多峰性関数
- 極大値や極小値が複数存在する関数
問.
図は以下の関数に関して 0 ≤ x ≤ 2 なる x の変域（定義域）におけるグラフである。
$$
y = \sqrt{(\sqrt{x} × sin(7x) × sin(3x) × cos(7x) + \frac{\sqrt{x}}{4})^2}
$$
import numpy as np
import matplotlib.pyplot as plt
# x の範囲
x = np.linspace(0, 2, 4000)
# 関数
y = np.sqrt(
(
np.sqrt(x)
* np.sin(7 * x)
* np.sin(3 * x)
* np.cos(7 * x)
+ np.sqrt(x) / 4
) ** 2
)
# グラフ描画
plt.figure(figsize=(8, 5))
plt.plot(x, y)
plt.title("complicated function")
plt.xlabel("x")
plt.ylabel("y")
plt.xlim(0, 2)
plt.grid(True)
plt.show()
このグラフを参考にして数値演算に関するプログラミングの問いに答えよ。
1)
関数$$y = \sqrt{(\sqrt{x} × sin(7x) × sin(3x) × cos(7x) + \frac{\sqrt{x}}{4})^2}$$
の 0 < x < 100 なる x の変域に関して、y の極大値と極小値の個数をできるだけ正しく求めるプログラムを作成し、プログラムと実行結果を提出せよ。
- 式変形
\begin{align}
y &= \sqrt{ \left( \sqrt{x}\sin(7x)\sin(3x)\cos(7x) +\frac{\sqrt{x}}4 \right)^2 }
\\ &= \left| \sqrt{x}\sin(7x)\sin(3x)\cos(7x) +\frac{\sqrt{x}}4 \right|
\\ &= \sqrt{x} \left| \sin(7x)\sin(3x)\cos(7x) +\frac14 \right| \qquad (x\ge0)
\end{align}
import math
# ---------------------------------------
# g(x)
# ---------------------------------------
def g(x):
return (
math.sin(7 * x)
* math.sin(3 * x)
* math.cos(7 * x)
+ 0.25
)
# ---------------------------------------
# f(x) = sqrt(x) * |g(x)|
# ---------------------------------------
def f(x):
return math.sqrt(x) * abs(g(x))
# ---------------------------------------
# g(x) = 0 の根を二分法で求める
# ---------------------------------------
def binary_root(left, right):
gl = g(left)
for _ in range(100):
mid = (left + right) / 2.0
gm = g(mid)
if abs(gm) < 1e-14:
return mid
if gl * gm <= 0:
right = mid
else:
left = mid
gl = gm
return (left + right) / 2.0
# ---------------------------------------
# f'(x)
#
# g(x) != 0 の範囲で数値微分する
# ---------------------------------------
def df(x):
h = 1e-6
return (
f(x + h) - f(x - h)
) / (2.0 * h)
# ---------------------------------------
# f'(x) = 0 の根を二分法で求める
# ---------------------------------------
def binary_derivative_root(left, right):
fl = df(left)
for _ in range(100):
mid = (left + right) / 2.0
fm = df(mid)
if abs(fm) < 1e-10:
return mid
if fl * fm <= 0:
right = mid
else:
left = mid
fl = fm
return (left + right) / 2.0
# ---------------------------------------
# メイン
# ---------------------------------------
def main():
# 探索範囲
start = 1e-6
end = 100.0
# 探索間隔
dx = 0.001
maxima = []
minima = []
# =====================================
# ① g(x) = 0 の点を探す
#
# 絶対値 |g(x)| による尖点なので
# 極小値の候補になる
# =====================================
x = start
g_prev = g(x)
while x < end:
next_x = min(x + dx, end)
g_curr = g(next_x)
if g_prev * g_curr < 0:
root = binary_root(x, next_x)
# 本当に g(root) = 0 か確認
if abs(g(root)) < 1e-10:
# 重複防止
if not minima or abs(root - minima[-1]) > 1e-4:
minima.append(root)
x = next_x
g_prev = g_curr
# =====================================
# ② f'(x) = 0 の点を探す
#
# g(x) = 0 の近くは避ける
# =====================================
x = start
