Greenwood press
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book-20600
RATES
99 The slope of the dotted line represents the average speed of the car from 12:34 PM to 1:57 PM , which is 79 miles per hour. The automobile has this rate at three other locations in this interval based on the equivalent slopes of the small thick lines (at the points denoted speeding). Besides finding the average rate as a means to describe varying speeds, it is possible to determine the instantaneous rate of an object using differential calcu- lus. If a total amount, such as distance or production levels, can be described as a function, then the rate at any moment can be determined by finding the deriv- ative of that function. Instead of finding the slope at the endpoints of an interval, a derivative is the slope of a line tangent to a curve at a particular point. The slope of the tangent line will describe the speed of the car at a specific moment in time. For example, in the above figure, a tangent line with a slope of 70 miles per hour is drawn on the curve at 1:34 PM , illustrating the speed of the car at that moment. In addition to automobile travel, the motion of falling objects shows variable rates. Since the earth pulls objects at a rate of 9.8 meters per second squared, falling objects are constantly accelerating. The position of a penny dropped off of a 400-meter-tall skyscraper can be represented by the function h = –4.9t 2 + 400, where h is the height of the penny above the ground in meters, and t is the time in seconds the penny is airborne. This function is a parabola. It will not have a constant slope, which means that the penny will not fall at the same rate towards the ground. However, the slope of the line tangent to the curve at any time, or the instantaneous rate, can be predicted by the derivative of this function, which is h ′ = –9.8t. This means that the penny will be falling at a rate of 9.8 meters per second after one second, 19.6 meters per second after two seconds, and so on. According to the position function, h = –4.9t 2 + 400, the penny will reach the ground at approximately t = 9 seconds, where h is equal to 0. According to the derivative of the position function, the velocity of the penny by the time it hit the ground would be h ′ = –9.8(9) = –88.2 meters per second, fast enough to fall straight through a person’s body. Hence, you are not likely to be permitted to drop objects from tall buildings! Human workforce productivity can have varying rates. In a factory, the work- ers may be less productive in the early morning because they are tired, and then reach an optimal work rate later in the morning when they are more awake. Later in the afternoon, they may become less productive again due to fatigue or bore- dom. Understanding the varying working rates of employees may help manage- ment determine an optimal time to take a break or to change work shifts. Know- ing the change in work rates would provide information to make smart decisions Download 1.81 Mb. Do'stlaringiz bilan baham: |
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