The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing, when the foot of the ladder is 4 m away from the wall ?
Sand is pouring from the pipe at the rate of 12 cm³/s. The falling sand forms a cone on a ground in such a way that the height of cone is always one-sixth of radius of the base. How fast is the height of sand cone increasing when the height is 4 cm ?
The total cost associated with provision of free mid-day meals to x students of a school in primary classes is given by C(x) = 0·005x³ – 0·02x² + 30x + 50. If the marginal cost is given by rate of change of total cost, write the marginal cost of food for 300 students.
The total expenditure (in ₹) required for providing the cheap edition of a book for poor and deserving students is given by R(x) = 3x² + 36x, where x is the number of set of books. If the marginal expenditure is defined as write the marginal expenditure required for 1200 such sets.
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.
The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupees) received from the sale of x units of a product is given by R(x) = 3x² + 36x + 5, find the marginal revenue, when x = 5.
If C = 0·003x³ + 0·02x² + 6x + 250 gives the amount of carbon pollution in air in an area on the entry of x number of vehicles, then find the marginal carbon pollution in the air, when 3 vehicles have entered in the area.
Find the equations of the normal to the curve y = 4x³ – 3x + 5 which are perpendicular to the line 9x – y + 5 = 0..
Show that the equation of normal at any point t on the curve x = 3 cos t – cos³t and y = 3 sin t – sin³t is 4(y cos2t – x sin³t) = 3 sin 4t
The equation of tangent at (2, 3) on the curve y² = ax³ + b is y = 4x – 5. Find the values of a and b.
Find the equations of the tangent and normal to the curve at the point .
For the curve y = 4x³ – 2x⁵, find all those points at which the tangent passes through the origin.
Find the equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point where x₁ = 2 and y₁ > 0.
Find the equations of the tangent and normal to the curve x = a sin³θ, y = b cos³θ at