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Graphical representations of mean values

This resource from the Institute for Math Learning at West Virginia University explores the concept of mean values, including the arithmetic, geometric, and harmonic means. It provides graphical representations and interactive examples to illustrate how these means can effectively summarize data. Learn how to calculate average speeds, interest rates, and apply concepts like the Mean Value Theorem in calculus. The resource is designed for students looking to deepen their understanding of mathematical means and their practical applications.

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Graphical representations of mean values

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  1. Graphical representations of mean values Mike Mays Institute for Math Learning West Virginia University

  2. Why means?

  3. Suppose you have a 79 on one test and an 87 on another, towards a midterm grade. B cutoff is 82. Do you have a B? A(a,b) = (a+b)/2 Arithmetic mean

  4. Suppose you earn 6% interest on a fund the first year, and 8% on the fund the second year. What is the average interest over the two year period? G(a,b) = Geometric mean

  5. Theorem: For a and b≥ 0, G(a,b) ≤ A(a,b), with equality iff a=b. h/a=b/h h2=a b h b a

  6. Interactive version http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AG.htm

  7. Morgantown is 120 miles from Slippery Rock. Suppose I drive 60mph on the way up and 40mph on the way back. What is my average speed for the trip? H(a,b) = 2ab/(a+b) Harmonic mean

  8. Fancier interactive version http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AGH.htm

  9. A mean is a symmetric function m(a,b) of two positive variables a and b satisfying the intermediacy property min(a,b) ≤ m(a,b) ≤ max(a,b) Homogeneity: m(a,b) = am(1,b/a)

  10. Examples A, G, H

  11. Algebraic approach 1: Powers

  12. Algebraic approach 2: Gini

  13. Graphical approach: Moskovitz Mf a b

  14. Fancier interactive version http://math.wvu.edu/~mays/AVdemo/deployed/Moskovitz.html

  15. Homogeneous Moskovitz means Mf is homogeneous, f (1)=1 iff f is multiplicative A1 G H x C 1/x

  16. Calculus: means and the MVT Mean Value Theorem for Integrals (special case): Suppose f(x) is continuous and strictly monotone on [a,b]. Then there is a unique c in (a,b) such that

  17. Special caseVs(a,b) from f(x) = xs • s → ∞ max • s = 1 A • s → 0 I • -1/2 (A+G)/2 • -1 L • -2 G • -3 (HG2)1/3 • s → -∞ min

  18. Numerical analysis 1: compounding

  19. Numerical analysis 2

  20. a0 = 2 b0 = 4 a1 = 2.8284 b1 = 3.3137 a2 = 3.06 b2 = 3.1825 a3 = 3.12 b3 = 3.1510 •  

  21. Thank you • math.wvu.edu/~mays/ • Beckenbach, E. F. and Bellman, R. Inequalities. New York: Springer-Verlag, 1983 • Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means and Their Inequalities. Dordrecht, Netherlands: Reidel, 1988.

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