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After Calculus I…

After Calculus I…. Glenn Ledder University of Nebraska-Lincoln gledder@math.unl.edu. Funded by the National Science Foundation. The Status Quo. Biology majors. Biochemistry majors. Calculus I (5 credits) Calculus II (5 credits) No statistics No partial derivatives.

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After Calculus I…

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  1. After Calculus I… Glenn Ledder University of Nebraska-Lincoln gledder@math.unl.edu Funded by the National Science Foundation

  2. The Status Quo Biology majors Biochemistry majors Calculus I (5 credits) Calculus II (5 credits) No statistics No partial derivatives • Calculus I (5 credits) • Baby Stats (3 credits)

  3. Design Requirements • Calculus I + a second course • Five credits each • Biologists want • Probability distributions • Dynamical systems • Biochemists want • Statistics • Chemical Kinetics

  4. My “Brilliant” Insight • The second course should NOT be Calculus II.

  5. My “Brilliant” Insight • The second course should NOT be Calculus II. • Instead: Mathematical Methods for Biology and Medicine

  6. Overview • Calculus (≈5%) • Models and Data (≈25%) • Probability (≈30%) • Dynamical Systems (≈40%)

  7. CALCULUSthe derivative • Slope of y=f(x) is f´(x) • Rate of increase of f(t) is • Gradient of f(x) with respect to x is

  8. CALCULUSthe definite integral • Area under y=f(x) is • Accumulation of F over time is • Aggregation of F in space is

  9. CALCULUSthe partial derivative • For fixed y, let F(x)=f(x;y). • Gradient of f(x,y) with respect to x is

  10. MODELS AND DATAmathematical models Equations Independent Variable(s) Dependent Variable(s) Narrow View

  11. MODELS AND DATAmathematical models Equations Independent Variable(s) Dependent Variable(s) Parameters Behavior Narrow View Broad View (see Ledder, PRIMUS, Feb 2008)

  12. MODELS AND DATAdescriptive statistics • Histograms • Population mean • Population standard deviation • Standard deviation for samples of size n

  13. MODELS AND DATAfitting parameters to data • Linear least squares • For y=b+mx, set X=x-x̄, Y=y-ȳ • Minimize • Nonlinear least squares • Minimize • Solve numerically

  14. MODELS AND DATAconstructing models • Empirical modeling • Statistical modeling • Trade-off between accuracy and complexity mediated by AICc

  15. MODELS AND DATAconstructing models • Empirical modeling • Statistical modeling • Trade-off between accuracy and complexity mediated by AICc • Mechanistic modeling • Absolute and relative rates of change • Dimensional reasoning

  16. Example: resource consumption

  17. Example: resource consumption • Time is split between searching and feeding S – food availability R(S) – overall feeding rate a – search speed C – feeding rate while eating

  18. Example: resource consumption • Time is split between searching and feeding S – food availability R(S) – overall feeding rate a – search speed C – feeding rate while eating food total t search t total t space search t food space ------- = --------- · --------- · -------

  19. Example: resource consumption • Time is split between searching and feeding S – food availability R(S) – overall feeding rate a – search speed C – feeding rate while eating food total t search t total t space search t food space ------- = --------- · --------- · ------- search t total t feed t total t --------- = 1 – -------

  20. MODELS AND DATAcharacterizing models • What does each parameter mean? • What behaviors are possible? • How does the parameter space map to the behavior space?

  21. MODELS AND DATAnondimensionalization and scaling

  22. PROBABILITYdistributions • Discrete distributions • Distribution functions • Mean and variance • Emphasis on computer experiments • (see Lock and Lock, PRIMUS, Feb 2008)

  23. PROBABILITYdistributions • Discrete distributions • Distribution functions • Mean and variance • Emphasis on computer experiments • (see Lock and Lock, PRIMUS, Feb 2008) • Continuous distributions • Visualize with histograms • Probability = Area

  24. PROBABILITYdistributions frequency width frequency width --------------- --------------- y = frequency/width means area stays fixed at 1.

  25. PROBABILITYindependence • Identically-distributed • 1 expt: mean μ, variance σ2, any type • n expts: mean nμ, variance nσ2, →normal

  26. PROBABILITYindependence • Identically-distributed • 1 expt: mean μ, variance σ2, any type • n expts: mean nμ, variance nσ2, →normal • Not identically-distributed

  27. PROBABILITYconditional

  28. DynamicalSystems1-variable • Discrete • Simulations • Cobweb diagrams • Stability • Continuous • Simulations • Phase line • Stability

  29. DynamicalSystemsdiscrete multivariable • Simulations • Matrix form • Linear algebra primer • Dominant eigenvalue • Eigenvector for dominant eigenvalue • Long-term behavior (linear) • Stable growth rate • Stable age distribution

  30. DynamicalSystemscontinuous multivariable • Phase plane • Nullclines • Linear stability • Nonlinear stability • Limit cycles

  31. For more information: gledder@math.unl.edu

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