The x - intercept of the tangent at any arbitrary point of the curve a x 2 + b y 2 = 1 is proportional to

cube of the abscissa of the point of tangency

square of the abscissa of the point of tangency

square root of the abscissa of the point of tangency

cube root of abscissa of the point of tangency

The figure shows a rightangled triangle with its hypotenuse OB along the yaxis and its vertex A on the parabola yx2 Let h represent the length of the hypotenuse which depends on the xcoordinate of the point A The value of limx0h is equal to

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 13

Number of positive integral value(s) of

a

for which the curve

y = a^{x}

intersects the line

y = x

is

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 14

Given,

f (x) = 4 - {(\frac{1}{2} - x)}^{\frac{2}{3}}, g (x) = \{\begin{matrix} \frac{\tan [x]}{x}, & x \neq 0 \\ 1 & x = 0 \end{matrix}

, h (x) = \{x\}, k (x) = 5^{l o g_{2} (x + 3)},

then in

[0, 1]

, Lagrange’s mean value theorem is not applicable to, where

[x]

and

\{x\}

denote the greatest integer and fraction part function.

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 15

If the function

f (x) = x^{4} + b x^{2} + 8 x + 1

has a horizontal tangent and a point of inflection for the same value of

x

, then the value of

b

is equal to

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 16

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

is

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 17

A horse runs along a circle with a speed of

20 k m / h

. A lantern is at the centre of the circle. A fence is there along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse moves along the fence at the moment when it covers

\frac{1}{8}

of the circle in

k m / h

is

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 18

Water runs into an inverted conical tent at the rate of

20 f t^{3} / m i n

and leaks out at the rate of

5 f t^{3} / m i n

. The height of the water is three times the radius of the water's surface. The rate at which the radius of the water surface is increasing when the radius is

5 f t

, is

HARD

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 19

Let

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

. If the equation

f (x) = k

has exactly one positive and one negative solution, then the value of

k

equals to

MEDIUM

JEE Main

IMPORTANT

Skills in Mathematics Differential Calculus for JEE Main & Advanced>Chapter 7 - dy/dx as a Rate Measurer & Tangents, Normals>Proficiency in 'dy/dx as a Rate Measurer & Tangents, Normals' Exercise 1>Q 20

The

x -

intercept of the tangent at any arbitrary point of the curve

\frac{a}{x^{2}} + \frac{b}{y^{2}} = 1

is proportional to

The figure shows a right-angled triangle with its hypotenuse $O B$ along the $y -$ axis and its vertex $A$ on the parabola $y = x^{2}$ . Let $h$ represent the length of the hypotenuse which depends on the $x$ -coordinate of the point $A$ . The value of $\lim_{x \to 0} (h)$ is equal to

Important Questions on dy/dx as a Rate Measurer & Tangents, Normals

Learn and Prepare for any exam you want !

The figure shows a right-angled triangle with its hypotenuse OB along the y-axis and its vertex A on the parabola y=x2. Let h represent the length of the hypotenuse which depends on the x-coordinate of the point A. The value of limx→0⁡h is equal to

Important Questions on dy/dx as a Rate Measurer & Tangents, Normals

Learn and Prepare for any exam you want !

The figure shows a right-angled triangle with its hypotenuse $O B$ along the $y -$ axis and its vertex $A$ on the parabola $y = x^{2}$ . Let $h$ represent the length of the hypotenuse which depends on the $x$ -coordinate of the point $A$ . The value of $\lim_{x \to 0} (h)$ is equal to