The function f x = x 3 − 3 x is

Increasing in − ∞ , − 1 ∪ 1 , ∞ and decreasing in - 1 , 1

Decreasing in − ∞ , − 1 ∪ 1 , ∞ and increasing in - 1 , 1

Increasing in 0 , ∞ and decreasing in - ∞ , 0

Decreasing in 0 , ∞ and increasing in − ∞ , 0

If the radius of a sphere is measured as 9 cm with an error of 0 . 03 cm , then find the approximate error in calculating its volume.

Consider the radius of the sphere be R and ∆ R be the error in measuring the radius. Given R = 9 cm and ∆ R = 0 . 03 cm . Now, the volume V of the sphere is given by V = 4 3 π R 3 . . . ( 1 ) Now, differentiating equation ( 1 ) , with respect to R , we get, d V d R = 4 π R 2 We have d V = d V d R ∆ R = 4 π R 2 ∆ R = 4 π 9 2 0 . 03 = 9 . 72 π cm 3 Thus, the approximate error in calculating the volume is 9 . 72 π cm 3 .

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A point on the parabola $y^{2} = 18 x$ at which the ordinate increases at twice the rate of the abscissa :

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Important Questions on Application of Derivatives

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 3

Coffee is coming out of a conical filter, with height and diameter both

15 c m

, into a cylindrical coffee pot with diameter

15 c m

. The rate at which coffee comes out from the filter into the pot is

100 c m^{3} / m i n

. The rate in

c m / m i n

at which the level in the pot is rising at the instance when the coffee in the pot is

10 c m

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 4

The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is

1 cm

the altitude is

6 cm

. When the radius is

6 cm

, the volume is increasing at the rate of

1 {cm}^{3} / s

. When the radius is

36 cm

, the volume is increasing at a rate of

n {cm}^{3} / s

. The value of '

n

' is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 5

A particle moves along the curve

y = x^{\frac{3}{2}}

in the first quadrant in such a way that its distance from the origin increases at the rate of

11

units per second. The value of

\frac{d x}{d t}

when

x = 3

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 6

A cube of ice melts without changing shape at the uniform rate of

4 {cm}^{3} / minute

. The rate of change of the surface area of the cube, in

{cm}^{3} / minute

, when the volume of the cube is

125 {cm}^{3}

, is

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 7

Two cyclists start from the junction of two perpendicular roads, their velocities being

3 v

metres/minute and

4 v

metres/minute. The rate at which the two cyclists are separating is

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 8

The interval on which

f (x) = 2 x^{3} + 9 x^{2} + 12 x - 1

is decreasing in

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 9

The exhaustive values of

x

for which

f (x) = \cot^{- 1} x + x

is increasing is

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 10

The function

f (x) = x^{3} - 3 x

A point on the parabola y2=18x at which the ordinate increases at twice the rate of the abscissa :

Important Questions on Application of Derivatives

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A point on the parabola $y^{2} = 18 x$ at which the ordinate increases at twice the rate of the abscissa :