A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) = 100 × (1− 0.1t)², 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?

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.

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

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

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
\(f(x)=|\frac{1}{x}|, x\in [-1,1]\)

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
\(f(x)=\sqrt{x}-\frac{x}{3}, x\in [0,9]\)

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

The volume of a cylinder is given by the formula V = πr² h. Find the greatest and least values of V if r + h = 6.

Evaluate the following limit, if necessary use  l ’Hôpital Rule
\(\underset { x\rightarrow \infty }{ lim } \frac { { 2x }^{ 2 }-3 }{ { x }^{ 2 }-5x+3 } \)

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

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

Find the point on the parabola y²=18x at which the ordinate increases at twice the rate of the abscissa.

Find the point at which the curve y-e^xy+x=0 has a vertical tangent.

Verify Rolle ’s Theorem for \(f(x)=\left| x-1 \right| ,O\le x\le 2\) 

Expand the polynomial f(x)=x²-3x+2 in power of (x-2)