chapter 1: functions & models 1.2 mathematical models: a catalog of essential functions
TRANSCRIPT
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Chapter 1: Functions & Models
1.2Mathematical 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
Mathematical Conclusions
Real-World Predictions
Formulate
Solve
Interpret
Test
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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 don’t 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 20⁰C and the temperature at a height of 1 km is 10⁰C, express the temperature T (in ⁰C) 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 3Use 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
012
21
1 ...)( axaxaxaxaxP nn
nn
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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
axxf )(
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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
nxxf )(
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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
nxxf1
)(
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Power Functions, case 3
• Where
• Called the reciprocal function
• Hyperbola with the coordinate axes as asymptotes
1)( xxf
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Rational Functions
• Ratio of two polynomials:
• Domain consists of all values such that Q(x) ≠ 0
)(
)()(
xQ
xPxf
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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 π
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Trig functions cont.• Sine and cosine are periodic functions• Period is 2π
• For all values of x, – Sin(x + 2π) = sin x– Cos(x + 2π) = 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 π
• For all values of x, tan (x + π) = tan x
• Don’t forget about the reciprocal functions
x
xxcos
sintan
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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
xaxf )(
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Logarithmic Functions
• Come in the form
• Base a is a positive constant
• Inverse functions of exponential functions
xxf alog)(
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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:
4
5
51)(
1
1)(
)(
5)(
tttu
x
xxh
xxg
xf x
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Homework
• P. 34
• 1-4, 9-17 odd, 21, 23, 25