prev_df = df(x)
while x < end:
next_x = min(x + dx, end)
curr_df = df(next_x)
# 微分値の符号が
# + -> - なら極大
if prev_df > 0 and curr_df < 0:
root = binary_derivative_root(x, next_x)
# g(x)=0 の近傍では |g(x)| による尖点が生じる。
# 本来は谷であるが、差分近似による数値誤差によって
# 極大点と誤判定される可能性があるため、
# g(root) ≈ 0 の点を除外して極大点のみを抽出する。
# なお、通常の滑らかな極小点は符号判定が安定しており問題になりにくい。
# (そもそもif prev_df > 0 and curr_df < 0:判定にかからない)
if abs(g(root)) > 1e-5:
if not maxima or abs(root - maxima[-1]) > 1e-4:
maxima.append(root)
# - -> + なら極小
elif prev_df < 0 and curr_df > 0:
root = binary_derivative_root(x, next_x)
# g(x)=0 の点ではないことを確認
if abs(g(root)) > 1e-5:
if not minima or abs(root - minima[-1]) > 1e-4:
minima.append(root)
x = next_x
prev_df = curr_df
# =====================================
# 結果表示
# =====================================
print("================================")
print("0 < x < 100")
print("================================")
print()
print("【極大値】")
print("個数 =", len(maxima))
for i, x in enumerate(maxima, 1):
print(
f"{i:3d}: "
f"x = {x:.10f}, "
f"f(x) = {f(x):.10f}"
)
print()
print("【極小値】")
print("個数 =", len(minima))
for i, x in enumerate(minima, 1):
print(
f"{i:3d}: "
f"x = {x:.10f}, "
f"f(x) = {f(x):.10f}"
)
if __name__ == "__main__":
main()
実行結果
================================
0 < x < 100
================================
【極大値】
個数 = 430
1: x = 0.1653407669, f(x) = 0.1728509662
2: x = 0.3483998544, f(x) = 0.1042525941
3: x = 0.5658279090, f(x) = 0.5602949497
4: x = 0.7724022068, f(x) = 0.0975598097
5: x = 0.9651283118, f(x) = 0.3426651000
6: x = 1.2580040029, f(x) = 0.5936497430
7: x = 1.4644604159, f(x) = 0.2700745103
8: x = 1.6801109821, f(x) = 0.9371035699
9: x = 1.8868122899, f(x) = 0.0406725392
10: x = 2.0570199118, f(x) = 0.3987012340
11: x = 2.3717100425, f(x) = 0.9408624301
12: x = 2.5827104486, f(x) = 0.3969222417
13: x = 2.7973350557, f(x) = 1.1321246316
14: x = 3.1425397056, f(x) = 0.4431468590
15: x = 3.4882680136, f(x) = 1.2642871341
16: x = 3.7012708120, f(x) = 0.4751744865
17: x = 3.9140731118, f(x) = 1.2088050628
18: x = 4.2285450944, f(x) = 0.5717271104
19: x = 4.3965402603, f(x) = 0.0620866120
20: x = 4.6060765718, f(x) = 1.5520569513
21: x = 4.8197387203, f(x) = 0.4900143717
22: x = 5.0290750214, f(x) = 1.1876364569
23: x = 5.3227823689, f(x) = 0.8055388056
24: x = 5.5118154353, f(x) = 0.2606882243
25: x = 5.7244739473, f(x) = 1.7871996254
26: x = 5.9377489164, f(x) = 0.4312467193
27: x = 6.1404329617, f(x) = 1.0876783014
28: x = 6.4272111938, f(x) = 1.1127903128
29: x = 6.6289348548, f(x) = 0.4556562840
30: x = 6.8430715273, f(x) = 1.9540445035
31: x = 7.0547808224, f(x) = 0.2949285489
32: x = 7.2448264507, f(x) = 0.9398049384
33: x = 7.5386055743, f(x) = 1.4541006069
34: x = 7.7470153910, f(x) = 0.6212516356
35: x = 7.9615588079, f(x) = 2.0405802406
36: x = 8.1699075789, f(x) = 0.0846353679
37: x = 8.3389947377, f(x) = 0.8029063546
38: x = 8.6537250009, f(x) = 1.7974842757
