BIOCALCULUS

Lectures on calculus for life science majors

Mike Mesterton-Gibbons
Professor
Department of Mathematics
Florida State University
Tallahassee, Florida 32306-4510
Phone: (850) 644-2580
Email: mesterto@math.fsu.edu

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Biocalculus introduces the fundamental ideas of calculus from the perspective of a biologist, i.e., it uses biological data to motivate and elucidate concepts that are essential for constructive use of Mathematica (or any other mathematical software) in solving biological problems. The approach is heuristic, but systematic. One can develop a great deal of mathematical maturity with remarkably little exposure to mathematical rigor, and in these lectures I encourage students to develop as much as possible of the first with as little exposure as possible to the second.

This is not, however, a no-brainer calculus for those who have drifted into biology under the illusion that it offers an escape from mathematics. On the contrary, calculus plays an increasingly central role in biology, and so its concepts must be mastered, despite -- or, rather, because of -- the widespread availability of powerful mathematical software. Every application of mathematics involves formulation (of a relevant problem), calculation (of requisite quantities) and interpretation (of results); and insofar as concepts and methods can be separated, formulation and interpretation require understanding of concepts, whereas calculation requires knowledge of method. Mathematica, in essence, is a magical black box for performing calculations. Although its graphical output may facilitate interpretation, Mathematica neither formulates nor interprets, and to that extent there is a greater need than ever before for biology majors to know the concepts thoroughly. This course addresses that need.

On the other hand, concepts and method are not so readily separated. Because all black boxes -- even magical black boxes -- are fallible, one cannot in general use mathematical software wisely unless one develops reliable instincts about whether it has truly yielded an answer to the problem one gave it (or meant to give it). Although using the software may refine such instincts, the only way to acquire them in the first instance is through extensive experience of solving problems without the software's help. In other words, there is still a need to enter the black box, which this course addresses as well.

An exception to the rule of fallibility, however, is that mathematical software is extraordinarily reliable for plotting simple graphs, in particular those of polynomials. I exploit that reliability by assuming at the outset that plotting graphs is a computer task, so that early exercises can introduce the software by requiring graphs to be drawn. In each set of exercises, an asterisked number indicates that a relevant Mathematica program can be downloaded from this site, and a number in bold refers to answers or hints at the end. All other problems can be solved by modifying an existing program or solution.

In sum, this course straddles the contentious divide between reform and tradition in calculus. Its goal is to mould biology majors into better scientists by enabling them to use Mathematica (or similar software) wisely, but its approach embodies a firm conviction that skill in using high technology for complex procedures requires skill in using low technology (e.g., pencil and paper) for simple procedures. So its outlook is thoroughly modern. But its style is deliberately old-fashioned.

Ordinary functions : a graphical perspective
Ordinary functions: an algebraic perspective.
Smoothness and concavity: a graphical perspective
Quotients, inverses and limits. Modelling photosynthesis
Ordinary sequences. Fibonacci's rapid rabbits
Discrete probability distributions. Sums of powers of integers
Function sequences. Compositions. The exponential and logarithm
Index functions. Area and signed area
From index function to ordinary function: ventricular recharge
Area as limit of a function sequence. D'Arcy Thompson's mini minnows
Arterial discharge: the area under a polynomial
From ventricular inflow to volume: integration
From ventricular volume to inflow: the derivative as growth rate
Smoothness and concavity: an algebraic perspective
Making joins smooth: the derivative of a piecewise-smooth function
Differential notation. The derivative of a sum or multiple
Sex allocation and the product rule
How flat must a flatworm be, not to have a heart? The fundamental theorem
Continuous probability distributions: the fundamental theorem again
Derivatives of compositions: the chain rule
Variation in rat pupil area. Implied distributions and integration by substitution
Properties of exponential and logarithm. The empirical basis of allometry
Periodic functions: models of rhythms in nature
Bivariate functions and their extrema: a graphical approach
More on bivariate functions: partial derivatives and integrals
The mean and median of a distribution
The variance. More on improper integrals
Symmetric distributions
Differential equations. The conceptual basis of allometry
Trigonometric function properties
The method of maximum likelihood

Download Course Materials

Click to download figures (mostly in Mathematica Notebooks, one or two in PDF files) or lectures. Lectures are all in PDF files, which you can view and/or print with Adobe Reader (free download); if you find that the page is upside down, select Rotate Clockwise twice from the View menu. Mathematica Notebooks can be viewed with Math Reader (also free download).

Lectures	Figures
Lecture 1	Figure 1 Figure 2 Figure 3 Figure 4 Exercise 4 Exercise 5 Exercise 6 Exercise 7
Lecture 2	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Exercise 2 Exercise 3 Exercise 7 Exercise 8
Lecture 3	Figure 1 Figure 2 Figure 3 Exercise 3
Lecture 4	Figure 1 Figure 2 Exercise 4 Exercise 5
Lecture 5	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Exercise 3 Exercise 5
Lecture 6	Figure 1
Lecture 7	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Exercise 2 Exercise 3 Exercise 5 Exercise 7 Exercise 8 Exercise 9 Exercise 11 Exercise 13
Lecture 8	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Exercise 1 Exercise 2
Lecture 9	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Exercise 2
Lecture 10	Figure 1 Figure 2 Figure 3 Figure 4 Exercise 3
Lecture 11	Figure 1 Figure 2 Figure 3 Exercise 1 Exercise 2
Lecture 12	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Exercise 4
Lecture 13	Figure 1 Exercise 4 Exercise 6
Lecture 14	Figure 1 Figure 2 Figure 3 Exercise 2 Exercise 6
Lecture 15	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Exercise 1 Exercise 10 Exercise 12 Exercise 13
Lecture 16	Figure 1 Figure 2 Figure 3 Figure 4
Lecture 17	Figure 1 Figure 2 Figure 3 Figure 4 All figures (PDF)
Lecture 18	Figure 1 Figure 2 Figure 3
Lecture 19	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5
Lecture 20	Figure 1 Figure 2
Lecture 21	Figure 1
Lecture 22	Figure 1 Figure 2 Figure 3 Figure 4
Lecture 23	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Exercise 4
Lecture 24	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10
Lecture 25	Figure 1 Figure 2
Lecture 26	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Exercise 7 Exercise 10
Lecture 27	Figure 1 Figure 2 Figure 3 Exercise 11
Lecture 28	Figure 1 Figure 2 Exercise 3
Lecture 29	Figure 1 Figure 2
Lecture 30	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6
Lecture 31	Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8