The amount of pollution content added in air in a city due to X diesel vehicles is given by P(x)=0.005x³+0.02x² +30x.Find the marginal increase   in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question?

A balloon which always remains spherical has a variable diameter \({3\over2}(2x+1)\). Find the rate of change of its volume with respect to x.

Find the point on the curve y=x²-4x+5,where tangent to the curve is parallel to the x-axis.

f(x)=-(x-1)²+10

f(x)=9x²+12x+2

Use differentials find the approximate value of \((127)^{1/3}\) .

Use differentials find the approximate value of \((0.037)^{1/2}\)

Find an angle \(\theta\) which increases twice as fast as its sine.

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

Find the equation of the normal to the curve \(ay^{ 2 }=x^{ 3 }\) at \((am^{ 2 },am^{ 3 })\)

Prove that the function f(x) = tan x - 4x is strictly decreasing on \(\left( -\frac { \pi }{ 3 } ,\frac { \pi }{ 3 } \right) \)

Gas is escaping from a spherical balloon at the rate of 900cm³ /sec. How fast is the surface area,radius of balloon shrinking when the radius of the balloon is 30cm?

A given rectangular area is to be fenced off, in a field whose length lies along the river. Show that the least length will be required when length of the field is twice its breadth.