ch7.4derivativesasratesofchange导数的改变率范例.ppt

7.4 Derivatives as rates of change The quantities x and y in a realtionship y=f(x) are often called variables,because x can stand for any number in the domain and y for any number in the range. When you draw the graph have a free choice of values of x,and then work out the values of y.So x is called the independent variable 自变量and y the dependent variable因变量. If are independent and dependent variables respectively ina functional relationship,then the derivative, measures the rate of change with respect to . If ,then Example 1: A sprinter in a womens 100-metre race reaches get top speed of 12 metres per second after she has run 36 metres. Up to that distance her speed is proportional to the square root of the distance she has run. Show that until she reaches full speed the rate of change of her speed with respect to distance is inversely proportional to her speed. Suppose that after she has run metres her speed is metres per second.You are told that,up to ,and also that Since can be written as . Example 2: A line of cars,each 5 metres long,is travelling along an open road at a steady speed of per hour. There is a recommended separation between each pair of cars given by the formula metres.At what speed should the cars travel to maximise the number of cars that the road can accommodate? A block,consisting of a cars length and the separation distance in front it,occupies metres of road,or For the largest number of blocks passing a checkpoint in an hour,the time (in hours) for a single block to pass the checkpoint should be as small as possible. Since the block is moving at speed per hour. Step 1 Since the speed must be positive,the domain is Step 2 The derivative is This shows that the cars flow best at a speed of just under 29 (Each block then takes approximately 0.00