which its volume is increasing with the radius when the later is 10 cm. 10. 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 2cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall? 11. A particle moves along the curve 6y=x +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate. 12. The radius of an air bubble is increasing at the rate of cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm? 13. A balloon, which always remains spherical, has a variable diameter 3 (2x+ (2x+1). Find the rate of change of its volume with respect to x. 14. Sand is pouring from a pipe at the rate of 12 cm/s. 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? 15. The total cost C (x) in Rupees associated with the production of x units of an item is given by C(x)=0.007x-0.003x + 15x + 4000. Find the marginal cost when 17 units are produced. 16. The total revenue in Rupees received from the sale of x units of a product is given by R(x) = 13x+26x + 15. Find the marginal revenue when x = 7. Choose the correct answer for questions 17 and 18. 17. The rate of change of the area of a circle with respect to its radius rat r = 6 cm is (A) 10 (B) 12 (C) 8 (D) 11 152 MATHEMATICS Rationalised 2023-24 18. The total revenue in Rupees received from the sale of x units of a product is given by R(x)=3x+36x + 5. The marginal revenue, when x = 15 is (A) 116 (B) 96 (C) 90 6.3 Increasing and Decreasing Functions (D) 126 In this section, we will use differentiation to find out whether a function is increasing or decreasing or none. Consider the function f given by f(x) = x, x = R. The graph of this function is a parabola as given in Fig 6.1. Values left to origin X f(x) = x -2 4 Values right to origin 2321 12 0 X f(x) = x 0 0 9 1 1 4 4 1 1 f(x) height of graph at .x, 1 1 4 X'