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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 3 2x 1 13 A balloon which always remains spherical has a variable diameter 2x 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 x in Rupe 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 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 r at r 6 cm is A 10T B 12 C 8T D 11T 152 MATHEMATICS 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 D 126 6 3 Increasing and Decreasing Functions In this section we will use differentiation to find out whether a function is increasing or decreasing or none Values left to origin f x x R x 13x 26x 15 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 X 2 3 2 1 1 2 4 9 4 Rationalised 2023 24 1 1 4 X 0 0 as we move from left to right the height of the graph decreases f x height of graph at x EX 2 10 Values right to origin f x x X 0 10 132 0 1 4 1 9 4 2 4 as we move from left to right the height of the graph increases

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