(*^

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                              Estimation and Limits -- A Descriptive Look

The cell below estimates the length of the curve  y = f(x)  for  a <= x <= b  using the method described in the browser window.  It uses the curve  y = Sqrt[1 - x^2]  for  0 <= x <= 1  using  n = 10 as an example.  
;[s]
14:0,1;73,0;125,1;133,0;140,1;151,0;223,1;227,2;240,1;241,0;247,3;258,0;267,4;273,0;291,-1;
5:6,13,9,Times,0,12,0,0,0;5,13,9,Times,1,12,0,0,0;1,13,9,Times,1,12,65535,0,65535;1,13,9,Times,1,12,0,0,65535;1,13,9,Times,1,12,65535,0,0;
:[font = input; preserveAspect; ]
Clear[f, a, b, n]

f[x_] := Sqrt[1 - x^2]
a := 0.0
b := 1.0

n := 10

h := (b - a)/n
x[i_] := a + i h

Sum[Sqrt[h^2 + (f[x[i]] - f[x[i-1]])^2], {i, 1, n}]
;[s]
7:0,0;28,1;41,0;42,2;59,0;61,3;68,0;155,-1;
4:4,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,65535,0,65535;1,12,10,Courier,1,12,0,0,65535;1,12,10,Courier,1,12,65535,0,0;
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The next cells can help you answer questions about tennis.  Evaluate the next three cells now.  The first cell defines the sequences  L[n]  and  W[n].  It is not necessary to understand the details of these definitions to answer the questions in the browser window.
;[s]
7:0,0;60,1;96,0;134,2;138,0;145,2;150,0;266,-1;
3:4,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;2,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect; ]
Clear[W, L, T]

YouMake := 0.85
OppMake := 0.80

L[1] := 1 - YouMake
W[1] := YouMake * (1 - OppMake)
T[n_] := T[n] = 1 - L[n] - W[n]

L[n_] := L[n] = 
    T[n - 1] * (1 - YouMake) + L[n - 1]
    
W[n_] := W[n] = 
    T[n - 1] * YouMake * (1 - OppMake) + W[n - 1]
:[font = input; preserveAspect; ]
L[1]
W[1]
T[1]
:[font = input; preserveAspect; ]
L[2]
W[2]
T[2]
^*)