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Unformatted text preview: Math021, week 7 Definition 7.1 When there is is a quantity changing over time and f ( t ) is that quantity at time t . f ( t ) is called the rate of change of the given quantity at time t . Example 7.2 In a chemical reaction chamber, the concentration of a certain reactant A after t seconds is [ A ]( t ) = 1 k ( t + c ) M. Here k and c are two fixed positive numbers. Show that the rate of change of the concentration of A is proportional to the square of the concentration of A . solution: The rate of change of the concentration of A (rate of reaction) is d dt [ A ] = d dt 1 k ( t + c ) =- 1 k ( t + c ) 2 =- k [ A ] 2 . Thus, the assertion is proved. Example 7.3 Water is pumped into a inverted conical vessel with semi-vertical angle π/ 6 at a rate of R m 3 s- 1 . Evaluate the rate at which water level in the vessel is raising at the moment when the water level is at H meters. solution: Let V ( t ) be the volume of water in the vessel after t seconds. h ( t ) is the water level in the vessel after t seconds. Then, V ( t ) = 1 3 h ( t ) π ( h ( t )tan π 6 ) 2 = π 9 h ( t ) 3 for all t. Taking derivatives on both sides of the equation yields V ( t ) = π 3 h ( t ) 2 h ( t ) . Suppose that after t seconds, the water level in the vessel is H meters. That is, h ( t ) = H . It is given that V ( t ) = R for all t . In particular, V ( t ) = R . We evaluate the previous equation at t and it yields R = π 3 H 2 h ( t ) 1 or h ( t ) = 3 R πH 2 . The water level is raising at a rate of 3 R πH 2 ms- 1 at the moment when the water level is H meters. Example 7.4 One is keeping an eye on a car which is moving in a straight line. At the moment when the car is cloest to the guy, the car is D meters away from the guys and it is moving at a speed of V meters per second. Evaluate the rate at which the guy turns his head at that moment.rate at which the guy turns his head at that moment....
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This note was uploaded on 02/13/2012 for the course MATH 021 taught by Professor Luxnu during the Fall '10 term at HKUST.
- Fall '10
- Rate Of Change