For the function f(x) = x², x∈ [0, 2] compute the average rate of changes in the subintervals [0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2] and the instantaneous rate of changes at the points x = 0.5,1, 1.5, 2

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.

If the volume of a cube of side length x is v = x³. Find the rate of change of the volume with respect to x when x = 5 units.

If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = \(\sqrt { 3 } x\) then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres.

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 points of x the curve y = x³ − 3x² + x − 2 at which the tangent is parallel to the line y = x 

Find the angle of intersection of the curve y = sin x with the positive x -axis.

Find the absolute extrem of the following function on the given closed interval
f(x) = x² -12x + 10; [1, 2]

Find the absolute extrema of the following function on the given closed interval
f(x) = 3x⁴-4x³ ;[-1, 2]

Prove using the Rolle’s theorem that between any two distinct real zeros of the polynomial \(a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\) there is a zero of the polynomial \(na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}\)

Find the absolute extrema of the following functions on the given closed interval.
\(f(x)=6x^{ \frac { 3 }{ 4 } }-3x^{ \frac { 1 }{ 3 } };\left[ -1,1 \right] \)

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

Find the absolute extrema of the following functions on the given closed interval.
\(f(x)=2cosx+sin2x;\left[ 0,\frac { \pi }{ 2 } \right] \)

Find two positive numbers whose sum is 12 and their product is maximum.

Find two positive numbers whose product is 20 and their sum is minimum.

Find the smallest possible value x²+y² given that x +y = 10.

A garden is to be laid out in a rectangular area and protected by wire fence. What is the largest possible area of the fenced garden with 40 metres of wire.

Find the asymptotes of the following curve \(f(x)=\frac { { x }^{ 2 } }{ { x }^{ 2 }-1 } \)

Find the asymptotes of the following curve. \(f(x)=\frac { { x }^{ 2 } }{ { x }^{ 2 }+1 } \)

Write down the Taylor series expansion, of the function log x about x =1 upto three nonzero terms for x > 0.

Find the asymptotes of the following curves \(f(x)=\frac { 3x }{ \sqrt { { x }^{ 2 }+2 } } \) 

Find the asymptotes of the following curve \(f(x)=\frac { { x }^{ 2 }-6x-1 }{ x+3 } \)

Evaluate  \(\underset{x\rightarrow 1}{lim}(\frac{x^{2}-3x+2}{x^{2}-4x+3})\).

Compute the limit  \(\underset{x\rightarrow a}{lim}(\frac{x^{n}-a^{n}}{x-a})\)

Find the asymptotes of the following curves :\(f(x)=\frac { { x }^{ 2 }+6x-4 }{ 3x-6 } \)

Evaluate the limit  \(\underset{x\rightarrow 0}{lim}(\frac{sin \ mx}{x})\)

Evaluate the limit \(\underset{x\rightarrow 0^{+}}{lim} (\frac{sin \ x}{x^{2}})\)

\(\underset{\theta \rightarrow 0}{lim} (\frac{1-cos \ m\theta}{1-cos \ n\theta})\) =1, then prove that, \(m=\pm n\) 

Evaluate: \(\underset{x\rightarrow \infty}{lim}(\frac{x^{2}+17x+29}{x^{4}})\).

Evaluate: : \(\underset{x\rightarrow 0^{+}}{lim}\) x log x.

Evaluate: \(\underset{x\rightarrow \infty}{lim}(\frac{e^{x}}{x^{m}}), m\in N\)

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

Prove that the function f (x) = x² − 2x − 3 is strictly increasing in \((2, \infty)\)

Find the local extremum of the function f (x) = x⁴ + 32x

Find the asymptotes of the function f(x) = \(\frac{1}{x}\)

Find the slant (oblique) asymptote for the function \(f(x)=\frac { { x }^{ 2 }-6x+7 }{ x+5 } \)