39: x = 8.8655635665, f(x) = 0.7354256656
40: x = 9.0795898336, f(x) = 2.0398694144
41: x = 9.4250937333, f(x) = 0.7675014596
42: x = 9.7707633423, f(x) = 2.1160902226
43: x = 9.9842502198, f(x) = 0.7804474934
44: x = 10.1966516416, f(x) = 1.9511645280
45: x = 10.5112677218, f(x) = 0.9014414160
46: x = 10.6796960342, f(x) = 0.0967660269
47: x = 10.8887999219, f(x) = 2.3864238457
48: x = 11.1027902122, f(x) = 0.7437336986
49: x = 11.3118226559, f(x) = 1.7812275749
50: x = 11.6055827679, f(x) = 1.1894934044
51: x = 11.7949332888, f(x) = 0.3813500209
52: x = 12.0073246085, f(x) = 2.5884380294
53: x = 12.2208492642, f(x) = 0.6186825854
54: x = 12.4232875743, f(x) = 1.5471341608
55: x = 12.7100901622, f(x) = 1.5648912322
56: x = 12.9120481878, f(x) = 0.6359380951
57: x = 13.1260006124, f(x) = 2.7063334513
58: x = 13.3379194884, f(x) = 0.4055273918
59: x = 13.5277670785, f(x) = 1.2842288420
60: x = 13.8215534218, f(x) = 1.9689407994
61: x = 14.0301348488, f(x) = 0.8360508462
62: x = 14.2445394606, f(x) = 2.7295005146
63: x = 14.4530811380, f(x) = 0.1125702886
64: x = 14.6220104659, f(x) = 1.0632002199
65: x = 14.9367238429, f(x) = 2.3615379300
66: x = 15.1486922271, f(x) = 0.9613339480
67: x = 15.3626059659, f(x) = 2.6534133860
68: x = 15.7081527357, f(x) = 0.9908348122
69: x = 16.0537985107, f(x) = 2.7124486238
70: x = 16.2673886806, f(x) = 0.9961980922
71: x = 16.4796930731, f(x) = 2.4805166139
72: x = 16.7943363130, f(x) = 1.1394449335
73: x = 16.9628736857, f(x) = 0.1219532292
74: x = 17.1718612417, f(x) = 2.9968725684
75: x = 17.3859384277, f(x) = 0.9306814459
76: x = 17.5948830863, f(x) = 2.2215100271
77: x = 17.8886532778, f(x) = 1.4767914033
78: x = 18.0780980256, f(x) = 0.4721203403
79: x = 18.2904051603, f(x) = 3.1946837895
80: x = 18.5040073132, f(x) = 0.7612906499
81: x = 18.7063645262, f(x) = 1.8984797920
82: x = 18.9931716230, f(x) = 1.9129784100
83: x = 19.1952086350, f(x) = 0.7753790108
84: x = 19.4090955892, f(x) = 3.2909326733
85: x = 19.6210880280, f(x) = 0.4918559405
86: x = 19.8108630414, f(x) = 1.5541118267
87: x = 20.1046495920, f(x) = 2.3746721865
88: x = 20.3132950402, f(x) = 1.0059868066
89: x = 20.5276454116, f(x) = 3.2766463280
90: x = 20.7362618168, f(x) = 0.1348369288
91: x = 20.9051281781, f(x) = 1.2712728817
92: x = 21.2198330587, f(x) = 2.8147457591
93: x = 21.4318541009, f(x) = 1.1434491780
94: x = 21.6457203358, f(x) = 3.1496286238
95: x = 21.9912839101, f(x) = 1.1723698287
96: x = 22.3369181198, f(x) = 3.1995213082
97: x = 22.5505532468, f(x) = 1.1729125220
98: x = 22.7628139612, f(x) = 2.9152868046
99: x = 23.0774684295, f(x) = 1.3356928329
100: x = 23.2460554756, f(x) = 0.1427638543
101: x = 23.4549889784, f(x) = 3.5024975353
102: x = 23.6691063365, f(x) = 1.0859081838
103: x = 23.8780092698, f(x) = 2.5879434193
104: x = 24.1717834239, f(x) = 1.7166638512
