上和與下和
Author: 藍邦偉  老師

移動四個藍色的點決定函數圖形，紅色兩點A,B決定範圍。

U

U

U

L

L

L

用幾何直觀說明上和與下和

P_4 = (1.6996540911660076, 6.909497185017182)
P_3 = (0.9382658154853522, 6.866602268880302)
P_2 = (0.506126523882818, 7.638710759344145)
P_1 = (0.1851087644066498, 2.5771106551922887)
u = Vector((x(Corner[1.0]), 0.0), (x(Corner[3.0]), 0.0))
f_1 = Polynomial({P_1, P_2, P_3, P_4})
v = Vector((0.0, y(Corner[1.0])), (0.0, y(Corner[3.0])))
A = Point(u)
B = Point(u)
A_1: (x(A), f_1(x(A)))
B_1: (x(B), f_1(x(B)))
n = 4 (slider from 3 to 5, step 1)
a = Segment(A, A_1)
b = Segment(B, B_1)
u_1 = UpperSum(f_1, x(A), x(B), n)
C = (0.12749019219297858, -6.25924206900502)
C_1: C + (0.0, 1.0)
C_2: C + (0.0, 2.0)
C_3: C + (0.0, 3.0)
D: C + (0.7, 0.0)
D_1: D + (0.0, 1.0)
D_2: D + (0.0, 2.0)
D_3: D + (0.0, 3.0)
E: D + (0.7, 0.0)
F: E + (0.7, 0.0)
G: F + (0.7, 0.0)
E_1: E + (0.0, 1.0)
F_1: F + (0.0, 1.0)
G_1: G + (0.0, 1.0)
E_2: E + (0.0, 2.0)
F_2: F + (0.0, 2.0)
G_2: G + (0.0, 2.0)
E_3: E + (0.0, 3.0)
F_3: F + (0.0, 3.0)
G_3: G + (0.0, 3.0)
h = Segment(C, C_3)
k_1 = Segment(D_3, D)
l = Segment(E_3, E)
m = Segment(F_3, F)
文字1: "" + n
u_2 = UpperSum(f_1, x(A), x(B), 2.0 * n)
c = false
o = false
文字5: "U_{" + n + "}=" + u_1
u_3 = UpperSum(f_1, x(A), x(B), 4.0 * n)
p = false
u_4 = UpperSum(f_1, x(A), x(B), n + 10.0)
l_1 = LowerSum(f_1, x(A), x(B), n)
r = false
l_2 = LowerSum(f_1, x(A), x(B), 2.0 * n)
l_3 = LowerSum(f_1, x(A), x(B), 4.0 * n)
l_4 = LowerSum(f_1, x(A), x(B), n + 3.0)
文字9: "L_{" + n + "}=" + l_1
s = false
t = false
area = Integral(f_1, x(A), x(B))
文字2: "" + (2.0 * n)
文字3: "" + (4.0 * n)
g = Segment(F, C)
j = Segment(F_1, C_1)
i = Segment(C_2, F_2)
d = Segment(C_3, F_3)
文字6: "U_{" + (2.0 * n) + "}=" + u_2
文字10: "L_{" + (2.0 * n) + "}=" + l_2
文字7: "U_{" + (4.0 * n) + "}=" + u_3
文字11: "L_{" + (4.0 * n) + "}=" + l_3
文字8: "" + n
文字13: "" + (4.0 * n)
文字12: "" + (2.0 * n)