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

Verify LMV theorem for f(x) = x³ - 2x² - x + 3 in [0, 1].

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 ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t² where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?

The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.

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

Find the equation of normal to the curve y⁴=ax²at(a,a)

Find the angle between two curves 2x²+y²=20 and x²-4y²+8=0

Apply Rolle’s theorem to find points on curve y=-1+cosx.whwre the tangent is parallel to x-axis in [0,2π]

Verify Rolle’s theorem for f(x)=ex sinx,\(0\le x\le \pi \)

A cylindrical hole 4 mm in diameter and 12 mm deep in a metal block is reboared to increase the diameter to 4.12mm. Estimate the amount of metal removed

Verify Mean Value theorem for f(x)=(x-1)(x-2)(x-3) in [0,4]

Obtain Maclaurin’s series for \(\frac { 1 }{ 1+x } \)

Write down the Taylor series expansion of the function cos x in ascending powers \(x-\frac { \pi }{ 4 } \) upto three non-zero terms.

Evaluate the following limits, if necessary use L’Hopitals rule
(i) \(\underset { x\rightarrow { 0 }^{ + } }{ lim } { x }^{ sinx }\)
(ii) \(\underset { x\rightarrow 0 }{ lim } \cfrac { cotx }{ cot2x } \) 
(iii) \(\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \left( tanx \right) ^{ cosx }\)