average rate of change

Question

A. Jacob wants to know the average rate of change of the car’s elevation with respect to time over the time interval [5, 7]. Use the piecewise function.

 

B. Identify an interval over which you will get an average rate of change of elevation with a sign opposite to the one you just found. Justify your answer by solving for that interval.

Answer 1

We are missing some key points of information over here with the following question, but I will provide an approach while having the above context.

A. When talking about the average rate of change of the car's elevation, I am assuming the graph of the problem displays the distance above ground level with respect to time, in which case the average rate of change would be the change in slope of the line that passes through the point at t=5 and t=7, not considering what is happening in between. The derivatives, or slopes of each minuscule time point, will average out within this period to the slope of the line as described above.

B. To find this interval, we can calculate which points will provide this result by utilizing the slope as a ratio between the change in time, which will be the interval, and the change in distance over that interval. This means that if we find that the slope of the line was 5 (or we can call the variable c), and the sample interval we are questioning is [7, 10], we know that the change in time is 3 since 10 - 7 = 3. As a result, the change in the distance will have to be 5 * 3 (or c * 3) = 15. If the change in distance is not 15, then that is not a valid interval. Note that to do this, we would have to manually determine the interval by calculating each change in distance at each interval.