Section 1.2 Mathematical Models
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Transcript of Section 1.2 Mathematical Models
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Section 1.2Mathematical Models
AP CalculusSeptember 4, 2008CASA
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Calculus, Section 1.2 2
Modeling Using Mathematics
Linear Polynomial Power Rational Algebraic Transcendental
Trigonometric Exponential Logarithmic
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Calculus, Section 1.2 3
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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Calculus, Section 1.2 4
Linear: Heights vs. Weight
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Calculus, Section 1.2 5
Does Height and Weight have a Linear Relationship?
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Calculus, Section 1.2 6
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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Calculus, Section 1.2 7
Find the LSR
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Calculus, Section 1.2 8
How closely to they match?
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Calculus, Section 1.2 9
How closely do they match?
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Calculus, Section 1.2 10
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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Calculus, Section 1.2 11
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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Calculus, Section 1.2 12
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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Calculus, Section 1.2 13
Power Functions
Can be classified as “odd” or “even”
,...5,4,3,2,1,)( nxxf n
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Calculus, Section 1.2 14
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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Calculus, Section 1.2 15
Rational Functions
Functions P and Q are both polynomials
)(
)()(
xQ
xPxf
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Calculus, Section 1.2 16
Trigonometric
Great for modeling periodic motion
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Calculus, Section 1.2 17
Assignment
Section 1.2, 1-19 odd