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Biocalculus
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| Organization: | Florida State University |
| Department: | Department of Mathematics |
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College
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This course straddles the contentious divide between reform and tradition in calculus. It's goal is to mold biology majors into better scientists by enabling them to use Mathematica wisely, but it's 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 it's outlook is thoroughly modern. But it's style is deliberately old-fashioned.
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Lectures and course notebooks posted on the web
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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 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 these lectures encourage students to develop as much as possible of the first with as little exposure as possible to the second.
Topics:
- 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
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http://www.math.fsu.edu/~mm-g/biocalc.html
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