Mathematics Integration Level: Misc Level

Prove that the curves xy - 4 and x^{2} + y^{2} - 8 touch each other.

Mathematics Integration Level: Misc Level

Prove that the curves y = x^{2} - 3x + 1 and x(y + 3) - 4 intersect at right angles at the poin (2, -1).

Mathematics Integration Level: Misc Level

Find the equations of the tangents to the curve y = x^{3} + 2x - 4 which are perpendicular to theline x + 14y + 3 = 0.

Mathematics Integration Level: Misc Level

Find the equations of all lines having slope 2 and that are tangents to the curve .

Mathematics Integration Level: Misc Level

Find the equations of the tangent to the curve x = Sin 3t, y = Cos 2t at .

Mathematics Integration Level: Misc Level

Find the equations of the tangent and the normal to the curve at (1,1).

Mathematics Integration Level: Misc Level

Find the equations of the tangent and the normal to the curve y = x^{4 }- 6x^{3}+13x^{2 }- 10x + 5 at (1,3).

Mathematics Integration Level: Misc Level

Find the points on the curve at which the tangents are.

(a) parallel to x-axis

(b) parallel to y-axis

Mathematics Integration Level: Misc Level

Determine the point of the curve y=3x^{2}-5 at which the tangent is perpendicular to the line whose slope is -1/3.

Mathematics Integration Level: Misc Level

If the tangent to the curve y=x3++ax+b at (1,-6) is parallel to the line 2x-2y+7=0, find a anb b.

Mathematics Integration Level: Misc Level

Show that the tangents to the curve at the points where x=2 and x=-2 are parallel.

Mathematics Integration Level: Misc Level

Find the slopes of the tangents at the three points where the curve cuts the x-axis.

Mathematics Integration Level: Misc Level

p(x)=0.03x^{3}+0.2x^{2}+15x+100 represents the air pollution in an industrial area due to smoke produced by x chimneys. Find the marginal value of air pollution when 3 chimneys are increased. Which value does this question indicate?

Mathematics Integration Level: Misc Level

Water is dripping out at a steady rate of 1 cu cm/sec through a tiny hole at the vertex of the conical vessel, whose axis is vertical. When the slant height of water in the vessel is 4 cm, find the rate of decrease of slant height, where the semi-vertical angle of the cone is

Mathematics Integration Level: Misc Level

A conical vessel whose height is 4 meters and the base radius 2 meters is being filled with water at the rate of 0.75 cubic meters per minute. Find the rate at which the level of the water is rising when the depth of water is 1.5 meters.