ACCORDING TO QUES,
DIFFERENCIATE: (x + 1)2 + (x + 2)3 + (x + 3)4
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Exercise 1
AB is a diameter of a circle and C is any point on the circle. Show that the area of Δ ABC is maximum, when it is isosceles.
Let consider AB be the width and C is any point on the circle with range r. \[\angle ACB\text{ }=\text{ }90o\] [angle in the semi-circle is 90o] Let \[AC\text{ }=\text{ }x\] Squaring on both the...
If 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?
We should accept x be to the edge and r be the span of the circle. Surface space of 3D square \[=\text{ }6x2\] Also, surface space of the circle \[=\text{ }4\pi r2\] Presently, their total is...
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. Solution:
We should believe x and y to be the length and expansiveness of given square shape ABCD. As per the inquiry, the square shape will be settled with regards to side AD which making a chamber with...
An open box with square base is to be made of a given quantity of card board of area c2. Show that the maximum volume of the box is c3/ 6√3 cubic units.
Leave x alone the length of the side of the square base of the cubical open box and y be its height Along these lines, the surface space of the open box
If the straight-line x cos α + y sin α = p touches the curve x2/a2 + y2/b2 = 1, then prove that a2 cos2 α + b2 sin2 α = p2.
The given bend is \[\mathbf{x2}/\mathbf{a2}\text{ }+\text{ }\mathbf{y2}/\mathbf{b2}\text{ }=\text{ }\mathbf{1}\] and the straight-line \[\mathbf{x}\text{ }\mathbf{cos}\text{ }\mathbf{\alpha }\text{...
A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?
We should consider that the organization expands the yearly membership by \[\mathbf{Rs}\text{ }\mathbf{x}.\] Along these lines, x is the quantity of supporters who end the administrations. ...
Find the points of local maxima, local minima and the points of inflection of the function f (x) = x5 – 5×4 + 5×3 – 1. Also find the corresponding local maximum and local minimum values.
Given, \[\mathbf{f}\text{ }\left( \mathbf{x} \right)\text{ }=\text{ }\mathbf{x5}\text{ }\text{ }\mathbf{5x4}\text{ }+\text{ }\mathbf{5x3}\text{ }\text{ }\mathbf{1}\] Separating the capacity,...
Prove that f (x) = sin x + √3 cos x has maximum value at x = π/6.
Let ∆ABC be the right-angled triangle in which \[\angle B\text{ }=\text{ }{{90}^{o}}\] Let \[\mathbf{AC}\text{ }=\text{ }\mathbf{x},\text{ }\mathbf{BC}\text{ }=\text{ }\mathbf{y}\] In this way,...
Prove that f (x) = sin x + √3 cos x has maximum value at x = π/6.
ACCORDING TO QUES, HENCE FUNCTION has maximum value at \[x\text{ }=\text{ }\pi /6\] and maximum value is \[2.\]
At what point, the slope of the curve y = – x3 + 3×2 + 9x – 27 is maximum? Also find the maximum slope.
Given, CURVE \[y\text{ }=\text{ }\text{ }x3\text{ }+\text{ }3x2\text{ }+\text{ }9x\text{ }\text{ }27\] Separating the two sides w.r.t. x, we get \[dy/dx\text{ }=\text{ }-\text{ }3x2\text{ }+\text{...
Show that f(x) = tan–1(sin x + cos x) is an increasing function in (0, π/4).
Given, \[f\left( x \right)\text{ }=\text{ }tan1\left( sin\text{ }x\text{ }+\text{ }cos\text{ }x \right)\text{ }in\text{ }\left( 0,\text{ }\pi /4 \right).\] DIFFERENTIATING the two sides w.r.t. x, we...
Show that for a ³ 1, f (x) = √3 sin x – cos x – 2ax + b is decreasing in R.
ACCORDING TO QUES, \[f~\left( x \right)\text{ }=\text{ }\surd 3\text{ }sin~x\text{ }~cos~x\text{ }~2ax\text{ }+\text{ }b,\text{ }a~{}^\text{3}\text{ }1\] differentiating both sides w.r.t. x WE HAVE...
Show that f (x) = 2x + cot-1 x + log [√(1 + x2) – x] is increasing in R.
Given, \[f\text{ }\left( x \right)\text{ }=\text{ }2x\text{ }+\text{ }bed\text{ }1\text{ }x\text{ }+\text{ }log\text{ }\left[ \surd \left( 1\text{ }+\text{ }x2 \right)\text{ }\text{ }x...
Show that the line x/a + y/b = 1, touches the curve y = b . e-x/a at the point where the curve intersects the axis of y.
Given curve condition, \[y\text{ }=\text{ }b\text{ }.\text{ }e-x/an\] and line condition \[x/a\text{ }+\text{ }y/b\text{ }=\text{ }1\] Presently, let the directions of where the curve meets the...
At what points on the curve x2 + y2 – 2x – 4y + 1 = 0, the tangents are parallel to the y-axis?
Given, the condition of the bend is \[x2\text{ }+\text{ }y2\text{ }\text{ }2x\text{ }\text{ }4y\text{ }+\text{ }1\text{ }=\text{ }0\text{ }\ldots \text{ }..\text{ }\left( I \right)\] Separating both...
