after calculus i
Download
Skip this Video
Download Presentation
After Calculus I…

Loading in 2 Seconds...

play fullscreen
1 / 31

After Calculus I… - PowerPoint PPT Presentation


  • 84 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
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

After Calculus I…

Glenn Ledder

University of Nebraska-Lincoln

[email protected]

Funded by the National Science Foundation

the status quo
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
Design Requirements
  • Calculus I + a second course
  • Five credits each
  • Biologists want
    • Probability distributions
    • Dynamical systems
  • Biochemists want
    • Statistics
    • Chemical Kinetics
my brilliant insight
My “Brilliant” Insight
  • The second course should NOT be Calculus II.
my brilliant insight5
My “Brilliant” Insight
  • The second course should NOT be Calculus II.
  • Instead: Mathematical Methods for Biology and Medicine
overview
Overview
  • Calculus (≈5%)
  • Models and Data (≈25%)
  • Probability (≈30%)
  • Dynamical Systems (≈40%)
calculus the derivative
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
calculus the definite integral
CALCULUSthe definite integral
  • Area under y=f(x) is
  • Accumulation of F over time is
  • Aggregation of F in space is
calculus the partial derivative
CALCULUSthe partial derivative
  • For fixed y, let F(x)=f(x;y).
  • Gradient of f(x,y) with respect to x is
models and data mathematical models
MODELS AND DATAmathematical models

Equations

Independent

Variable(s)

Dependent

Variable(s)

Narrow View

models and data mathematical models11
MODELS AND DATAmathematical models

Equations

Independent

Variable(s)

Dependent

Variable(s)

Parameters

Behavior

Narrow View

Broad View

(see Ledder, PRIMUS, Feb 2008)

models and data descriptive statistics
MODELS AND DATAdescriptive statistics
  • Histograms
  • Population mean
  • Population standard deviation
  • Standard deviation for samples of size n
models and data fitting parameters to data
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 data constructing models
MODELS AND DATAconstructing models
  • Empirical modeling
  • Statistical modeling
    • Trade-off between accuracy and complexity mediated by AICc
models and data constructing models15
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 consumption17
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 consumption18
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 consumption19
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 data characterizing models
MODELS AND DATAcharacterizing models
  • What does each parameter mean?
  • What behaviors are possible?
  • How does the parameter space map to the behavior space?
probability distributions
PROBABILITYdistributions
  • Discrete distributions
    • Distribution functions
    • Mean and variance
    • Emphasis on computer experiments
      • (see Lock and Lock, PRIMUS, Feb 2008)
probability distributions23
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
probability distributions24
PROBABILITYdistributions

frequency

width

frequency

width

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

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

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

probability independence
PROBABILITYindependence
  • Identically-distributed
    • 1 expt: mean μ, variance σ2, any type
    • n expts: mean nμ, variance nσ2, →normal
probability independence26
PROBABILITYindependence
  • Identically-distributed
    • 1 expt: mean μ, variance σ2, any type
    • n expts: mean nμ, variance nσ2, →normal
  • Not identically-distributed
dynamical systems 1 variable
DynamicalSystems1-variable
  • Discrete
    • Simulations
    • Cobweb diagrams
    • Stability
  • Continuous
    • Simulations
    • Phase line
    • Stability
slide29
DynamicalSystemsdiscrete multivariable
  • Simulations
  • Matrix form
  • Linear algebra primer
    • Dominant eigenvalue
    • Eigenvector for dominant eigenvalue
  • Long-term behavior (linear)
    • Stable growth rate
    • Stable age distribution
slide30
DynamicalSystemscontinuous multivariable
  • Phase plane
  • Nullclines
  • Linear stability
  • Nonlinear stability
  • Limit cycles
ad