The volume of a sphere is increasing in volume at the rate of 3 πcm³ / sec. The rate of change of its radius when radius is \(\frac { 1 }{ 2 } \) cm

The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is 

The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

The tangent to the curve y² - xy + 9 = 0 is vertical when 

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\) 

A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds
What is the average velocity with which the camera falls during the last 2 seconds?

A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds

A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
(i) How fast is the top of the ladder moving down the wall?
(ii) At what rate, the area of the triangle formed by the ladder, wall and the floor is changing?

A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.

Find the points on the curve y² - 4xy = x² + 5 for which the tangent is horizontal.

A particle moves so that the distance moved is according to the law s(t) = \(s(t)=\frac{t^{3}}{3}-t^{2}+3\). At what time the velocity and acceleration are zero.

A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) = 128t −16t².
(1) Compute the maximum height of the particle reached.
(2) What is the velocity when the particle hits the ground?

Salt is poured from a conveyer belt at a rate of 30 cubic metre per minute forming a conical pile with a circular base whose height and diameter of base are always equal. How fast is the height of the pile increasing when the pile is 10 metre high?

Find the values in the interval (1, 2) of the mean value theorem satisfied by the function f (x) = x − x² for 1 ≤ x ≤ 2

Using the l’Hôpital Rule prove that, \(\underset{x\rightarrow 0^{+}}{lim}(1+x)^{\frac{1}{x}}=e\)

Prove that the ellipse x² + 4y² = 8 and the hyperbola x²-2y² = 4 intersect orthogonally.

Prove that the function f (x) = x² + 2 is strictly increasing in the interval (2,7) and strictly decreasing in the interval (−2, 0)