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
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
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
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)
A, G, H
Mf
a
b
http://math.wvu.edu/~mays/AVdemo/deployed/Moskovitz.html
Mf is homogeneous, f (1)=1 iff f is multiplicative
A1
G
Hx
C1/x
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
a0 = 2b0 = 4
a1 = 2.8284b1 = 3.3137
a2 = 3.06b2 = 3.1825
a3 = 3.12b3 = 3.1510