Application of Derivatives · Mathematics · MHT CET

The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm , is _________ $\mathrm{cm}^3 / \mathrm{cm}^2$....

Water is being poured at the rate of $36 \mathrm{~m}^3 / \mathrm{min}$ into a cylindrical vessel, whose circular base is of radius 3 meters. Then the ...

The equation of the normal to the curve $y=x \log x$ parallel to $2 x-2 y+3=0$ is

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x$, then

If $\mathrm{f}(1)=1, \mathrm{f}^{\prime}(1)=3$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$ is

If $\theta$ denotes the acute angle between the curves $y=10-x^2$ and $y=2+x^2$, at a point of the intersection, then $|\tan \theta|$ is equal to...

If $y=a \log x+b x^2+x$ has its extremum values at $x=-1$ and $x=2$, then

The set of all points, for which $f(x)=x^2 e^{-x}$ strictly increases, is

The abscissa of the point on the curve $y=\mathrm{a}\left(\mathrm{e}^{\frac{x}{a}}+\mathrm{e}^{-\frac{x}{a}}\right)$ where the tangent is parallel to ...

The Number of values of C that satisfy the conclusion of Rolle's theorem in case of following function $\mathrm{f}(x)=\sin 2 \pi x, x \in[-1,1]$ is...

An open tank with a square bottom, to contain 4000 cubic cm . of liquid, is to be constructed. The dimensions of the tank, so that the surface area of...

The function $\mathrm{f}(x)=2 x^3-9 x^2+12 x+2$ is decreasing in

The equation of motion of a particle is $s=a t^2+b t+c$. If the displacement after 1 second is 20 m , velocity after 2 seconds is $30 \mathrm{~m} / \m...

If $\mathrm{f}(x)=x^3-10 x^2+200 x-10$, then

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq.m) of the flowerbed is...

A ladder 5 m in length is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall, at the rate of $2 \mathrm{~m...

If $y=\mathrm{a} \log x+\mathrm{b} x^2+x$ has its extreme values at $x=-1$ and $x=2$, then the value of $\left(\frac{a}{b}+\frac{b}{a}\right)$ is...

The curve $y=a x^3+b x^2+c x+5$ touches the X - axis at $(-2,0)$ and cuts the Y -axis at a point Q where its gradient is 3 , then values of $a, b, c$ ...

The maximum value of the function $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^3-15 x^2+36 x-48$ on the set $A=\left\{x / x^2+20 \leq 9 x\right\}$ is

If the normal to the curve $y=\mathrm{f}(x)$ at the point $(3,4)$ makes an angle of $\left(\frac{3 \pi}{4}\right)$ with the positive $X$-axis, then th...

If $\mathrm{f}(x)=x^3-6 x^2+9 x+3$ is monotonically decreasing function, then $x$ lies in

If equation of normal to the curve $x=\sqrt{t}$, $y=\mathrm{t}-\frac{1}{\sqrt{\mathrm{t}}}$ at $\mathrm{t}=4$ is

The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm , is

The distance ' $s$ ' in meters covered by a body in $t$ seconds is given by $s=3 t^2-8 t+5$. The body will stop after

The volume of a ball is increasing at the rate of $4 \pi \mathrm{cc} / \mathrm{sec}$. The rate of increase of the radius, when the volume is $288 \pi ...

Let $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$. Values of C will be __________ if L.M.V.T. (Lagrange's Mean Value Theorem) can be applied.

If $y=4 x-5$ is a tangent to the curve $y^2=p x^3+q$ at $(2,3)$, then the values of $p$ and $q$ are respectively

Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when dept...

A point moves along the arc of parabola $y=2 x^2$. Its abscissa increases uniformly at the rate of 2 units $/ \mathrm{sec}$. At the instant, the point...

The equation of the normal to the curve $y=x \log x$, which is parallel to the line $2 x-2 y+3=0$, is

The function $\mathrm{f}(x)=2 x^3-6 x+5$ is an increasing function, if

A square plate is contracting at the uniform rate $3 \mathrm{~cm}^2 / \mathrm{sec}$, then the rate at which the perimeter is decreasing, when the side...

A poster is to be printed on a rectangular sheet of paper of area $18 \mathrm{~m}^2$. The margins at the top and bottom of 75 cm each and at the sides...

