- 84 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'After Calculus I ' - teague

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### After Calculus I…

Glenn Ledder

University of Nebraska-Lincoln

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

- Calculus I (5 credits)
- Baby Stats (3 credits)

Design Requirements

- Calculus I + a second course
- Five credits each
- Biologists want
- Probability distributions
- Dynamical systems

- Biochemists want
- Statistics
- Chemical Kinetics

My “Brilliant” Insight

- The second course should NOT be Calculus II.

My “Brilliant” Insight

- The second course should NOT be Calculus II.
- Instead: Mathematical Methods for Biology and Medicine

Overview

- 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

Broad View

(see Ledder, PRIMUS, Feb 2008)

MODELS AND DATAdescriptive statistics

- 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

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

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

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

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 – -------

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

Download Presentation

Connecting to Server..