Your task is to estimate how far an object traveled during t

Your task is to estimate how far an object traveled during the time interval 0 lessthanorequalto t lessthanorequalto 8, but you only have the following data about the velocity of the object. To get an idea of what the velocity function might look like, you pick up a black pen. plot the data points, and connect them by curves. Your sketch looks something like tax black curve in the graph below. Using the same data, you also decide to estimate how far the object traveled using a right endpoint Niemen sum. So, you sketch the curve again with a black pen. and draw rectangles whose height is determined by the velocity measurement at the right endpoint of each one-second interval. You decide to use a left endpoint Niemen sum to estimate the total displacement. So, [Right approximation you pick up a blue pen and draw rectangles whose height is determined by the velocity measurement at the left endpoint of each one -second interval. By using the left endpoint Niemen sum as an approximation. you are assuming that the actual velocity endpoint Niemen sum. find approximately how far the object is approximately constant on each one-second interval (or. Equivalently, that the actual traveled. Your answers must include the correct units. acceleration is approximately zero on each one-second Interval), and that the velocity Total displacement = and acceleration have discontinuous jumps every second. This assumption is probably Total distance traveled = incorrect because it is likely that the velocity and acceleration change continuously over time. However, you decide to use this approximation anyway since it seems like a reasonable approximation to the actual velocity given the limited amount of data. (A) Using the left endpoint Rican sum. find approximately how far the object traveled. Your answers must include the correct units. Total displacement = Total distance traveled =

Solution

Riemann left sum Time t 0 1 2 3 4 5 6 7 Velocity v -4 -1 -4 -2 -1 -4 1 2 |v*t| 4 1 8 6 4 20 6 14 63 Total distance travelled under left sum 63 Right sum Time t 1 2 3 4 5 6 7 8 Velocity v -1 -4 -2 -1 -4 1 2 4 |v*t| 1 8 6 4 20 6 14 32 91 Total distance travelled under left sum 91