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Which mode of transport is the cheapest for transporting goods?

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Important Questions on Application of Derivatives

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Find two positive numbers whose sum is

15

and the sum of whose squares is minimum.

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2 m$ and volume is $8 m^{3}$ . If building of tank costs $₹ 70$ per square metre for the base and $₹ 45$ per square metre for the sides, what is the cost of least expensive tank $?$

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Show that the height of the right circular cylinder of greatest volume which can be inscribed in a right circular cone of height

h

and radius

r

is one-third of the height of the cone, and the greatest volume of the cylinder is

\frac{4}{9}

times the volume of the cone.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

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 :

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The point on the curve

x^{2} = 2 y

which is nearest to the point

(0, 5)

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

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:

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ?

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Find the rate of change of the area of a circle with respect to its radius $r$ when $r = 3 c m$

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

S_{1}

and

S_{2}

are respectively the sets of local minimum and local maximum points of the function,

f (x) = 9 x^{4} + 12 x^{3} - 36 x^{2} + 25, x \in R,

then

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

f (x)

is a non-zero polynomial of degree four, having local extreme points at

x = - 1, 0, 1;

then the set

S = {x \in R : f (x) = f (0)}

contains exactly

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

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:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

From the top of a

64

metres high tower, a stone is thrown upwards vertically with the velocity of

48 m / s .

The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration

g = 32 m / s^{2}

, is:

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The total cost

C (x)

associated with the production of

x

units of an item is given by

C (x) = 0 \cdot 005 x^{3} - 0 \cdot 02 x^{2} + 30 x + 5000

. Find the marginal cost when

3

units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Acid used in Lead Batteries is:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

The radius of a circle is increasing at the uniform rate of

3 cm / sec

. At the instant when the radius of the circle is

2 cm

, its area increases at the rate of

k π

{cm}^{2}

, then

k =

_____

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

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

r

r = 6 cm .

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

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

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Suppose

f (x)

is a polynomial of degree four having critical points at

- 1, 0, 1

. If

T = \{x \in R | f (x) = f (0)\}

, then the sum of squares of all the elements of

T

is :

EASY

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

Let

M

and

m

be respectively the absolute maximum and the absolute minimum values of the function,

f (x) = 2 x^{3} - 9 x^{2} + 12 x + 5

in the interval

[0, 3]

. Then

M - m

is equal to

HARD

Mathematics>Differential Calculus>Application of Derivatives>Maxima and Minima

If the function

f

given by

f (x) = x^{3} - 3 (a - 2) x^{2} + 3 a x + 7

, for some

a \in R

is increasing in

(0, 1]

and decreasing in

[1, 5)

, then a root of the equation,

\frac{f (x) - 14}{{(x - 1)}^{2}} = 0, (x \neq 1)

is :

Which mode of transport is the cheapest for transporting goods?

Important Questions on Application of Derivatives

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