The equation of the tangent to the curve $x=\operatorname{acos}^3 \theta, y=\operatorname{asin}^3 \theta$ at $\theta=\frac{\pi}{4}$ is

Let C be a curve given by $y(x)=1+\sqrt{4 x-3}$, $x>\frac{3}{4}$. If P is a point on C , such that the tangent at P has slope $\frac{2}{3}$, then a po...

If $\mathrm{f}(x)=\frac{\log x}{x}(x>0)$, then it is increasing in

The maximum value of $\frac{\log x}{x}$ is

If the curves $y^2=6 x, 9 x^2+\mathrm{b} y^2=16$ intersect each other at right angles, then the value of $b$ is

If Mean value theorem holds for the function $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$ then the values of $c$ as per the theorem are

A ladder 5 m long rests against a vertical wall. If its top slides downwards at the rate of $10 \mathrm{~cm} / \mathrm{sec}$., then the foot of the la...

The equation of the tangent to the curve $y=1-\mathrm{e}^{\frac{x}{3}}$ at the point of intersection with Y -axis is

If $8 \mathrm{f}(x)+6 \mathrm{f}\left(\frac{1}{x}\right)=x+5$ and $y=x^2 \mathrm{f}(x)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=-1$ is

The function $f(x)=\frac{\log _e(\pi+x)}{\log _e(e+x)}$ is

The sum of intercepts on coordinate axes made by tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is

A wire of length 2 units is cut into two parts, which are bent respectively to form a square of side $x$ units and a circle of radius of r units. If t...

If sum of two numbers is 3 , then the maximum value of the product of first number and square of the second number is

The curve $y=a x^3+b x^2+c x+5$ touches the $x$-axis at $(-2,0)$ and cuts the $y$-axis at a point Q where its gradient is 3 , then the value of $\math...

If $y=a \log x+b x^2+x$ has its extreme value at $x=-1$ and $x=2$, then the value of $a+b$ is

If $\mathrm{f}(x)=x^3+b x^2+c x+d$ and $0

The minimum value of the function $\mathrm{f}(x)=2 x^3-15 x^2+36 x-48$ on the set $\mathrm{A}=\left\{x \mid x^2+20 \leqslant 9 x\right\}$ is

The value of c for which Rolle's theorem for the function $\mathrm{f}(x)=x^3-3 x^2+2 x$ in the interval $[0,2]$ are

A triangular park is enclosed on two sides by a fence and on the third side a straight river bank. The two sides having fence are of same length $x$. ...

A stone is dropped into a quiet lake and waves move in circles at speed of $8 \mathrm{~cm} / \mathrm{sec}$. At the instant when the radius of the circ...

A bullet is shot horizontally and its distance S cm at time t second is given by $\mathrm{S}=1200 \mathrm{t}-15 \mathrm{t}^2$, then the distance cover...

The equation of the normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

The co-ordinates of a point on the curve $y=x \log x$ at which the normal is parallel to the line $2 x-2 y=3$ are

The value of C for which Mean value Theorem holds for the function $\mathrm{f}(x)=\log _e x$ on the interval $[1,3]$ is

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}$ is...

The equation of normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

The length of the longest interval, in which the function $3 \sin x-4 \sin ^3 x$ is increasing, is

If Rolle's theorem holds for the function $\mathrm{f}(x)=x^3+\mathrm{bx}{ }^2+\mathrm{ax}+5$ on $[1,3]$ with $\mathrm{c}=2+\frac{1}{\sqrt{3}}$, then t...

The normal to the curve, $y(x-2)(x-3)=x+6$ at the point, where the curve intersects the Y-axis, passes through the point

The function $$\mathrm{f}(x)=x^3-6 x^2+9 x+2$$ has maximum value when $$x$$ is

If $$y=4 x-5$$ is a tangent to the curve $$y^2=\mathrm{p} x^3+\mathrm{q}$$ at $$(2,3)$$, then $$\mathrm{p}-\mathrm{q}$$ is

The diagonal of a square is changing at the rate of $$0.5 \mathrm{~cm} / \mathrm{sec}$$. Then the rate of change of area when the area is $$400 \mathr...

Let $$x_0$$ be the point of local minima of $$\mathrm{f}(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$$ where $$...

Let the curve be represented by $$x=2(\cos t+t \sin t), y=2(\sin t-t \cos t)$$. Then normal at any point '$$t$$' of the curve is at a distance of ____...