Find the equation of the normal lines to the curve 3×2 – y2 = 8 which are parallel to the line x + 3y = 4
Given curve, \[3x2\text{ }\text{ }y2\text{ }=\text{ }8\] Separating the two sides w.r.t. x, we get \[6x\text{ }\text{ }2y.\text{ }dy/dx\text{ }=\text{ }0\Rightarrow -\text{ }2y\left( dy/dx...
Prove that the curves y2 = 4x and x2 + y2 – 6x + 1 = 0 touch each other at the point (1, 2).
Given bend conditions are: \[y2\text{ }=\text{ }4x\text{ }\ldots \text{ }.\text{ }\left( 1 \right)\text{ }and\text{ }x2\text{ }+\text{ }y2\text{ }\text{ }6x\text{ }+\text{ }1\text{ }=\text{ }0\text{...
Find the angle of intersection of the curves y = 4 – x2 and y = x2.
The given bends are \[y\text{ }=\text{ }4\text{ }\text{ }x2\text{ }\ldots \text{ }.\text{ }\left( I \right)\text{ }and\text{ }y\text{ }=\text{ }x2\text{ }\ldots \text{ }\left( ii \right)\] Also, we...
Find the co-ordinates of the point on the curve √x + √y = 4 at which tangent is equally inclined to the axes.
ACCORDING TO EQUATION OF CURVE, \[\surd x\text{ }+\text{ }\surd y\text{ }=\text{ }4\] Presently, let (x1, y1) be he required point on the bend SO , \[\surd x1\text{ }+\text{ }\surd y1\text{ }=\text{...
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
Given bends are conditions of two circles, \[xy\text{ }=\text{ }4\text{ }\ldots \text{ }..\text{ }\left( I \right)\] and \[x2\text{ }+\text{ }y2\text{ }=\text{ }8\text{ }\ldots \text{ }.\text{...
Find the condition that the curves 2x = y2 and 2xy = k intersect orthogonally.
It's seen that the given bends are condition of two circles. \[2x\text{ }=\text{ }y2\text{ }\ldots \text{ }..\text{ }\left( 1 \right)\] and \[2xy\text{ }=\text{ }k\text{ }\ldots \text{ }..\text{...
x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of second square with respect to the area of first square.
How about we consider the space of the primary square \[A1\text{ }=\text{ }x2\] Furthermore, space of the subsequent square be \[A2\text{ }=\text{ }y2\] Presently, \[A1\text{ }=\text{ }x2\text{...
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.
We should expect x to be the length of the block. In this way, the volume of the solid shape \[V\text{ }=\text{ }x3\text{ }\ldots \text{ }.\text{ }\left( 1 \right)\] Considering that, \[dV/dt\text{...
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 seconds? What is the average rate at which the water flows out during the first 5 seconds?
Given, \[L\text{ }=\text{ }200\left( 10\text{ }\text{ }t \right)2\] where L addresses the quantity of liters of water in the pool. On separating both the sides w.r.t, t, we get \[dL/dt\text{...
A man, 2m tall, walks at the rate of m/s towards a street light which is m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the shadow changing when he is m from the base of the light?
Let AB is the stature of streetlamp post and CD is the tallness of the man with the end goal that \[AB\text{ }=\text{ }5\left( 1/3 \right)\text{ }=\text{ }16/3\text{ }m\text{ }and\text{ }CD\text{...
Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm, respectively.
Given, The interior range $$ \[r\text{ }=\text{ }3\text{ }cm\] What's more, outside sweep \[R\text{ }=\text{ }r\text{ }+\text{ }r\text{ }=3.0005\text{ }cm\] \[r\text{ }=\text{ }3.0005\text{...
Find the approximate value of (1.999)5.
\[\left( 1.999 \right)5\text{ }=\text{ }\left( 2\text{ }\text{ }0.001 \right)5\] Let \[x\text{ }=\text{ }2\text{ }and\text{ }x\text{ }=\text{ }-\text{ }0.001\] Likewise, let \[y\text{ }=\text{ }x5\]...
Find the approximate value of (1.999)5.
question suggests, angle is \[\pi /3.\]
Find an angle q, 0 < q < /2, which increases twice as fast as its sine.
As per the inquiry, we have In this manner, the necessary point is \[\pi /3.\]
Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being separated.
How about we believe P to be any point where the two streets are leaned at a point of \[45o.\] Presently, two men An and B are moving along the streets PA and PB separately with same speed 'V'....
A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.
Speed of the kite(V) \[=\text{ }10\text{ }m/s\] Leave FD alone the tallness of the kite and AB be the stature of the kite and AB be the tallness of the kid. Presently, let AF \[=\text{ }x\text{ }m\]...
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 proportional to the surface. Prove that the radius is decreasing at a constant rate.
Given, a round ball of salt Then, at that point, the volume of ball \[V\text{ }=\text{ }4/3\text{ }\pi r3\] where r = sweep of the ball Presently, as per the inquiry we have \[dV/dt\propto S\] ,...