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

The temperature T in celsius in a long rod of length 10 m, insulated at both ends, is a function of length x given by T = x(10 − x). Prove that the rate of change of temperature at the midpoint of the rod is zero.

A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t³ − 6t² +9 t +1, where s is measured in metres and t in seconds?
(1) At what time the particle is at rest?
(2) At what time the particle changes its direction?
(3) Find the total distance travelled by the particle in the first 2 seconds.

The price of a product is related to the number of units available (supply) by the equation Px + 3P −16x = 234, where P is the price of the product per unit in Rupees(Rs) and x is the number of units. Find the rate at which the price is changing with respect to time when 90 units are available and the supply is increasing at a rate of 15 units/week.

Compute the value of 'c' satisfied by the Rolle’s theorem for the function f (x) = x² (1 - x)², x ∈ [0,1]

Without actually solving show that the equation x⁴+2x³-2 = 0 has only one real root in the interval (0, 1).

Prove that there is a zero of the polynomial \(2x^{3}-9x^{2}-11x+12\) in the interval (2, 7) given that 2 and 7 are the zeros of the polynomial \(x^{4}-6x^{3}-11x^{2}+24x+28\)

Prove, using mean value theorem, that \(|sin \alpha-sin\beta|\le |\alpha-\beta|, \alpha, \beta \in R\)

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

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

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

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

Find the local maximum and minimum of the function x² y² on the line x + y = 10

Sketch the curve y = f (x) = x³−6x-9

Sketch the graph of the function \(y=\frac { 3x }{ { x }^{ 2 }-1 } \)