Mathematics 136, section 1 -- Calculus 2

Discussion 1 -- Total Change of a function, or

A Balloon Trip with the Montgolfier Brothers

February 3, 2020

Goals

As an application of the FTC, we want to consider the relationship between the rate of change of a function, and the total change of the function over some interval of values of the independent variable.

The Story

The Montgolfier brothers, Joseph and Etienne, were pioneers in hot-air ballooning in the late 1700's in France. If they had possessed appropriate instruments, they might have left a record of one of their early experiments as shown in the graph below. The graph shows the vertical velocity v of their balloon as a function of time t. (This means v = dy/dt, the rate of change of the height of the balloon, as a function of t, positive values for v mean the balloon is going up; negative values, down. Their acceleration at time t is a = dv/dt = d2 y/dt2.)

Discussion Questions

  1. Over what intervals was the acceleration positive, negative. Was it ever zero? What happened at t = 40? (Hint: How does a hot-air balloon work?)
  2. The balloon started out at ground level. Let's say that is y(0) = 0. At what time was the greatest altitude, y achieved? (Be careful!) Call that time t = b.
  3. What definite integral would compute the total change in height y(b) - y(0)? (Hint: Think FTC, Part I.)
  4. We don't have a formula for v(t), but we do have the graph above. How could you estimate the greatest altitude -- the total change in y over the interval t = 0 to t = a? Recall, rate x time = distance. For instance, what would happen if you subdivided the interval of t-values [0,a] into n equal smaller periods 0 < t1 < t2 < ... < tn-1 < tn = a and "sampled" the v(t) function at the start of each interval. Explain why the total distance traveled over the jth interval is approximately v(tj-1) Delta t, j = 1, ... , n. Use this to get an estimate of the total distance traveled over the whole interval. (Give both a numerical value and a general formula or recipe for what you are doing.)
  5. What happened at the end of this flight? Did the balloon end up caught in a tree (or at the top of a hill), in a valley, or at the same height it started? How can you tell?

Assignment

Group write-ups due in class no later than Wednesday, February 5.