The graph below shows the total distance s in miles traveled

The graph below shows the total distance s, in miles, traveled by a bicyclist after t hours. Estimate the bicyclist\'s average velocity over the following time intervals. Over [0, 1], the average velocity is approximately Over [1, 2,5|, the average velocity is approximately Over [2,5,3,5], the average velocity is approximately Estimate the bicyclist\'s instantaneous velocity at the following times (As a guideline, I wilt tell you that all of the values below should be multiples of 5.) At t = 1/2 hour, the instantaneous velocity is approximately At t = 2 hours, the instantaneous velocity is approximately At t = 3 hours, the instantaneous velocity is approximately Estimate the bicyclist\'s maximum velocity and the specific time at which it occurs. The maximum velocity is approximately which occurs when t

Solution

a) average velocity = distance on y axis / coresponding intervel length

   if the interval length is (0,1) the correponding distance on y axis = y2-y1 / x2 -x1

                                                                                                         = 15mi - 0/ 1 hr -0 = 15 mi/hr

   simillarly for the interval length is (1,2.5) average velocity = 20mi - 15mi / 2.5 - 1 = 5 mi / 1.5 hr = 3.33 mi /hr

simillarly for the interval length is (2.5,3.5) average velocity = 40mi - 20mi / 3.5 - 2.5 = 20 mi / 1 hr = 20 mi /hr

b) instantaneous velocity is the solpe of the curve at the point of time

at t = 1/2 hr , we can easily notice that the solpe of the curve is unchanged from t = 0 to t= 1/2

             instantaneous velocity = slope of the curve at particular instance = height / base = 10/0.5 = 20 mi / hr

at t = 2hrs , the slope of the curve at this point is 0 since the bicycle do not cover any distance at this instance of time   instantaneous velocity = 0

at t = 3hrs , from 2.5 to 3.5 the slope remain constant , so the velovity between 2.5 and 3.5 will be the instataneous velocity         = 40 - 20 / 3.5 - 2.5 = 20 / 1 = 20 mi /hr

c) From the instataneous velocity measurements we had measured velocitys at different instants and from above values it is clear that the bycycle velocity is maximum at 3.0 hr

     which is = 20 mi / hr