1 / 13

Functions & Models

Understand, predict, and transform phenomena using analytical tools to create reliable models. Explore behaviors over time, growth patterns, periodic behavior, and more. Study populations like diseases, chemicals, and learning capacities.

ajanet
Download Presentation

Functions & Models

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. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Functions & Models What’s in a name?

  2. Model • Studying an analytically measurable phenomenon • Want to understand it well enough to use, predicting the future and/or past in order to make reasoned decisions • Re-interpret/transform phenomenon using analytical tools to a simpler and “computable” model that captures enough of the phenomenon’s characteristics from which to make reliable decisions

  3. For Our Course . . . we will almost always assume that the phenomenon we are studying can be thought of as a population ( wide interpretation ). • A disease in a well defined region • The concentration of a chemical compound in a closed environment • The learning capacity of children • Amount of radioactive substance in a specified region • A species of butterfly on a tropical island • The portion of a specified group of people who are directly adversely affected by a specific political policy

  4. Characteristics of the Phenomenon/Population to be studied • In general, how does it “behave” over time • Does it reach a maximum/largest value/size and/or a minimum/smallest value/size? If so, when? • As “time” progresses, is the population growing, shrinking, or remaining constant? • How is the population growing/shrinking?

  5. Characteristics, cont. • When will the population reach some specified size? • Will the population grow or shrink without bound (population explosion/extinction)? • Does the population exhibit periodic behavior (“repeats itself” )? • Can the population be characterized as being “the same” as another (known) population?

  6. Analytical Tools Used • Computer simulations • Time series analysis • Statistical representations • Stochastic representations • Partial Differential Equations Basis for all of these approaches is the notion of function ! ! !

  7. Function • Well-defined rule that, when given some initial component “state” values (variables), returns a single/unique value. • In the “models” context this return value (value ofthe function) represents the overall “state” of the population/phenomenon. • For our course we are interested only in functions that require only a single input value (functionsofasinglevariable).

  8. Representations of Functions • Contextually – sentence(s) that describes the essence of the rule • Graphically - pie charts, line charts, bar charts, tables as seen in newspapers, tv, magazines, etc. • Symbolically – mathematical equations, i.e. y=mx+b None of these representations are inherently better than the others. In fact the representation that we have seen the most in recent math/science classes (symbolic) is the least frequently used and hardest to correctly find (if not impossible).

  9. Functional Notation • The symbol P(t) represents the state of the population P at the “time” t. • There is NOTHING sacred about the choice of letters, neither in the “name” of the population (in this case P) nor in the single “input” value or variable (in this case t). • Our text often uses x rather than t as the variable, and this changes NOTHING in terms of the actual function • P(t) = 2t + 7.03448 and P(x) = 2x + 7.03448 represent theexactsamefunction

  10. Functional Notation, cont. • In earlier experiences we were given function equations like y = 2x + 1 and the associated homework problems of the form x = 3, y = (fill in the green box, 7 is the correct answer) With functional notation we rewrite the function as f(x) = 2x + 1 and the homework problem as f(3) = 7.

  11. Functional Notation, cont. • The expression f() means that we should replace every instance of the function’s variable with , NO MATTER WHAT  IS!!! • If f(x) = 3x + x2 then f() = 3* + 2, f(-3.9887) = 3*(-3.9887) + (-3.9887)2, f(1+h) = 3(1+h) + (1+h)2, f(2x) = 3*(2x) + (2x)2, f(-x) = 3*(-x) + (-x)2, f(x+h) = 3*(x+h) + (x+h)2 NOTICE THE USE OF THE PARENTHESES !!!

  12. Domain and Range • The values that a function’s variable cantake is called the domain of the function. Sometimes it is easier to describe those values that the function’s variable cannottake. • The domain of a function “restricts” the set of useable variable values • Sometimes restricted by the symbolic representation ( no zeroes in the denominator, no negatives under even roots, etc. ) • Sometimes restricted by the context of the model the function is being used in (sides of a rectangle cannot be negative in length, even though the symbolic function for area allows negatives, area = base * height ).

  13. Domain & Range, cont. • We will learn how to compute the domain of functions as an important aspect of creating models. • The set of all possible values of a function is called its range (or in more formal contexts its co-domain). We will study issues related to the range of a function as we develop specific families of functions.

More Related