Application of Derivatives

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 between the ladder and the floor is decreasing at the rate of __________ radians/second when...

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

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

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 and at the sides \(50 \mathrm{~cm}\) each are to be left. Then the dimensions i.e. height ...

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

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

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 \(x\) is given by \(\frac{\mathrm{dp}}{\mathrm{d} x}=100-12 \sqrt{x}\).

If the firm empl...

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

\(A\) 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 from \(O\), it is moving away at the rate of \(3 \mathrm{feet} / \mathrm{sec}\). At this i...

If \(a\) and \(b\) are positive number such that \(a>b\), then the minimum value of \(a \sec \theta-b \tan \theta\left(0 < \theta < \frac{\pi}{2}\right)\) is

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 in \([1,5)\), then a root of the equation \(\frac{\mathrm{f}(x)-14}{(x-1)^2}=0(x \neq 1)\) ...

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 \(39 \mathrm{~m} / \mathrm{sec}\), then the rate at which the string is being out, is

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

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 rate of \(1 \mathrm{~m} / \mathrm{sec}\)., then when it is \(8 \mathrm{~m}\) away from th...

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 of the square is \(20 \mathrm{~cm}\), is

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 minimum area of the sheet metal used in its construction, are

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 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the n...

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 \(\alpha \hat{\mathrm{i}}+\hat{\mathrm{k}}\) becomes minimum, is

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 of increase in its volume in \(\mathrm{cm}^3 / \mathrm{sec}\) 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 is zero, is

Let \(\mathrm{f}(x)=\mathrm{e}^x-x\) and \(\mathrm{g}(x)=x^2-x, \forall x \in \mathrm{R}\), then the set of all \(x \in \mathrm{R}\), where the function \(\mathrm{h}(x)=(\mathrm{fog})(x)\) is increasing is

The value of \(\mathrm{c}\) for the function \(\mathrm{f}(x)=\log x\) on [\(1\), e] if LMVT can be applied, 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 \(50 \mathrm{~cm}^3 / \mathrm{min}\). If the thickness of ice is \(5 \mathrm{~cm}\), then the rate at which the...

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

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 given by \(\frac{\mathrm{d} P}{\mathrm{~d} x}=100-12 \sqrt{x}\). If the firm employs 25 mo...

If \(y=\cos \left(\sin x^2\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=\sqrt{\frac{\pi}{2}}\) 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 \(f^{\prime}(3)\) is equal to

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 distance travelled in \(3 \mathrm{~sec}\) is

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

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

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

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

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

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

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 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 rate at which the area increases 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 radius when radius is \(2 \mathrm{~cm}\).

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

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

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

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

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 rate of increase in its area, at the instant when its radius is 10 cm, is _________ cm$$^...

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 temperature of the body is \(0^{\circ} \mathrm{F}\) and after 20 minutes the temperature of t...

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

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