105: x = 24.3612733726, f(x) = 0.5480578969
106: x = 24.5735392781, f(x) = 3.7029734733
107: x = 24.7871791820, f(x) = 0.8811123549
108: x = 24.9894960018, f(x) = 2.1942725851
109: x = 25.2763046767, f(x) = 2.2068287656
110: x = 25.4783813428, f(x) = 0.8933130201
111: x = 25.6922347486, f(x) = 3.7863237179
112: x = 25.9042647022, f(x) = 0.5651478719
113: x = 26.0940020589, f(x) = 1.7836160021
114: x = 26.3877881895, f(x) = 2.7205540005
115: x = 26.5964670979, f(x) = 1.1511030311
116: x = 26.8107885613, f(x) = 3.7446840969
117: x = 27.0194446482, f(x) = 0.1539153202
118: x = 27.1882771440, f(x) = 1.4497848046
119: x = 27.5029770293, f(x) = 3.2044863041
120: x = 27.7150265995, f(x) = 1.3003035838
121: x = 27.9288666295, f(x) = 3.5776704225
122: x = 28.2744391431, f(x) = 1.3293416254
123: x = 28.6200665689, f(x) = 3.6216711995
124: x = 28.8337268525, f(x) = 1.3262869996
125: x = 29.0459627277, f(x) = 3.2931499870
126: x = 29.3606233033, f(x) = 1.5065911228
127: x = 29.5292387623, f(x) = 0.1609050861
128: x = 29.7381410395, f(x) = 3.9438229093
129: x = 29.9522815450, f(x) = 1.2215667789
130: x = 30.1611600968, f(x) = 2.9085733507
131: x = 30.4549363175, f(x) = 1.9269037586
132: x = 30.6444528037, f(x) = 0.6146847372
133: x = 30.8566942418, f(x) = 4.1494609758
134: x = 31.0703564862, f(x) = 0.9864859835
135: x = 31.2726491201, f(x) = 2.4546773552
136: x = 31.5594585254, f(x) = 2.4659079482
137: x = 31.7615590353, f(x) = 0.9973985019
138: x = 31.9753920445, f(x) = 4.2240101554
139: x = 32.1874447470, f(x) = 0.6299698316
140: x = 32.3771590520, f(x) = 1.9867837354
141: x = 32.6709447459, f(x) = 3.0271707361
142: x = 32.8796442193, f(x) = 1.2798699338
143: x = 33.0939477204, f(x) = 4.1603979375
144: x = 33.3026284139, f(x) = 0.1708767235
145: x = 33.4714397579, f(x) = 1.6086073936
146: x = 33.7861363704, f(x) = 3.5517142520
147: x = 33.9982038323, f(x) = 1.4401746219
148: x = 34.2120272555, f(x) = 3.9597088428
149: x = 34.5576053121, f(x) = 1.4696420111
150: x = 34.9032282858, f(x) = 3.9995088441
151: x = 35.1169046455, f(x) = 1.4636771345
152: x = 35.3291244947, f(x) = 3.6319111975
153: x = 35.6437889034, f(x) = 1.6599875582
154: x = 35.8124227579, f(x) = 0.1771986898
155: x = 36.0213046976, f(x) = 4.3405059991
156: x = 36.2354602558, f(x) = 1.3435975194
157: x = 36.4443228170, f(x) = 3.1972094706
158: x = 36.7381002928, f(x) = 2.1163605760
159: x = 36.9276342343, f(x) = 0.6747645562
160: x = 37.1398594719, f(x) = 4.5523667580
161: x = 37.3535364828, f(x) = 1.0816423861
162: x = 37.5558130137, f(x) = 2.6899908357
163: x = 37.8426228157, f(x) = 2.7002429194
164: x = 38.0447392428, f(x) = 1.0916040631
165: x = 38.2585585412, f(x) = 4.6204197706
166: x = 38.4706265108, f(x) = 0.6887175553
167: x = 38.6603252526, f(x) = 2.1710213589
168: x = 38.9541105740, f(x) = 3.3054670718
169: x = 39.1628239672, f(x) = 1.3968163702
170: x = 39.3771152247, f(x) = 4.5381896383
