Abstract
Chapter 1 is a brief development of one-variable calculus. The opening section presents the concept of the parameter, which is essential in modeling and helpful in calculus. Differential calculus occupies Sections 1.2–1.6, with one section each on the derivative concept, derivative computation, local behavior and linear approximation, optimization, and related rates. The final three sections introduce the definite integral as the accumulation of change, present the properties of definite integrals, and calculate definite integrals using the fundamental theorem of calculus and the substitution technique. Examples with biological relevance include atmospheric carbon dioxide, organism growth, optimal foraging, the Deepwater Horizon oil spill, and population growth and demographics. The problem sets contain some modeling problems, including several on population demographics and two that explore explanations of optimal fetching behavior observed in dogs.
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Notes
- 1.
We’ll consider more general versions of the model in Examples 1.1.3–1.1.5.
- 2.
This notation means that the parameter m can be any real number.
- 3.
We use the word “illustrate” rather than “plot” or “graph” here because we can’t simultaneously plot all lines y = mx. Instead, we plot several representative examples together and infer the properties of the whole family.
- 4.
It is pronounced BYE-fur-ca-shun.
- 5.
See Problem 7.1.3.
- 6.
The mathematical epidemiologists Valerie Tweedle and Robert J. Smith? (The question mark in “Smith?” is part of the spelling of his name. I do not know whether this is to distinguish him from all the other Robert J. Smiths in the world or to disprove the popular belief that mathematicians are the most uninteresting people alive. I prefer not to ask.) claim that the most infectious “disease” is the social disease they call “Bieber fever,” whose basic reproductive number they estimate to be about 27 [2].
- 7.
Note that we are assuming the patient has been taking the drug for several days already. The results would be more complicated if our study began with the very first dose.
- 8.
This seems very strange. The ordering of two weightlifters who competed in the 2004 Olympics might have changed after the 2008 Olympics, even though the two performances being compared were already historical. This is not the only strange thing about Olympic weightlifting. The International Weightlifting Federation changed the weight classes in 1993 and again in 1998, so currently recognized world records are all for performances after 1997, in spite of obvious problems. For example, the pre-1993 world record for the 60 kg class is a full 5% larger than the currently accepted world record for the 62 kg class. An additional quirk deals with the rules for breaking ties, under which the world record in the 85 kg class was claimed by the silver medalist in the 2008 Olympics. Andrei Rybakou posted the score of 394 kg a few minutes before fellow competitor Lu Yong posted the same score. Rybakou got the world record because he was the first to get the score, but Yong was awarded the gold medal by virtue of being 0.28 kg smaller (about 0.5 pounds) than Rybakou at the official weigh-in. For the sake of completeness, we note that a “world record” score was also achieved in the 105 kg weight class by gold medalist Andrei Aramnau; the quotation marks are used because Aramnau’s score at 105 kg was less than the pre-1993 world record for the 100 kg class.
- 9.
This is the pattern as measured at Mauna Loa.
- 10.
Discrete changes occurring in continuous time are best dealt with in mathematical modeling, rather than mathematics per se. This topic is discussed in some detail in Section 2.2.
- 11.
The limit is a very subtle mathematical concept that was not rigorously defined until many years after the original development of calculus. The creators of calculus were able to manage without a rigorous definition of the limit, and so can we.
- 12.
The actual meaning of the distinct symbols dy and dt will be considered later. For now, the reader should think of \(\frac{dy} {dt}\) as a single symbol rather than a quotient.
- 13.
This is the reason for the assertion in Section 1.1 that understanding parameters is very helpful for understanding calculus.
- 14.
In practice, too small a value of h causes difficulties because of the way real numbers must be stored in computers. This is not a serious practical difficulty; we just have to avoid ridiculously small values of h.
- 15.
It seems silly to have the extra factor a in the denominator of the fraction, since it must be balanced by an extra factor a in the front. The point will be clear in the calculation that follows. Mathematics contains many instances where something that appears to make extra work turns out to be helpful by the end.
- 16.
Problem 1.3.31.
- 17.
Section 1.4.
- 18.
See Problem 1.3.32.
- 19.
In some contexts, it is important to have a partial derivative notation that is distinguishable from ordinary derivative notation.
- 20.
Don’t be intimidated by the word “theorem.” Mathematicians use the word to indicate a significant mathematical result that follows logically from a relatively simple set of mathematical assumptions. The student should think of a theorem as a statement that combines some quantitative or qualitative result with a set of conditions that are sufficient to ensure that the result is correct.
- 21.
In most cases where a decimal value is obtained by a calculator or computer from a formula that includes functions other than polynomials, the computed value is a numerical approximation. The point here is that the numerical approximation obtained by a computer is almost always more accurate than the linear approximation.
- 22.
