A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is propotional to the surface. Prove that the radius is decreasing at a constant rate.
If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius.
A kite is moving horizontally at a height of 151.5 m. If the speed of kite is $10 \mathrm{~m} / \mathrm{s}$, how fast is the string being let out, when the kite is 250 m away from the boy who is flying the kite, if the height of boy is 1.5 m ?
Two men $A$ and $B$ start with velocities $v$ at the same time from the junction of two roads inclined at $45^{\circ}$ to each other. If they travel by different roads, then find the rate at which they are being separated.
Find an angle $\theta$, where $0<\theta<\frac{\pi}{2}$, which increases twice as fast as its sine.
Find the approximate value of $(1.999)^5$.
Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm , respectively.
A man, 2 m tall, walks at the rate of $1 \frac{2}{3} \mathrm{~m} / \mathrm{s}$ towards a street light which is $5 \frac{1}{3} \mathrm{~m}$ above the ground. At what rate is the tip of his shadow moving and at what rate is the length of the shadow changing when he is $3 \frac{1}{3} \mathrm{~m}$ from the base of the light?
A swimming pool is to be drained for cleaning. If $L$ represents the number of litres of water in the pool $t$ seconds after the pool has been plugged off to drain and $L=200(10-t)^2$. How fast is the water running out at the end of 5 s and what is the average rate at which the water flows out during the first 5 s?
The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.
If $x$ and $y$ are the sides of two squares such that $y=x-x^2$, then find the rate of change of the area of second square with respect to the area of first square.
Find the condition that curves $2 x=y^2$ and $2 x y=k$ intersect orthogonally.
Prove that the curves $x y=4$ and $x^2+y^2=8$ touch each other.
Find the coordinates of the point on the curve $\sqrt{x}+\sqrt{y}=4$ at which tangent is equally inclined to the axes.
Find the angle of intersection of the curves $y=4-x^2$ and $y=x^2$.
Prove that the curves $y^2=4 x$ and $x^2+y^2-6 x+1=0$ touch each other at the point $(1,2)$.
Find the equation of the normal lines to the curve $3 x^2-y^2=8$ which are parallel to the line $x+3 y=4$.
At what points on the curve $x^2+y^2-2 x-4 y+1=0$, the tangents are parallel to the $Y$-axis?
Show that the line $\frac{x}{a}+\frac{y}{b}=1$, touches the curve $y=b \cdot e^{-x / a}$ at the point, where the curve intersects the axis of $Y$.
Show that $f(x)=2 x+\cot ^{-1} x+\log \left(\sqrt{1+x^2}-x\right)$ is increasing in $R$.
Show that for $a \geq 1, f(x)=\sqrt{3} \sin x-\cos x-2 a x+b$ is decreasing in $R$.
Show that $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function in $\left(0, \frac{\pi}{4}\right)$.
At what point, the slope of the curve $y=-x^3+3 x^2+9 x-27$ is maximum? Also, find the maximum slope.
Prove that $f(x)=\sin x+\sqrt{3} \cos x$ has maximum value at $x=\frac{\pi}{6}$.
If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, then show that the area of triangle is maximum, when the angle between them is $\frac{\pi}{3}$.
Find the points of local maxima, local minima and the points of inflection of the function $f(x)=x^5-5 x^4+5 x^3-1$. Also, find the corresponding local maximum and local minimum values.
A telephone company in a town has 500 subscribers on its list and collects fixed charges of ₹ 300 per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of ₹ 1 per one subscriber will discontinue the service. Find what increase will bring maximum profit?
If the straight line $x \cos \alpha+y \sin \alpha=p$ touches the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then prove that $a^2 \cos ^2 \alpha+b^2 \sin ^2 \alpha=p^2$.
If an open box with square base is to be made of a given quantity of card board of area $c^2$, then show that the maximum volume of the box is $\frac{c^3}{6 \sqrt{3}}$ cu units.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
I the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sum of their volumes is minimum?
If $A B$ is a diameter of a circle and $C$ is any point on the circle, then show that the area of $\triangle A B C$ is maximum, when it is isosceles.
A metal box with a square base and vertical sides is to contain $1024 \mathrm{~cm}^3$. If the material for the top and bottom costs ₹ $5 \mathrm{per} \mathrm{cm}^2$ and the material for the sides costs ₹ 2.50 per $\mathrm{cm}^2$. Then, find the least cost of the box.
The sum of surface areas of a rectangular parallelopiped with sides $x$, $2 x$ and $\frac{x}{3}$ and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if $x$ is equal to three times the radius of the sphere. Also, find the minimum value of the sum of their volumes.
If the sides of an equilateral triangle are increasing at the rate of 2 $\mathrm{cm} / \mathrm{s}$ then the rate at which the area increases, when side is 10 cm , is
A ladder, 5 m long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of $10 \mathrm{~cm} / \mathrm{s}$, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 m from the wall is
The curve $y=x^{1 / 5}$ has at $(0,0)$
The equation of normal to the curve $3 x^2-y^2=8$ which is parallel to the line $x+3 y=8$ is
If the curve $a y+x^2=7$ and $x^3=y$, cut orthogonally at $(1,1)$, then the value of $a$ is
If $y=x^4-10$ and $x$ changes from 2 to 1.99, then what is the change in $y$ ?
The equation of tangent to the curve $y\left(1+x^2\right)=2-x$, where it crosses $X$-axis, is
The points at which the tangents to the curve $y=x^3-12 x+18$ are parallel to $X$-axis are
The tangent to the curve $y=e^{2 x}$ at the point $(0,1)$ meets $X$-axis at
The slope of tangent to the curve $x=t^2+3 t-8$ and $y=2 t^2-2 t-5$ at the point $(2,-1)$ is
Two curves $x^3-3 x y^2+2=0$ and $3 x^2 y-y^3-2=0$ intersect at an angle of
The interval on which the function $f(x)=2 x^3+9 x^2+12 x-1$ is decreasing is
If $f: R \rightarrow R$ be defined by $f(x)=2 x+\cos x$, then $f$
If $y=x(x-3)^2$ decreases for the values of $x$ given by
The function $f(x)=4 \sin ^3 x-6 \sin ^2 x+12 \sin x+100$ is strictly
Which of the following functions is decreasing on $\left(0, \frac{\pi}{2}\right)$ ?
The function $f(x)=\tan x-x$
If $x$ is real, then the minimum value of $x^2-8 x+17$ is
The smallest value of polynomial $x^3-18 x^2+96 x$ in $[0,9]$ is
The function $f(x)=2 x^3-3 x^2-12 x+4$, has
The maximum value of $\sin x \cdot \cos x$ is
At $x=\frac{5 \pi}{6}, f(x)=2 \sin 3 x+3 \cos 3 x$ is
The maximum slope of curve $y=-x^3+3 x^2+9 x-27$ is
The functin $f(x)=x^x$ has a stationary point at
The maximum value of $\left(\frac{1}{x}\right)^x$ is
The curves $y=4 x^2+2 x-8$ and $y=x^3-x+13$ touch each other at the point ............. .
The equation of normal to the curve $y=\tan x$ at $(0,0)$ is .............. .
The values of $a$ for which the function $f(x)=\sin x-a x+b$ increases on $R$ are ............ .
The function $f(x)=\frac{2 x^2-1}{x^4}$, (where, $x>0$ ) decreases in the interval ............ .
The least value of function $f(x)=a x+\frac{b}{x}($ where $, a>0, b>0, x>0)$ is .............. .