The rate of change of the area of a circle with respect to its radius r at r = 6 cm is

The total revenue in Rupees received from the sale of x units of a product is given by
R(x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is

Which of the following functions are decreasing on 0, \(\frac{\pi}{2}\)?

On which of the following intervals is the function f given by f (x) = x¹⁰⁰ + sin x^–1 decreasing?

The interval in which y = x² e^–x is increasing is

The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is

The line y = x + 1 is a tangent to the curve y² = 4x at the point

If f(x) = 3x² + 15x + 5, then the approximate value of f (3.02) is

The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is

The point on the curve x² = 2y which is nearest to the point (0, 5) is

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

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

The maximum value of \({ [x(x-1)+1] }^{ \frac { 1 }{ 3 } }\), \(0\le x\le 1\) is

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

The slope of the tangent to the curve x = t² + 3t – 8, y = 2t² – 2t – 5 at the point (2,– 1) is

The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is

The normal at the point (1,1) on the curve 2y + x² = 3 is

The normal to the curve x² = 4y passing (1,2) is

The points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are

The total cost C(x) associated with provision of free mid-day meals to x students of a school in primary classes is given by C(x)=0.005x³-0.02x²+30x+50
If the marginal cost is given by rate of change \(dC\over dX\)of total cost,write the marginal cost of food for 300students.What value is shown here?

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8cm and y = 6cm, find the rates of change of
(a) the perimeter,
(b) the area of the rectangle.

Sand is pouring from a pipe at the rate of 12 cm³/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand-cone increasing, when the height is 4 cm?

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

The volume of a cube is increasing at the rate of 8 cm³/sec. How fast is the surface area increasing when the length of an edge is 12 cm?

A particle moves along the curve 6y = x³+2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as x-coordinate .

Find the values of x for which f(x) = [x(x-2)]² is an increasing function.

Prove that \(y={4sin\theta\over2+cos\theta}-\theta\) is an increasing function of \(\theta\) in \([0,{\pi\over2}]\).

Show that \(y=log(1+x)-{2x\over2+x},x>-1\) is an increasing function of x, throughout its domain.

Find the equation of the tangent to the curve \(y=\frac { x-7 }{ \left( x-2 \right) \left( x-3 \right) } \)at the point where it cuts the x-axis.

Show that the normal at any point \(\theta\) to the curve x = a cos  \(\theta\)+ a\(\theta\) sin \(\theta\); y = a sin \(\theta\) - a  \(\theta\) cos  \(\theta\) is at a constant distance from origin.

Find the equation of all lines having slope -1 that are tangents to the curve \(y={1\over x-1},x\neq 1\).

If the radius of a sphere is measured as 9cm with an error of 0.03cm, then find the approximate error in calculating its surface area

Show that the right circular cone of least curved surface and volume has an altitude equal to \(\sqrt{2}\) times the radius of the base.

A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

Show that the right circular cylinder of given volume open at the top has minimum total surface area,provided its height is equal to radius of its base.

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is \(8\over27\)of the volume of the sphere.

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is sin^-1\(1\over3\).

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is \(2R\over \sqrt{3}\). Also find the maximum volume.

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m³. If building of tank costs Rs. 70 per sq. metres for the base and Rs. 45 per square metre for sides. What is the cost of least expensive tank?

Manufacturer can sell x items at price of Rs. \((5-{x\over100})\)  each. The cost price of x items is Rs. \(({x\over5}+500)\). Find the number of items he should sell to earn maximum profit.

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

A square piece of tin of side 18cm is to be made into a box without top by a cutting square from each corner and folding up the flaps to from a box. What should be the side of the square to be cut off so that the volume of the box is maximum?. Also find the maximum volume.

Show that the semi-vertical angle of the right circular cone of maximum volume and given slant height, is tan^-1\(\sqrt{2}\)

Find two positive numbers whose sum is 15 and the sum of whose squares is minimum

Find the shortest distance of the point (0, c) from the parabola y = x², where \(\frac{1}{2}\) \(0\le c \le 5.\)

Let AP and BQ be two vertical poles at points A and B respectively. If Ap = 16m, PQ = 22m, Ab = 20m. Then find the distance of a point  R on a point A such that RP² + RQ² is minimum.

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: \(f(x)=\sin x+\cos x, x \in[0, \pi]\)

Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals:
\(f(x)=4 x-\frac{1}{2} x^2, x \in\left[-2, \frac{9}{2}\right]\)

An open topped box is to be constructed by removing equal squares from each corner of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.

Find the point on the curve \(y=x^3-11x+5\) at which the equation of tangent is \(y=x-11.\)

Using differentials, find the approximate value of \(\sqrt {49\cdot5}\)

Find the intervals in which \(f(x) =\frac{3}{10}x^4-\frac{4}{5}x^3+\frac{36}{5}x+11\)is (a) strictly increasing (b) strictly decreasing.

Find the equations of the tangent and normal to the curve \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } -\frac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) at the point \((\sqrt { 2 } a,b)\).

Using differentials, find the approximate value of (3.968)^3/2.

Find the intervals in which the function f given by
\(f(x)=\frac{4 \sin x-2 x-x \cos x}{2+\cos x}\)
is (i) increasing (ii) decreasing

Find the intervals in which the following function is increasing or decreasing
\(f(x)=x^{3}+\frac{1}{x^{3}}, x \neq 0\)

Find the equations of the normal to the curve \(y=x^{3}+2 x+6\) ,which are parallel to line x+ 14y + 4 = 0.

Using differential, find the approximate value of the following \((255)^{1 / 4}\)

Using differential, find the approximate value of the following
\((0.999)^{1 / 10}\)

Using differential, find the approximate value of the following.
\((81.5)^{1 / 4}\)

Find the points at which the function f given by \(f(x)=(x-4)^{4}(x+1)^{3}\) 
(i) local maxima.
(ii) local minima.
(iii) point of inflection.

It is given that at x = 1, the function \(x^{4}-62 x^{2}+a x+9\) attains maximum value on the interval [0, 2]. Find the value of a.

Find the maximum and minimum values, if any of the following function. 
\(f(x)=|x+2|-1\)

Find the maximum and minimum value, if any of the following functions
\(g(x)=-|x+1|+3\)

Find the maximum and minimum values, if any of the following function
\(h(x)=x+1, x \in(-1,1)\)

Find two positive numbers whose sum is 16 and the sum of whose squares is minimum.

Find the maximum area of an isosceles triangle inscribed in the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with its vertex at one end of the major axis

A point on the hypotenuse of a triangle is at distances a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is \(\left(a^{2 / 3}+b^{2 / 3}\right)^{3 / 2}\).