Maglev Train

The table below gives the speed of a maglev train at different times:

Time in minutes [t]: 0 1 2 3 4 5 6 7 8
Speed in mph [ v(t) ]: 0 3 12 27 48 75 108 147 192

  1. For each time interval, you will find an upper bound and a lower bound for the number of miles the train logged during that time period. Note: you may make the following assumption: the speed of the train is increasing during the entire 8 minute period in question.

    Consider the time interval [0,2]. This is mathematical notation for the time interval between t=0 and t=2, i.e., for the interval 0 ≤ t ≤ 2. What is an upper bound for the number of miles the train covered during the time interval [0,2]?

    Although you do not know what the speed of the train is at all times (e.g. you are not told what the speed is when t=4.5), you have enough information to find an upper bound for how many miles were logged during [0,2]. Think about what you do know about the train's speed during this time interval, and about the length of the time interval.

    (a) An upper bound for the number of miles logged during [0,2] is:__________. (Warning: remember units! 1 minute is 1/60 hours) (b) Similarly, An upper bound for the number of miles logged during [1,3] is:__________. (c) and a lower bound for the number of miles logged during [1,3] is:__________.

    Explain in careful, precise, full sentences (which a classmate would understand) how you got these answers, and why they are valid:









    Explain where, and how, you used the assumption about increasing speed. Why would your answers not (necessarily) be correct were this assumption violated?














    (Make sure you understand everything before you continue. For example, an upper bound for the elapsed distance during the time interval [3,5] is 75*(2/60) = 2.5 miles)

  2. Suppose you were a consultant for your local county's train station and you had to estimate the total number of miles logged by the train during the time interval [1,3]. Could you do it? If the above was the only information you had, and if your job depended on making sure that your "error" is as small as possible, what would your estimate be? (the error is defined to be the difference between your estimate and the actual, correct, and often unknown answer). Given your estimate, can you guarantee that your error is "no larger than _____ (some value)"? Yes, you can!

    Your estimate of the number of miles logged during [1,3]: __________ and you can make a guarantee to the train station operators. Namely, for your estimate, although it may not be exactly equal to the correct total number of miles the train logged during [1,3], you can guarantee that your error is no larger than _________miles. That is, the difference between your estimate of __________miles and the number of miles the train actually logged between t=1 and t=3 is no more than __________ (your last two fill-in-the-blank answers are of course the same as your previous two).

    Now, explain (as always, this means in full, complete, accurate, and specific sentences which others can follow, and which are convincing!) how you chose your estimate, and prove that (i.e., give a convincing argument that) your error "is no more than _____ (whatever answer you gave)":








    Now fill in the following table, in which "lower bound" means "lower bound for miles logged this interval" and similarly for upper bound. "Your Estimate" is short for "Your estimate of how many miles the train logged (traveled) during this time interval", and "Your error-bound" is short for "you guarantee that your estimate for this interval is no more than _____ miles off"

    Time interval: Lower bound: Upper bound: Your estimate: Your Error-bound:
    [0,1]     
    [1,2]     
    [2,3]   0.325 miles0.125 miles
    [3,4]     
    [4,5]     
    [5,6]     
    [6,7]     
    [7,8]     

  3. After a little thinking, you should now also be able to fill in:

    Time interval: Lower bound: Upper bound: Your estimate: Your Guarantee:
    [0,8]     

  4. You can now fill in the second and third columns of the following table, leaving the fourth column blank for now (Note that the odometer reading at t=2 is equal to the distance logged during [0,1] plus the distance logged during [1,2], for example. Here, an "error-bound" of B means your are saying that your "approximate odometer reading" is no more than B miles off from the true odometer reading)

    Time:
    (in minutes)     		Approximate
    odometer reading:     		Error-bound: Exact
    odometer reading:
    t = 0    
    t = 1    
    t = 2    
    t = 3 0.475 miles  
    t = 4    
    t = 5    
    t = 6    
    t = 7    
    t = 8    

  5. We now reveal some new information: although it was not possible to know this from the limited data you were initially given, it can now be revealed that the train moved so that the odometer at time t (in minutes) had a value of (1/60)*t3 miles. Use this to fill in the last column in the above chart. Then verify that the "error-bounds" you promised in the third column live up to their words. Do they? How so?





    Using your knowledge of Differential Calculus, you know that since the distance from the train's base station at time t is given by (1/60)* t3, it must be that the velocity at time t, or v(t), is given by the formula __________ in miles per minute; thus, v(t) = __________ in miles per hour Verify this last equation by looking at the data in the initial page 1 chart of speeds for t=0 through t=8 -- check it!

  6. Label the y-axis (which is v(t)) below and sketch your v(t):
    
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         1       2       3       4       5       6       7       8 (in minutes)
    Now draw rectangles whose areas are equal to the "lower bound" for each interval [0,1], [1,2], [2,3], etc. Then draw (in another color if possible) rectangles with a greater height, whose area equals each of your "upper bound" answers. Can you find an area representation for your "error-bound" answers (we are talking about the "elapsed distance" table here, NOT the odometer readings table).

  7. Homework: For your final challenge, use your formula for v(t) to create a chart like the opening one, only with readings every half minute, and then fill out new charts like in parts II through IV with readings every half minute. Are your estimates better this time around? In other words, can you give "error-bounds" which are smaller for both part II and part IV?

    Without necessarily carrying out the exercise, what would happen if you used your v(t) to create an odometer chart with estimates every 0.25 miles. Would your odometer estimates (i.e., your estimates of "elapsed distance") be any better? Why or why not?

    Finally, explain how your estimates involving trains, speeds, odometers, etc, when looked at graphically, look like the areas of rectangles being used as an approximation for areas under the graphs of a function.





    Copyleft notice: Copyright © 2001, Harel Barzilai. Non-profit educational use explicitly allowed and encouraged.
    May not use for any other purpose without written permission. Inspired by Time and Speed activity in Calculus: An Active Approach with Projects.