Section 1.2Mathematical Models

AP CalculusSeptember 4, 2008CASA

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Modeling Using Mathematics

Linear Polynomial Power Rational Algebraic Transcendental

Trigonometric Exponential Logarithmic

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Linear

Very common Very easy Slope (m) is a rate of

change Something per

Something

Vertical Intercept (b) is a starting point

bmxxf )(

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Linear: Heights vs. Weight

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Does Height and Weight have a Linear Relationship?

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Using LSR to find the Equation

We often use “Least Squares Regression” to find an equation for line that best represents the data.

It is considered the “best fit” line because it minimizes the differences between the actual data and the predicted line.

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Find the LSR

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How closely to they match?

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How closely do they match?

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Advantages & Limitations

The linear model allows us to make predictions about “appropriate” weight of a player, given their height.

The predictions don’t work well outside the range of the data.

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Polynomials

Model the path of object pulled by gravity.

Equations can be found using data and “QuadReg”, or “CubicReg”, or “QuartReg”

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Polynomials

“QuadReg” produces an equation in the form ax2+bx+c (2nd degree poly.)

“CubicReg” produces and equation in the form ax3+bx2+cx+d (3rd degree poly.)

“QuartReg” produces an equation in the form ax4+bx3+cx2+dx+e (4th degree poly.)

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Power Functions

Can be classified as “odd” or “even”

,...5,4,3,2,1,)( nxxf n

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Root Functions

“even” versions exist only in first quadrant ,...5,4,3,2,1,)( 1 nxxf n

,...5,4,3,2,1,)( nxxf n

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Rational Functions

Functions P and Q are both polynomials

)(

)()(

xQ

xPxf

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Trigonometric

Great for modeling periodic motion

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Assignment

Section 1.2, 1-19 odd