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# After Calculus I - PowerPoint PPT Presentation

After Calculus I…. Glenn Ledder University of Nebraska-Lincoln [email protected] 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…

Glenn Ledder

Funded by the National Science Foundation

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)

• Calculus I + a second course

• Five credits each

• Biologists want

• Probability distributions

• Dynamical systems

• Biochemists want

• Statistics

• Chemical Kinetics

• The second course should NOT be Calculus II.

• The second course should NOT be Calculus II.

• Instead: Mathematical Methods for Biology and Medicine

• Calculus (≈5%)

• Models and Data (≈25%)

• Probability (≈30%)

• Dynamical Systems (≈40%)

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

CALCULUSthe definite integral

• Area under y=f(x) is

• Accumulation of F over time is

• Aggregation of F in space is

CALCULUSthe partial derivative

• For fixed y, let F(x)=f(x;y).

• Gradient of f(x,y) with respect to x is

MODELS AND DATAmathematical models

Equations

Independent

Variable(s)

Dependent

Variable(s)

Narrow View

MODELS AND DATAmathematical models

Equations

Independent

Variable(s)

Dependent

Variable(s)

Parameters

Behavior

Narrow View

(see Ledder, PRIMUS, Feb 2008)

• Histograms

• Population mean

• Population standard deviation

• Standard deviation for samples of size n

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

MODELS AND DATAconstructing models

• Empirical modeling

• Statistical modeling

• Trade-off between accuracy and complexity mediated by AICc

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

• Time is split between searching and feeding

S – food availability R(S) – overall feeding rate

a – search speed C – feeding rate while eating

• 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

------- = --------- · --------- · -------

• 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 – -------

MODELS AND DATAcharacterizing models

• What does each parameter mean?

• What behaviors are possible?

• How does the parameter space map to the behavior space?

MODELS AND DATAnondimensionalization and scaling

PROBABILITYdistributions

• Discrete distributions

• Distribution functions

• Mean and variance

• Emphasis on computer experiments

• (see Lock and Lock, PRIMUS, Feb 2008)

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

PROBABILITYdistributions

frequency

width

frequency

width

---------------

---------------

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

PROBABILITYindependence

• Identically-distributed

• 1 expt: mean μ, variance σ2, any type

• n expts: mean nμ, variance nσ2, →normal

PROBABILITYindependence

• Identically-distributed

• 1 expt: mean μ, variance σ2, any type

• n expts: mean nμ, variance nσ2, →normal

• Not identically-distributed

PROBABILITYconditional

DynamicalSystems1-variable

• Discrete

• Simulations

• Cobweb diagrams

• Stability

• Continuous

• Simulations

• Phase line

• Stability

DynamicalSystemsdiscrete multivariable

• Simulations

• Matrix form

• Linear algebra primer

• Dominant eigenvalue

• Eigenvector for dominant eigenvalue

• Long-term behavior (linear)

• Stable growth rate

• Stable age distribution

DynamicalSystemscontinuous multivariable

• Phase plane

• Nullclines

• Linear stability

• Nonlinear stability

• Limit cycles