Let $$\mathrm{B} \equiv(0,3)$$ and $$\mathrm{C} \equiv(4,0)$$. The point $$\mathrm{A}$$ is moving on the line $$y=2 x$$ at the rate of 2 units/second....

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathrm{R} / x^2+30 \leq 11 x\right\}$$ is

Slope of the tangent to the curve $$y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)$$, where $$0 \leq ...

If slope of the tangent to the curve $$x y+a x+b y=0$$ at the point $$(1,1)$$ on it is 2, then the value of $$3 a+b$$ is

$$A(1,-3), B(4,3)$$ are two points on the curve $$y=x-\frac{4}{x}$$. The points on the curve, the tangents at which are parallel to the chord $$A B$$,...

Water is running in a hemispherical bowl of radius $$180 \mathrm{~cm}$$ at the rate of 108 cubic decimeters per minute. How fast the water level is ri...

If Rolle's theorem holds for the function $$f(x)=x^3+b x^2+a x+5$$ on $$[1,3]$$ with $$c=2+\frac{1}{\sqrt{3}}$$, then the values of $$a$$ and $$b$$ re...

If $$\mathrm{f}(x)=x^3+\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}$$ and $$0

The slope of the normal to the curve $$x=\sqrt{t}$$ and $$y=t-\frac{1}{\sqrt{t}}$$ at $$t=4$$ is

Values of $$c$$ as per Rolle's theorem for $$f(x)=\sin x+\cos x+6$$ on $$[0,2 \pi]$$ are

A poster is to be printed on a rectangular sheet of paper of area $$18 \mathrm{~m}^2$$. The margins at the top and bottom of $$75 \mathrm{~cm}$$ each ...

The equation of the normal to the curve $$3 x^2-y^2=8$$, which is parallel to the line $$x+3 y=10$$, is

If the curves $$y^2=6 x$$ and $$9 x^2+b y^2=16$$ intersect each other at right angle, then value of '$$b$$' is

A ladder, 5 meters long, rests against a vertical wall. If its top slides downwards at the rate of $$10 \mathrm{~cm} / \mathrm{s}$$, then the angle be...

A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 4 meter and volume is 36 cubic meters....

The angle between the tangents to the curves $$y=2 x^2$$ and $$x=2 y^2$$ at $$(1,1)$$ is

The function $$\mathrm{f}(x)=\sin ^4 x+\cos ^4 x$$ is increasing in

A ladder 5 meters long rests against a vertical wall. If its top slides downwards at the rate of $$10 \mathrm{~cm} / \mathrm{s}$$, then the angle betw...

If the function $$f$$ is given by $$f(x)=x^3-3(a-2) x^2+3 a x+7$$, for some $$\mathrm{a} \in \mathbb{R}$$, is increasing in $$(0,1]$$ and decreasing i...

If $$a$$ and $$b$$ are positive number such that $$a>b$$, then the minimum value of $$a \sec \theta-b \tan \theta\left(0 ...

$$A$$ rod $$A B, 13$$ feet long moves with its ends $$A$$ and $$B$$ on two perpendicular lines $$O X$$ and $$O Y$$ respectively. When $$A$$ is 5 feet ...

The equation of the tangent to the curve $$y=\sqrt{9-2 x^2}$$, at the point where the ordinate and abscissa are equal, is

At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production $$\mathrm{P}$$ w.r.t. additional number of worker...

Value of $$c$$ satisfying the conditions and conclusions of Rolle's theorem for the function $$\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]$$ is

If $$\mathrm{f}(x)=x \mathrm{e}^{x(1-x)}, x \in \mathrm{R}$$, then $$\mathrm{f}(x)$$ is

The value of $$\mathrm{c}$$ for the function $$\mathrm{f}(x)=\log x$$ on [$$1$$, e] if LMVT can be applied, is

The displacement '$$\mathrm{S}$$' of a moving particle at a time $$t$$ is given by $$S=5+\frac{48}{t}+t^3$$. Then its acceleration when the velocity i...

If the surface area of a spherical balloon of radius $$6 \mathrm{~cm}$$ is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate o...

The value of $$\alpha$$, so that the volume of parallelopiped formed by $$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$ and $$\alpha \hat{\...

In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 1...