See Example 1.6.3 for justification of these assumptions.
- 23.
See Problem 1.5.8.
- 24.
These experiments were simply to determine data points for the function n(x); as such they did not have anything to do with the independent measurement of the crows’ behavior.
- 25.
See Section 2.5 on mechanistic modeling.
- 26.
The parameter v is determined by the focal power of the cornea and the distance from the lens to the retina. If your focal length f 0 is smaller than v, then you need glasses.
- 27.
As of this writing, some of this discussion can be found in the Wikipedia entry for “Fetch (game).”
- 28.
While any discussion of dogs solving calculus problems is obviously tongue in cheek, it should not be at all surprising that animals can choose near-optimal strategies for finding objects quickly. The wild ancestors of today’s dogs needed to hunt efficiently to survive, so natural selection developed whatever behavior strategies today’s dogs now use to fetch balls thrown into water.
- 29.
This is enough to cover the entire state of Delaware to a depth of 1.2 cm. On the other hand, if the spill were spread across the entire Gulf of Mexico, it would only reach a depth of 0.05 mm. The enormity of the spill is a matter of perspective, and the environmental damage from the spill was highly dependent on concentration at any particular location.
- 30.
Indices have been omitted from the sum because we are thinking of an arbitrarily large number of subintervals rather than a fixed number.
- 31.
Fecundity is the expected reproductive capacity, usually given as a function of the age of the mother.
- 32.
The name Euler is pronounced “oiler.”
- 33.
Leonhard Euler (1707–1783) was one of the most prolific mathematicians of all time. He remained productive even after becoming blind in 1766. Alfred J. Lotka (1880–1949) did a lot of pioneering work in mathematical population dynamics, including (1.7.3).
- 34.
In general, definite integrals can be used to determine the total amount of stuff aggregated over two-dimensional and three-dimensional regions, even when the density of stuff needs to be given per unit area or per unit volume. These multiple integrals are beyond the scope of our treatment. Details can be found in any calculus book that includes multivariable calculus.
- 35.
Among the more subtle effects of global warming are changes in the development times for plants and animals. Some species have evolved to develop in conjunction with a host or food species. Different degree-day requirements and threshold temperatures mean that synchronized development of two species may be lost when the mean temperature changes.
- 36.
The formula from this problem could be used as part of a model, but it is not complete as it is. In any practical setting, the rates of photosynthesis, root uptake, and respiration would not be known functions of time; instead, they would be functions of the plant mass. The model equation would then have m on both sides and would be a problem to be solved rather than a quantity to be calculated.
- 37.
There is a subtle distinction between “total change in position” and “distance traveled” that matters if you travel both forward and backward. For simplicity, we assume that you only travel forward; however, we use the language that is still correct if backward travel is allowed. This is also why we use mileposts to measure distance rather than using the car’s odometer.
- 38.
Equation (1.7.4).
- 39.
The function N represents the standard normal distribution. See Section 3.6.
References
Alexander, RM. Optima for Animals. Princeton University Press, Princeton, NJ (1996)
Atlantic magazine. http://www.theatlantic.com/health/archive/2012/06/science-confirms-bieber-fever-is-more-contagious-than-the-measles/258460/.CitedSep2012
Belovsky GE. Diet optimization in a generalist herbivore: The moose. Theo. Pop. Bio., 14, 105–134 (1978)
CDC and the World Health Organization. History and epidemiology of global smallpox eradication. In Smallpox: Disease, Prevention, and Intervention. http://www.bt.cdc.gov/agent/smallpox/training/overview/.CitedSep2012
Heffernan JM and MJ Keeling. An in-host model of acute infection: Measles as a case study. Theo. Pop. Bio., 73, 134–147 (1978)
National Incident Command, Interagency Solutions Group, Flow Rate Technical Group. Assessment of Flow Rate Estimates for the Deepwater Horizon / Macondo Well Oil Spill. U.S. Department of the Interior (2011)
National Oceanic and Atmospheric Administration. Mauna Loa CO2 data. http://www.esrl.noaa.gov/gmd/ccgg/trends/#mlo_data.CitedSep2012
Pennings, TJ. Do dogs know calculus? College Math. J., 34, 178–182 (2003)
Perruchet P and J Gallego. Do dogs know related rates rather than optimization? College Math. J., 37, 16–18. (2006)
Stacey DA and MDE Fellowes. Temperature and the development rates of thrips: Evidence for a constraint on local adaptation? Eur. J. Entomol., 99, 399–404 (2002)
Zach R. Shell dropping: Decision-making and optimal foraging in Northwestern crows. Behaviour, 68, 106–117 (1979)
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Ledder, G. (2013). A Brief Summary of Calculus. In: Mathematics for the Life Sciences. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7276-6_1
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