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THEOREM 1 Net Change as the Integral of a Rate

THEOREM 1 Net Change as the Integral of a Rate The net change in s ( t ) over an interval [ t 1 , t 2 ] is given by the integral. Water leaks from a tank at a rate of 2 + 5 t liters/hour , where t is the number of hours after 7 AM. How much water is lost between 9 and 11 AM?.

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THEOREM 1 Net Change as the Integral of a Rate

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  1. THEOREM 1 Net Change as the Integral of a Rate The net change in s (t) over an interval [t1, t2] is given by the integral

  2. Water leaks from a tank at a rate of 2 + 5tliters/hour, where t is the number of hours after 7 AM. How much water is lost between 9 and 11 AM?

  3. Traffic FlowThe number of cars per hour passing an observation point along a highway is called the traffic flow rate q (t) (in cars per hour). (a) Which quantity is represented by the integral

  4. Traffic FlowThe number of cars per hour passing an observation point along a highway is called the traffic flow rate q (t) (in cars per hour). (b) The flow rate is recorded at 15-min intervals between 7:00 and 9:00 AM. Estimate the number of cars using the highway during this 2-hour period.

  5. The Integral of Velocity Let s (t) be the position at time t of an object in linear motion. Then the object’s velocity is v (t) = s (t), and the integral of v (t) is equal to the net change in position or displacement over a time interval [t1, t2]: We must distinguish between displacement and distance traveled. If you travel 10 km and return to your starting point, your displacement is zero but your distance traveled is 20 km. To compute distance traveled rather than displacement, integrate the speed |v (t)|.

  6. THEOREM 2 The Integral of Velocity For an object in linear motion with velocity v (t),

  7. A particlehas velocityv (t) = t3 −10t2 + 24t m/s. Compute: (a) Displacement over [0, 6]

  8. A particlehas velocityv (t) = t3 −10t2 + 24t m/s. Compute: (b) Total distance traveled over [0, 6]

  9. Total versus Marginal Cost Consider the cost function C (x) of a manufacturer (the dollar cost of producing x units of a particular product or commodity). The derivativeC (x) is called the marginal cost. The cost of increasing production from a to b is the net change C (b) − C (a), which is equal to the integral of the marginal cost: The marginal cost of producing x computer chips (in units of 1000) is (a) Find the cost of increasing production from 10,000 to 15,000 chips.

  10. Total versus Marginal Cost Consider the cost function C (x) of a manufacturer (the dollar cost of producing x units of a particular product or commodity). The derivativeC (x) is called the marginal cost. The cost of increasing production from a to b is the net change C (b) − C (a), which is equal to the integral of the marginal cost: The marginal cost of producing x computer chips (in units of 1000) is (b) Determine the total cost of producing 15,000 chips, assuming that it costs $30,000 to set up the manufacturing run [that is, C (0) = 30,000].

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