An open metallic tank is to be constructed, with a square base and vertical sides, having volume 500 cubic meter. Then the dimensions of the tank, for...

A square plate is contracting at the uniform rate $$4 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate at which the perimeter is decreasing, when side o...

A ladder of length $$17 \mathrm{~m}$$ rests with one end against a vertical wall and the other on the level ground. If the lower end slips away at the...

If the line $$a x+b y+c=0$$ is a normal to the curve $$x y=1$$, then

A kite is $$120 \mathrm{~m}$$ high and $$130 \mathrm{~m}$$ of string is out. If the kite is moving away horizontally at the rate of $$39 \mathrm{~m} /...

If slope of a tangent to the curve $$x y+a x+b y=0$$ at the point $$(1,1)$$ on it is 2, then a - b is

The equation $$x^3+x-1=0$$ has

The range of values of $$x$$ for which $$f(x)=x^3+6 x^2-36 x+7$$ is increasing in

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}$$ is

A water tank has a shape of inverted right circular cone whose semi-vertical angle is $$\tan ^{-1}\left(\frac{1}{2}\right)$$. Water is poured into it ...

Let $$\mathrm{f}(0)=-3$$ and $$\mathrm{f}^{\prime}(x) \leq 5$$ for all real values of $$x$$. The $$\mathrm{f}(2)$$ can have possible maximum value as...

The value of $$c$$ of Lagrange's mean value theorem for $$f(x)=\sqrt{25-x^2}$$ on $$[1,5]$$ is

The value of $$\alpha$$, so that the volume of the parallelopiped formed by $$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$ and $$\alpha \h...

The maximum value of xy when x + 2y = 8 is

An object is moving in the clockwise direction around the unit circle $$x^2+y^2=1$$. As it passes through the point $$\left(\frac{1}{2}, \frac{\sqrt{3...

A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is $$3 \mathrm{~mm}$$ and 1 hour later it reduces...

If the function $$f(x)=x^3-3(a-2) x^2+3 a x+7$$, for some $$a \in I R$$ is increasing in $$(0,1]$$ and decreasing in $$[1,5)$$, then a root of the equ...

A firm is manufacturing 2000 items. It is estimated that the rate of change of production $$P$$ with respect to additional number of workers $$x$$ is ...

If the normal to the curve $$y=f(x)$$ at the point $$(3,4)$$ makes an angle $$\left(\frac{3 \pi}{4}\right)^c$$ with positive $$X$$-axis, then $$f^{\pr...

If $$y=\cos \left(\sin x^2\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ at $$x=\sqrt{\frac{\pi}{2}}$$ is

A spherical iron ball of $$10 \mathrm{~cm}$$ radius is coated with a layer of ice of uniform thickness that melts at the rate of $$50 \mathrm{~cm}^3 /...

The distance 's' in meters covered by a particle in t seconds is given by $$s=2+27 t-t^3$$. The particle will stop after _________ distance.

The curve $$y=a x^3+b x^2+c x+5$$ touches $$X$$-axis at $$P(-2,0)$$ and cuts $$Y$$-axis at a point $$Q$$, where its gradient is 3, then

The minimum value of the function f(x) = x log x is

The maximum area of the rectangle that can be inscribed in a circle of radius $$r$$ is

$$f(x)=\log |\sin x|$$, where $$x \in(0, \pi)$$ is strictly increasing on

The velocity of a particle at time $$t$$ is given by the relation $$v=6 t-\frac{t^2}{6}$$. Its displacement S is zero at $$\mathrm{t}=0$$, then the di...

The radius of a circular plate is increasing at the rate of $$0.01 \mathrm{~cm} / \mathrm{sec}$$, when the radius is $$12 \mathrm{~cm}$$. Then the rat...

If $$x=a(\theta+\sin \theta)$$ and $$y=a(1-\cos \theta)$$ then $$\left(\frac{d^2 y}{d x^2}\right)_{at~ \theta=\pi / 2}=$$

The equation of tangent to the curve $$y=\sqrt{2} \sin \left(2 x+\frac{\pi}{4}\right)$$ at $$x=\frac{\pi}{4}$$, is

Function $$f(x)=e^{-1 / x}$$ is strictly increasing for all $$x$$ where

If $$x=-2$$ and $$x=4$$ are the extreme points of $$y=x^3-\alpha x^2-\beta x+5$$, then

10 is divided into two parts such that the sum of double of the first and square of the other is minimum, then the numbers are respectively

The function $$f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}$$ is increasing, if

If $$f(x)=x^2+a x+b$$ has minima at $$x=3$$ whose value is 5 , then the values of $$a$$ and $$b$$ are respectively.

