A Note dy dx is positive if y increases as x increases and is negative if y decreases as x increases. Example The length x of a rectangle is decreasing at the rate of cm/minute and the width y is increasing at the rate of 2cm/minute. When x =10cm and y = 6cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle. Solution Since the length x is decreasing and the width y is increasing with respect to time, we have cm/min dx dt = − and cm/min dy dt = (a) The perimeter P of a rectangle is given by P = ( x + y ) Therefore P dt = dx dy = −+ = − cm/min (b) The area A of the rectangle is given by A = x .
y Therefore A dt = dx dy y ⋅ ⋅ = – ( ) + ( ) (as x = cm and y = cm) = cm /min Example The total cost C( x ) in Rupees, associated with the production of x units of an item is given by C( x ) = . x – . x + x + Find the marginal cost when units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. Solution Since marginal cost is the rate of change of total cost with respect to the output, we have Marginal cost (MC) = .
– . + = . Hence, the required marginal cost is ` . (nearly).
Example The total revenue in Rupees received from the sale of x units of a product is given by R( x ) = x + x + . Find the marginal revenue, when x = , where by marginal revenue we mean