Application of Derivatives

The value of K in order that f(x) = sin x \(-\) cos x \(-\) kx + 5 decreases for all positive real values of x is given by

The chord of the curve \(y = {x^2} + 2ax + b\), joining the points where x = \(\alpha\) and x = \(\beta\), is parallel to the tangent to the curve at abscissa x is equal to

If f(x) is a function such that f'(x) = (x \(-\) 1)2(4 \(-\) x), then

Time period T of a simple pendulum of length l is given by \(T = 2\pi \sqrt {{l \over g}}\). If the length is increased by 2%, then an approximate change in the time period is

If the area of a rectangle is 64 square units, find the minimum value possible for its perimeter.

The acceleration of a particle starting from rest moving in a straight line with uniform acceleration is 8m/sec2. The time taken by the particle to move the second metre is

Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is \(\left( { - {1 \over 2},{1 \over 2}} \right)\).

The co-ordinates of the point on the curve \(y = {x^2} - 3x + 2\) where the tangent is perpendicular to the straight line y = x are

The point in the interval [0, 2\(\pi\)], where \(f(x) = {e^x}\sin x\) has maximum slope, is

The equation of normal of \({x^2} + {y^2} - 2x + 4y - 5 = 0\) at (2, 1) is

If the normal to the curve y = f(x) at the point (3, 4) makes an angle 3\(\pi\)/4 with the positive x-axis, then f'(3) is

A train moving with constant acceleration takes t seconds to pass a certain fixed point and the front and back end of the train pass the fixed point with velocities u and v respectively. Show that the length of the train is $${1 \over 2}(u ...

Angle between y2 = x and x2 = y at the origin is

If the rate of increase of the radius of a circle is 5 cm/sec., then the rate of increase of its area, when the radius is 20 cm, will be

The distance covered by a particle in t seconds is given by x = 3 + 8t \(-\) 4t2. After 1 second its velocity will be

The Rolle's theorem is applicable in the interval \(-\)1 \(\le\) x \(\le\) 1 for the function

A particle is moving in a straight line. At time t, the distance between the particle from its starting point is given by x = t \(-\) 6t2 + t3. Its acceleration will be zero at

The equation of the tangent to the conic \({x^2} - {y^2} - 8x + 2y + 11 = 0\) at (2, 1) is

A particle is projected vertically upwards and is at a height h after t1 seconds and again after t2 seconds then