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Section 1.2 Mathematical Models

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This section delves into mathematical modeling using various functions in AP Calculus, covering linear, polynomial, power, rational, algebraic, transcendental, trigonometric, exponential, and logarithmic functions. It highlights the importance of linear regression for data representation and predictions, particularly in the context of height and weight relationships. Advantages and limitations of linear modeling are discussed, along with polynomial modeling techniques like QuadReg and CubicReg. This comprehensive overview equips students with essential modeling strategies for calculus applications.

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Section 1.2 Mathematical Models

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  1. Section 1.2Mathematical Models AP Calculus September 4, 2008 CASA

  2. Modeling Using Mathematics • Linear • Polynomial • Power • Rational • Algebraic • Transcendental • Trigonometric • Exponential • Logarithmic 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 Calculus, Section 1.2

  4. Linear: Heights vs. Weight Calculus, Section 1.2

  5. Does Height and Weight have a Linear Relationship? 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. Calculus, Section 1.2

  7. Find the LSR Calculus, Section 1.2

  8. How closely to they match? Calculus, Section 1.2

  9. How closely do they match? 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. 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” 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.) Calculus, Section 1.2

  13. Power Functions • Can be classified as “odd” or “even” Calculus, Section 1.2

  14. Root Functions • “even” versions exist only in first quadrant Calculus, Section 1.2

  15. Rational Functions • Functions P and Q are both polynomials Calculus, Section 1.2

  16. Trigonometric • Great for modeling periodic motion Calculus, Section 1.2

  17. Assignment • Section 1.2, 1-19 odd Calculus, Section 1.2

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