Mathematics : Application of Derivatives

Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of t heradius of the base. How fast is the sand cone increasing when the height is 4 cm?

Find the points on the curve  x² + y² – 2x – 3= 0  at  whichthe tangents are parallel to x-axis.

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Find the point on the curve  y = x³ – 11x + 5  at which the equation of tangent is  y = x – 11.

Using differentials, find the approximate value of  $\sqrt{ 49.5}.$

Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.

On the ellipse 4x² + 9y² = 1, the points at which the tangents are parallel to the line 8x = 9y, are

• $\left(\frac{2}{5}, \frac{1}{5}\right)$

• $\left(- \frac{2}{5}, \frac{1}{5}\right)$

• $\left(- \frac{2}{5}, - \frac{1}{5}\right)$

• $\left(\frac{2}{5}, - \frac{1}{5}\right)$

A container s the shape of an inverted cone. Its height is 6 m and radius is 4m at the top. If it is filled with water at the rate of 3m/min then the rate of change of height of water(in mt/min) when the water level is 3 m is

• $\frac{3}{4\mathrm{\pi }}$

• $\frac{2}{9\mathrm{\pi }}$

• $16\mathrm{\pi }$

• $2\mathrm{\pi }$

$\mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{the} \mathrm{lengths} \mathrm{of} \mathrm{the} \mathrm{tangents} \mathrm{from} \mathrm{the} \mathrm{vertices} \mathrm{of} \mathrm{a} \mathrm{triangle} \mathrm{to} \mathrm{its} \mathrm{incircle}. \mathrm{Then}$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \frac{1}{{\mathrm{r}}^2}\left(\mathrm{\alpha \beta \gamma }\right)$

• $\mathrm{\alpha } + \mathrm{\beta } + \mathrm{\gamma } = \frac{1}{\mathrm{r}}\left(\mathrm{\alpha \beta \gamma }\right)$

• $\frac{1}{\mathrm{\alpha }} + \frac{1}{\mathrm{\beta }} + \frac{1}{\mathrm{\gamma }} = \mathrm{r}\left(\mathrm{\alpha \beta \gamma }\right)$

• ${\mathrm{\alpha }}^2 + {\mathrm{\beta }}^2 + {\mathrm{\gamma }}^2 = \frac{2}{\mathrm{r}}\left(\mathrm{\alpha \beta \gamma }\right)$

If a cylindrical vessel of given volume V with no lid on the top is to be made from a sheet of metal, then the radius (r) and height(h) of the vessel so that the metal sheet used is minimum is

• $\mathrm{r} = \sqrt[3]{\frac{\mathrm{\pi }}{\mathrm{V}}}, \mathrm{h} = \sqrt[3]{\frac{\mathrm{\pi }}{\mathrm{V}}}$

• $\mathrm{r} = \sqrt{\mathrm{\pi V}}, \mathrm{h} = \sqrt{\mathrm{\pi V}}$

• $\mathrm{r} = \sqrt[3]{\frac{\mathrm{V}}{\mathrm{\pi }}}, \mathrm{h} = \sqrt[3]{\frac{\mathrm{V}}{\mathrm{\pi }}}$

• $\mathrm{r} = \sqrt{\frac{\mathrm{V}}{\mathrm{\pi }}}, \mathrm{h} = \sqrt{\frac{\mathrm{V}}{\mathrm{\pi }}}$