Mathematics 136 -- Calculus 2

Discussion 1 -- Integrals and Net Change; Another Application

September 13, 2016

Background

Suppose we apply the Fundamental Theorem, part II to an integral of the form ∫ab f '(x) dx, where the integrand is the derivative of another function f(x). Then f(x) will play the role of the antiderivative, and we have

∫ab f '(x) dx = f(b) - f(a)

We will call this the net change of f(x) over the interval from a to b. For instance if f(t) represents the height of a balloon above the ground as a function of time (see question 4 below), then f '(t) = v(t) is the balloon's vertical velocity, and
∫ab f '(t) dt = f(b) - f(a)

gives the net change in the height of the balloon from t = a to t = b.

Discussion Questions

  1. If w'(t) represents the rate of growth of a child in pounds per year, explain in words what the value of the integral
    ∫48 w'(t) dt

    represents. (No calculations -- just give a one sentence description of what the result would mean, in real world terms.)
  2. The number n(t) of honeybees in a population starts at n(0) = 200 bees and increases at a rate of n'(t) bees per week from the start of the counting until t = 10 weeks. Write an expression for the number of bees in the population at the end of the 10 weeks using a definite integral (and anything else you need).
  3. If f '(x) represents the slope of a trail at x miles from the start of the trail, what does the value of
    ∫26 f '(x) dx

    represent in real world terms? (You can assume the trail follows a straight line path if you like. Also, don't worry about trying to take the curvature of the earth into account!)
  4. 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. We want to use the graph of the velocity to try to estimate the total change in height over the trip using the information in the graph.
    1. Over what time intervals was the balloon rising? Over what intervals was it falling? At what time was the greatest altitude achieved on this trip?
    2. What apparently happened at t = 40? (Think about how a hot-air balloon works.)
    3. What apparently happened right before t = 60?
    4. Did the balloon end up higher, lower, or at the same height it started? Explain carefully how can you tell using the value of a definite integral. Give enough details to convince a skeptical calculus teacher(!)

Assignment

Group write-ups due in class on Wednesday, September 14.