Chapter 1: Functions

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Chapter 1 Functions Models

• 1.2
• Mathematical Models A Catalog of Essential
  Functions

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Mathematical Model

• A mathematical description of a real-world
  phenomenon
• Uses a function or an equation

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The modeling process
Real-World Problem
Mathematical Model
Formulate
Solve
Test
Real-World Predictions
Mathematical Conclusions
Interpret
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Stage One

• Formulate a mathematical model by identifying and
  naming the independent and dependent variables
• Make assumptions that simplify the phenomenon
  enough to make it mathematically tractable
• May need a graphical representation of the data

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Stage Two

• Apply the mathematics we know to the model to
  derive mathematical conclusions

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Stage Three

• Interpret the mathematical conclusions about the
  original real-world phenomenon by way of offering
  explanations or making predictions

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Stage Four

• Test our predictions against new real data
• If the predictions dont compare well, we revisit
  and revise

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Linear Models

• Linear functions
• The graph of the function is a line
• Use slope-intercept form of the equation of a
  line
• Grow at a constant rate

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Example 1

• (a) As dry air moves upward, it expands and
  cools. If the ground temperature is 20C and the
  temperature at a height of 1 km is 10C, express
  the temperature T (in C) as a function of the
  height h (in km), assuming that a linear model is
  appropriate.

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Example 1

• (b) Draw the graph of the function in part (a).
  What does the slope represent?

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Example 1

• (c) What is the temperature at a height of 2.5 km?

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Empirical Model

• Used if there is no physical law of principle to
  help us formulate a model
• Based entirely on collected data
• Use a curve that fits the data (it catches the
  basic trend of the data points)

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Example 2

• Table 1 on page 26 lists the average carbon
  dioxide level in the atmosphere, measured in
  parts per million at Mauna Loa Observatory from
  1980 to 2002. Use the data in Table 1 to find a
  model for the carbon dioxide level.

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Example 3

• Use the linear model given by C 1.55192t
  2734.55 to estimate the average CO2 level for
  1987 and to predict the level for the year 2010.
  According to this model, when will the CO2 level
  exceed 400 parts per million?

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Polynomials

• A function P is called a polynomial if
• Where n is a nonnegative integer
• a constants called coefficients of the
  polynomial
• Domain
• Degree of polynomial is n

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Example 4

• A ball is dropped from the upper observation deck
  of the CN Tower, 450 m above the ground, and its
  height h above the ground is recorded at 1-second
  intervals in Table 2 on pg 29. Find a model to
  fit the data and use the model to predict the
  time at which the ball hits the ground.

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Power Functions

• A function of the form
• Where a is a constant

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Power Functions, case 1

• Where and n is a positive integer
• The general shape depends on whether n is even or
  odd
• As n increases, the graph becomes flatter near 0
  and steeper when x 1

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Power Functions, case 2

• Where and n is a positive integer
• These are root functions
• If n is even, the domain is all positive numbers
• If n is odd, the domain is all real numbers

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Power Functions, case 3

• Where
• Called the reciprocal function
• Hyperbola with the coordinate axes as asymptotes

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Rational Functions

• Ratio of two polynomials
• Domain consists of all values such that Q(x) ? 0

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Algebraic Functions

• A function constructed using algebraic operations
  starting with polynomials
• Any rational function is automatically an
  algebraic function
• Graphs can be a variety of shapes

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Trigonometric Functions

• Radian measure always used unless otherwise
  indicated
• Domain for sine and cosine curves are all real
  numbers
• Range is closed interval -1,1
• The zeroes of the sine function occur at the
  integer multiples of p

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Trig functions cont.

• Sine and cosine are periodic functions
• Period is 2p
• For all values of x,
• Sin(x 2p) sin x
• Cos(x 2p) cos x
• Use sine and cosine functions to model repetitive
  phenomena
• Tides, vibrating springs, sound waves

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Trig functions cont.

• Remember
• Tangent function has period of p
• For all values of x, tan (x p) tan x
• Dont forget about the reciprocal functions

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Exponential Functions

• Functions of the form
• The base a is a positive constant
• Used to model natural phenomena
• Population growth, radioactive decay

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Logarithmic Functions

• Come in the form
• Base a is a positive constant
• Inverse functions of exponential functions

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Transcendental Functions

• Not algebraic
• Includes trigonometric, inverse trigonometric,
  exponential, and logarithmic functions
• Comes back in chapter 11 (if you take calculus BC
  in college!)

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Example 5

• Classify the following functions as one of the
  types of functions

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Homework

• P. 34
• 1-4, 9-17 odd, 21, 23, 25