Chapter 9A SBM Modeling Our World

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Chapter 9A SBMModeling Our World

• Functions The Building Blocks of Mathematical
  Models

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Functions

• A function describes how a dependent variable
  changes with respect to one or more independent
  variables. When there are only 2 variables, we
  often summarize them as an order pair (x,y).
• (independent variable, dependent variable)
• We say that the dependent variable is a function
  of the independent variable and y is the
  dependent variable. We write the function as
• y f (x)

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Domain and Range

• The domain of a function is the set of values
  that both make sense and are of interest for the
  independent variable.
• The range of a function consists of the values of
  the dependent variable that correspond to the
  values in the domain.

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Creating Using Graphs of Functions

• Step 1 Identify the independent/dependent
  variables in the function.
• Step 2 Identify the domain range of the
  function. (Use this info. To choose
  the scale and labels for the axes).
• Step 3 Make a graph using the given data.
• Step 4 Before accepting any predictions of the
  model, be sure to evaluate the data
  assumptions from which the model was built.

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Some basic questions

• What is the purpose of a mathematical model?
• To represent something real and help us to
  understand it.
• Name the three basic ways to represent a
  function.
• Data table, graph, and equation
• What is meant by the domain of a function?
• The set of values that both make sense and are of
  interest for the independent variable

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More basic questions

• What is meant by the range of a function?
• The values of the dependent variable that
  correspond to the values in the domain
• What type of function repeats a particular
  pattern over and over?
• A periodic function

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Describing a Function

• In words, describe a function involving the
  altitude, over time, of a ball thrown into the
  air.
• Possible answer At time 0, the ball is at
  initial height 2 ft as time goes on, the
  ball's height increases to a maximum of 66 ft at
  time 2 sec then the ball's height decreases
  again.

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More on Functions

• In words, describe a function involving the
  elevation, over time, of a piece of paper
  floating on water disturbed by regular waves.
• Possible answer At time 0, the paper is at
  initial height 0. As time goes on, the paper's
  height increases to a maximimum of 4 in. then
  the paper's height decreases again to 0, then
  drops to -4 in., then rises to 0 and repeats the
  cycle.

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Dependent Independent

• Use the notation (independent variable, dependent
  variable) to characterize a function involving
  the altitude, over time, of a ball thrown into
  the air.
• (time in seconds, altitude in feet)

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Graphing a Function

• Assuming that the complete function includes all
  other data values that make sense, draw the
  graph of the function.

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The Graph
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Questions from the Table?

• What is the benefit of representing a function
  with a graph?
• A graph is easy to interpret and consolidates a
  great deal of information.
• Use a graph of the function to find the day on
  which the concentration is 6.6 ppm.
• Day 8

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More Questions from the Table?

• What are the domain and range of this function?
• Domain all t gt 0
• Range all values c such that 5 lt c
• Use words to describe the function.
• The concentration starts at 5 ppm and steadily
  rises over the course of 14 days.
• Use a graph of the function to find the
  concentration on day 6.
• 6.2 ppm

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Homework

• 9A 2, 4 8, 12 15, 17 19, 21 - 24