The slant height of a right circular cone is $$3 \mathrm{~cm}$$. The height of the cone for maximum volume is

The point on the curve $$y^2=2(x-3)$$ at which the normal is parallel to the line $$y-2 x+1=0$$ is

A sperical snow ball is forming so that its volume is increasing at the rate of $$8 \mathrm{~cm}^3 / \mathrm{sec}$$. Find the rate of increase of radi...

The abscissa of the points, where the tangent to the curve $$y=x^3-3 x^2-9 x+5$$ is parallel to $$X$$ axis are

For all real $$x$$, the minimum value of the function $$f(x)=\frac{1-x+x^2}{1+x+x^2}$$ is

The function $$f(x)=\log (1+x)-\frac{2 x}{2+x}$$ is increasing on

A body at an unknown temperature is placed in a room which is held at a constant temperature of $$30^{\circ} \mathrm{F}$$. If after 10 minutes the tem...

A stone is thrown into a quite lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of 4 cm/sec, then the...

The function $$f(x)=\cot ^{-1} x+x$$ is increasing in the interval.

The curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect each other orthogonally, then $$a^2=$$

The surface area of a spherical balloon is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$. Then rate of increase in the volume of the ball...

If $$f(x)=2x^3-15x^2-144x-7$$, then $$f(x)$$ is strictly decreasing in

The equation of tangent to the circle $$x^2+y^2=64$$ at the point $$\mathrm{P\left(\frac{2\pi}{3}\right)}$$ is

Water is being poured at the rate of 36 m$$^3$$/min. into a cylindrical vessel, whose circular base is of radius 3 m. Then the wate level in the cyli...

A wire of length 20 units is divided into two parts such that the product of one part and cube of the other part is maximum, then product of these par...

A particle is moving on a straight line. The distance $$\mathrm{S}$$ travelled in time $$\mathrm{t}$$ is given by $$\mathrm{S=a t^2+b t+6}$$. If the p...

The equation of the tangent to the curve $$y = 4x{e^x}$$ at $$\left( { - 1,{{ - 4} \over e}} \right)$$ is

The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is

The equation of the normal to the curve $2 x^2+y^2=12$ at the point $(2,2)$ is

The area of the square increases at the rate of $0.5 \mathrm{~cm}^2 / \mathrm{sec}$. The rate at which its perimeter is increasing when the side of th...

The equation of normal to the curve $$y=\sin \left(\frac{\pi x}{4}\right)$$ at the point $$(2,5)$$ is

$$\text { If } \sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right] \text {, then } \frac{dy}{dx}=$$

For every value of $$x$$, the function $$f(x)=\frac{1}{a^x}, a>$$ 0 is,

If $$f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$$, then value of '$$c$$' by applying LMVT is

The equation of tangent at $$P(-4,-4)$$ on the curve $$x^2=-4 y$$ is

- The edge of a cube is decreasing at the rate of $0.04 \mathrm{~cm} / \mathrm{sec}$. If the edge of the cube is 10 cms , then rate of decrease of sur...

If $r$ is the radius of spherical balloon at time $t$ and the surface area of balloon changes at a constant rate $K$, then......

The slope of normal to the curve $x=\sqrt{t}$ and $y=t-\frac{1}{\sqrt{t}}$ at $t=4$ is ..........

If $f(x)=x+\frac{1}{x}, x \neq 0$, then local maximum and minimum values of function $f$ are respectively.......

The function $f(x)=x^3-3 x$ is ............

Using differentiation, approximate value of $f(x)=x^2-2 x+1$ at $x=2.99$ is ............

A particle moves so that $x=2+27 t-t^3$. The direction of motion reverses after moving a distance of ....... units.

A stone is dropped into a pond. Waves in the form of circles are generated and radius of outermost ripple increases at the rate of $5 \mathrm{~cm} / \...

The equation of normal to the curve $y=\log _\theta x$ at the point $P(1,0)$ is ............

If $f(x)=3 x^3-9 x^2-27 x+15$, then the maximum value of $f(x)$ is ...........