91Ó°ÊÓ

The volume of a cube is increasing at the rate of \(8 \mathrm{~cm}^{3} / \mathrm{s}\). How fast is the surface area increasing when the length of an edge is \(12 \mathrm{~cm}\) ?

Find the slope of the tangent to curve \(y=x^{3}-x+1\) at the point whose \(x\) -coordinate is 2 .

Find the approximate change in the volume \(\mathrm{V}\) of a cube of side \(x\) metres caused by increasing the side by \(1 \%\).

Find the intervals in which the function \(f\) given by \(f(x)=2 x^{3}-3 x^{2}-36 x+7\) is (a) strictly increasing (b) strictly decreasing

A stone is dropped into a quiet lake and waves move in circles at the speed of \(5 \mathrm{~cm} / \mathrm{s}\). At the instant when the radius of the circular wave is \(8 \mathrm{~cm}\), how fast is the enclosed area increasing?

Show that \(y=\log (1+x)-\frac{2 x}{2+x}, x>-1\), is an increasing function of \(x\) throughout its domain.

At what points in the interval \([0,2 \pi]\), does the function \(\sin 2 x\) attain its maximum value?

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is \(2 \mathrm{~m}\) and volume is \(8 \mathrm{~m}^{3}\). If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

Find the points at which the function \(f\) given by \(f(x)=(x-2)^{4}(x+1)^{3}\) has (i) local maxima (ii) local minima (iii) point of inflexion

Let \(f\) be a function defined on \([a, b]\) such that \(f^{\prime}(x)>0\), for all \(x \in(a, b)\). Then prove that \(f\) is an increasing function on \((a, b)\).