For the curve y = 5x – 2x³, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3.
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.
If the radius of sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.
The volume of a cube is increasing at the rate of 9 cm³/sec. How fast is the surface area increasing when the length of an edge is 10 cm.
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 ?
The volume of a sphere is increasing at the rate of 8 cm³/sec. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.
Find the equation of the normal at a point on the curve x² = 4y which passes through the point (1, 2). Also find the equation of the corresponding tangent.
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
Probability of solving specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem, independently, then find the probability that
(i) the problem is solved
(ii) exactly one of them solves the problem
Solve the differential equation : given that when x = 0, y =
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.
A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.
Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.