Important Questions on Application of Derivatives

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

Let

f (x + y) = f (x) f (y)

for all

x

and

y .

Then what is

f^{'} (5)

equal to

[

where

f^{'} (x)

is the derivative of

f (x)] ?

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

The sides of an equilateral triangle are increasing at the rate of

4 cm / \sec

. The rate at which its area is increasing, when the side is

14 cm

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

The position of a moving car at time

t

is given by

f (t) = a t^{2} + b t + c, t > 0

, where

a, b and c

are real numbers greater than

1

. Then the average speed of the car over the time interval

[t_{1}, t_{2}]

is attained at the point:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If the side of a cube is increased by

5 %

, then the surface area of a cube is increased by

HARD

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is

{t a n}^{- 1} (\frac{1}{2}) .

Water is poured into it at a constant rate of

5 cubic m / \min .

Then the rate

(

m / \min),

at which the level of water is rising at the instant when the depth of water in the tank is

10 m;

is:

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

An inverted conical flask is being filled with water at the rate of

3 {cm}^{3} \sec^{- 1}

. The height of the flask is

10 cm

and the radius of the base is

5 cm

. How fast is the water level rising when the level is

4 cm ?

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If the surface area of a cube is increasing at a rate of

3.6 {cm}^{2} / \sec

, retaining its shape; then the rate of change of its volume (in

{cm}^{3} / \sec

), when the length of a side of the cube is

10 cm

, is:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A ladder of

5 m

long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate

3 m / \sec .

The height of the upper end (in meters) while it is descending at the rate of

4 m / \sec,

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A spherical iron ball of radius

10 c m

is coated with a layer of ice of uniform thickness that melts at a rate of

50 c m^{3} / m i n .

When the thickness of the ice is

5 c m,

then the rate at which the thickness

(

c m / m i n)

of the ice decreases, is :

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

2

m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate

25 c m / s e c

, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is

1

m above the ground is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

The radius of a circle is increasing at the rate

2 cm / \sec

. The rate at which its area is increasing, when the radius of the circle is

5

decimeters, is

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If the volume of spherical ball is increasing at the rate of

4 π

{cm}^{3} / \sec

then the rate of change of its surface area when the volume is

288 π {cm}^{3}

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

The displacement of a particle at the time

t

is given by

s = \sqrt{1 + t},

then its acceleration

a

is proportional to

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

The radius of a right circular cylinder increases at the rate of

0.1 cm / \min

, and the height decreases at the rate of

0.2 cm / \min

. The rate of change of the volume of the cylinder in

{cm}^{3} / \min

, when the radius is

2 cm

and the height is

3 cm

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Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

Read the following data carefully

Parameter	Values
Initial population of lions	$400$
Number of new births in one year	$100$
Number of deaths in one year	$40$

Imagine the lion population grows logistically where the carrying capacity is $800 .$ Then the per capita birth rate $(b)$ , per capita death rate $(d)$ , intrinsic rate of natural increase $(r)$ , increase or decrease in population during the unit time period $(\frac{d N}{d t})$ are respectively

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A point is in motion along a hyperbola

y = \frac{10}{x}

so that its abscissa

x

increases uniformly at a rate of

1

unit per second. Then, the rate of change of its ordinate when the point passes through

(5, 2)

HARD

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If the radius of a spherical balloon increases by

0.1 %,

then its volume increases approximately by

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A spherical iron ball of

10 c m

radius is coated with a layer of ice of uniform thickness that melts at a rate of

50 {c m}^{3} / m i n

. When the thickness of ice is

5 c m

, then the rate (in

c m / m i n

.) at which the thickness of ice decreases, is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

If the volume of a spherical ball is increasing at the rate of

4 π cc / \sec

then the rate of increase of its radius

(in cm / \sec),

when the volume is

288 π cc

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Rate of Change of Quantities

A man

6^{'}

tall moves away from a source of light

20^{'}

above the ground level, his rate of walking being

4 m / h .

At what rate is the tip of his shadow moving?

An oil tank is being filled. The oil volume V in litres after t minutes is given by:V=3t+t2.How fast is the volume increasing when there is 10 litres of oil in the tank?

Important Questions on Application of Derivatives

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An oil tank is being filled. The oil volume $V$ in litres after $t$ minutes is given by: $V = 3 t + t^{2}$ .How fast is the volume increasing when there is $10 l i t r e s$ of oil in the tank?