I. Exponential increase
- Recall the exponential formula (e.g. see Moore’s Law and Space Exploration: New Insights and Next Steps, equation (1) on p. 15. Draw the general shape of the curve.
- What does increasing a do to the curve?
- Suppose a = b = 1. In the year 1961 what is y?
- In the year 1962 what is y?
- In the year 1963 what is y?
- In the year 1960 what is y?
- What is the doubling time for y? (This builds on the preceding questions)
- Suppose b = 2. What are the values for y in 1960, 1961, 1962, and 1963?What is the doubling time for y now?
- Suppose b = 3. What is the doubling time for y?
- What does the parameter b represent?
- Suppose the 2 was a 3, and b = 1. How does y change from one year to the next?
- 2 (or 3) is called the "base" of the exponential function. What concept or meaning does the base provide?
II. Exponential decay
- Suppose the number of satellites launched in 2019 is N. As these satellites fail, N will decrease. Sketch* a curve showing N(t), on the x and y axes where y = N(t) and x = t (alternatively you could use a vertical N axis and a horizontal t axis and it would be the same thing). *By "sketch" is meant a hand drawn curve showing the correct shape but no numbers. For example, is the curve increasing, decreasing, increasing then decreasing, decreasing then increasing, etc.? Does it accelerate upward, accelerate downward, go up but slower and slower, down but slower and slower, some combination of these, etc.?
- Consider https://en.wikipedia.org/wiki/Exponential_decay (more specifically https://en.wikipedia.org/w/index.php?title=Exponential_decay&oldid=962680362), and the equation therein, dN/dt= -lamba * N. If we are interested in understanding the reliability of satellites launched in 2019, what does N represent?
- Consider N and dN/dt. Which of these represents a quantity of something and which represents the rate at which the quantity changes?
- Suppose N starts out at some largish number. Then we wait until N decreases to half of its starting value. So N is different from what it used to be. Is dN/dt different from what it used to be? How different?
- How can we still use dN/dt when the underlying process N(t) does not change smoothly, but is actually constant for a short time, then jumps to a new value? For example, the number of satellites jumps up when one is launched, jumps down when one fails, and is constant the rest of the time. A radioactive element with a half life decays in steps: every time an atom disintegrates the number of atoms decreases by one, and the rest of the time the number is constant. How and why would we even use the dN/dt concept? Hint: see https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-1/v/secant-lines-and-average-rate-of-change. Advanced follow-on question: If dN/dt is a simplifying model of a more complex underlying physical process, what makes it a reasonable model (or not reasonable)?
- What does t represent?
- What is the relationship between Δt and dt?
- Which is bigger, Δt or dt?
- What is the relationship between ΔN/Δt and dN/dt?
- How many points on the curve do you need to determine ΔN/Δt?
- How many points on the curve do you need to determine dN/dt?
- Consider ΔN/Δt. For a smooth curve, if you make Δt smaller, what happens to ΔN?
- What is the name for Δt if you make it so small that it is almost zero?
- How long is dt ? For reference, see e.g. https://en.wikipedia.org/wiki/Differential_calculus. Also see Khan Academy, https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-1-new/ab-2-1/v/derivative-as-a-concept.
- How big is dN ?
- How big is dN/dt in words?
- Please explain the meaning of the concept dN/dt.
- Recall the equation from earlier, dN/dt= -lamba * N, that is, dN/dt = -λ*N. If satellites were more reliable this year than last year, would dN/dt this year be smaller or larger than last year?
- Recall the equation from earlier, dN/dt = -λ * N. Suppose someone was surprised to find out that satellites launched in 2019 were more reliable than they guessed before analyzing the data. Then what about the value of lambda would also surprise them?
- Suppose the negative sign in the equation was a plus sign instead. How would the equation behave differently?
- Supposing we wait until half the satellites have failed. How does this affect the value of dN/dt ?
- How does it affect number of satellites failing each year?
- How does it affect the number of satellites failing each hour?
- How does it affect the probability that a satellite will fail while you are figuring out the answer to this question?
- Suppose we wish to "solve" the equation, by which let us say we mean to write another equation that models the same phenomena but without using the term dN/dt. The general approach used is i________, not algebra. (Fill in the blank.)
- The solution equation is N(t) = N0 *e^(-λ * t), which introduces a couple of new symbols. What is N0 ?
- What is e?
- Let Tau=1/lambda. What is the meaning of Tau? See the Wikipedia page.
- If we wanted to determine the halving time, we would use 2 instead of e, which would require changing the value of lambda to a different value. This would give us an equation like N(t) = N0*2^(-bt). If b was doubled what would that do to the halving time?
- For N(t) = N0*2^(-bt), what is the halving time?
- Suppose we rewrote N(t) = N0*2^(-bt) using another parameter, a, giving N(t) = N0*2^(-t/a). What is the halving time in terms of a?
- Getting back to Tau, algebraically solve N(t) = N0 *e^(-λ * t) in terms of lambda. What do you get?
- Convert your answer to use Tau instead of lambda. What is the formula for Tau?
- What fraction of satellites launched in 2014 have failed? What fraction have not failed? For now, use arbitrary numbers that you make up, if you wish.
- What is the mean lifespan of satellites launched in 2014?
- What is the mean lifespan of satellites launched in each year?
- Does the mean lifespan increase over time?
- Does it increase exponentially over time? Linearly? According to Wright's law?
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