Graphical representations of mean values

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# Graphical representations of mean values - PowerPoint PPT Presentation

Graphical representations of mean values. Mike Mays Institute for Math Learning West Virginia University. Why means?. 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.

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### Graphical representations of mean values

Mike Mays

Institute for Math Learning

West Virginia University

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

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

h/a=b/h

h2=a b

h

b

a

Interactive version

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

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

Fancier interactive version

http://jacobi.math.wvu.edu/~mays/AVdemo/Labs/AGH.htm

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)

Examples

A, G, H

Fancier interactive version

http://math.wvu.edu/~mays/AVdemo/deployed/Moskovitz.html

Homogeneous Moskovitz means

Mf is homogeneous, f (1)=1 iff f is multiplicative

A1

G

H x

C 1/x

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

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

a0 = 2 b0 = 4

a1 = 2.8284 b1 = 3.3137

a2 = 3.06 b2 = 3.1825

a3 = 3.12 b3 = 3.1510

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.