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?

For what value of m,The function f(x) = mx + c, is decreasing for x \(\in\) R.

Show that f(x) = (x-1) e^x+1 is an increasing function for x > 0.

For the function y=x³, if x=5 and \(\Delta \)x=0.01,find \(\Delta \)y

f(x)=9x²+12x+2

The side of an equilateral triangle is increasing at the rate of 5 cm/sec. At what rate its area increasing when the side of the triangle is 10 cm.

The volume of a cube increasing at the rate of 9 cm³/sec. How fast is the surface area increasing when the length of an edge is 10 cm.

Find the slope and tangent and normal to the curve \(x^2+2y+y^2=0\ at \ (-1,2).\)

Find the value of a if tangent to curve y = x²-ax + 7 is parallel to the line 2x - y + 9 = 0 at (- 1, 1).

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

An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long?

\(f(x)=x^{ 2 }-x+1\) is neither  increasing nor decreasing strictly on (-1, 1)

Find the equations of the tangent  and normal to the curve x = 1-\(cos\theta ,y-=\theta -sin\theta \) at \(\theta =\frac { \pi }{ 4 } \)

Show that the height of the cylinder, open at the top, of given surface area and greatest volume is equal to the radius of its base.

Show that a cylinder of given volume open at the top has maximum total surface provided it height is equal to radius of its base.

A stone is dropped into a quiet lake and waves move in circles at a speed of 5cm per second.At the instant when the radius of the circular wave is 8cm,how fast is the enclosed area increasing?

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

At what point of the ellipse 16x²+9y²=400,does the ordinate decrease at the same rate at which the abscissa increases?

\(\sqrt{80}\)

A right circular cylinder inscribed in a given cone.Show that the curved surface area of cylinder is maximum when diameter of cylinder is equal to radius of base of cone.

Find the intervals in which \(f(x)-\sin3x-\cos3x,0 is strictly increasing or strictly decreasing.

The sum of surface areas of a sphere and a cuboid with sides \(\frac{x}{3},x\) and 2x is constant. Show that the sum of their volumes is minimum if x is equal to the three times the radius of shpere.