A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by

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

The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

The point on the curve y = x² is the tangent parallel to X-axis is __________

The value of \(\underset { x\rightarrow \infty }{ lim } { e }^{ -x }\) is __________

A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.

A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.

Find the values in the interval \((\frac{1}{2},2)\) satisfied by the Rolle's theorem for the function \(f(x)=x+\frac{1}{x}, x\in[\frac{1}{2},2]\)

A particle moves in a line so that x =\(\sqrt { t } \). Show that the acceleration is negative and proportional to the cube of the velocity.

Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.

A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?

Find the point on the curve y = x² − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.

Prove that \(\frac { x }{ 1+x } \) < < log (1+ x) for x > 0.

Find the equation of normal to the cure y = sin²x at \(\left( \frac { \pi }{ 3 } ,\frac { 3 }{ 4 } \right) \).

The ends of a rod AB which is 5 m long moves along two grooves OX, OY which at the right angles. If A moves at a constant speed of \(\frac { 1 }{ 2 } \) m/sec, what is the speed of B, when it is 4m from O?

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.

If f(x) = a log x + bx²+ x has entreme values at x = - 1 and x = 2, then find a and b.

Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan^-1(\(\sqrt { 2 } \)).