171: x = 39.5858126689, f(x) = 0.1863002515
172: x = 39.7546095469, f(x) = 1.7531000516
173: x = 40.0693038529, f(x) = 3.8678957204
174: x = 40.2813835840, f(x) = 1.5676149240
175: x = 40.4951955416, f(x) = 4.3080000398
176: x = 40.8407773697, f(x) = 1.5976690234
177: x = 41.1863971993, f(x) = 4.3446106108
178: x = 41.4000847193, f(x) = 1.5892338948
179: x = 41.6122933720, f(x) = 3.9416649253
180: x = 41.9269604085, f(x) = 1.8003612981
181: x = 42.0956071451, f(x) = 0.1921153424
182: x = 42.3044747859, f(x) = 4.7038545044
183: x = 42.5186409162, f(x) = 1.4554323212
184: x = 42.7274921821, f(x) = 3.4618634856
185: x = 43.0212704966, f(x) = 2.2901979085
186: x = 43.2108167922, f(x) = 0.7299158126
187: x = 43.4230305120, f(x) = 4.9224040296
188: x = 43.6367180088, f(x) = 1.1690790953
189: x = 43.8389830478, f(x) = 2.9063139998
190: x = 44.1257930891, f(x) = 2.9158055860
191: x = 44.3279208959, f(x) = 1.1783017917
192: x = 44.5417303449, f(x) = 4.9854084487
193: x = 44.7538092696, f(x) = 0.7428336014
194: x = 44.9434967975, f(x) = 2.3408028955
195: x = 45.2372818115, f(x) = 3.5620869190
196: x = 45.4460052522, f(x) = 1.5047009454
197: x = 45.6602876289, f(x) = 4.8868620473
198: x = 45.8689972122, f(x) = 0.2005410504
199: x = 46.0377835727, f(x) = 1.8865582416
200: x = 46.3524761669, f(x) = 4.1601155490
201: x = 46.5645648350, f(x) = 1.6854465890
202: x = 46.7783684007, f(x) = 4.6301658300
203: x = 47.1239529605, f(x) = 1.7161716366
204: x = 47.4695704520, f(x) = 4.6642483553
205: x = 47.6832661723, f(x) = 1.7055726407
206: x = 47.8954665608, f(x) = 4.2287902082
207: x = 48.2101355105, f(x) = 1.9305551153
208: x = 48.3787917712, f(x) = 0.2059544464
209: x = 48.5876488100, f(x) = 5.0410815405
210: x = 48.8018227731, f(x) = 1.5592665644
211: x = 49.0106656359, f(x) = 3.7076744410
212: x = 49.3044445484, f(x) = 2.4517404160
213: x = 49.4940000482, f(x) = 0.7811830675
214: x = 49.7062051589, f(x) = 5.2665055556
215: x = 49.9199004867, f(x) = 1.2504165809
216: x = 50.1221569124, f(x) = 3.1076151311
217: x = 50.4089671089, f(x) = 3.1164936977
218: x = 50.6111034563, f(x) = 1.2590436086
219: x = 50.8249054874, f(x) = 5.3254405335
220: x = 51.0369926558, f(x) = 0.7932664442
221: x = 51.2266717196, f(x) = 2.4990764006
222: x = 51.5204564800, f(x) = 3.8014226752
223: x = 51.7291875144, f(x) = 1.6053516507
224: x = 51.9434631549, f(x) = 5.2122621732
225: x = 52.1521819395, f(x) = 0.2138355493
226: x = 52.3209603086, f(x) = 2.0111798308
227: x = 52.6356515823, f(x) = 4.4331147107
228: x = 52.8477470504, f(x) = 1.7955622763
229: x = 53.0615442082, f(x) = 4.9313291879
230: x = 53.4071308375, f(x) = 1.8270040875
231: x = 53.7527465225, f(x) = 4.9633441071
232: x = 53.9664485228, f(x) = 1.8144673380
233: x = 54.1786425609, f(x) = 4.4976228601
234: x = 54.4933129654, f(x) = 2.0525070996
235: x = 54.6619765539, f(x) = 0.2189204497
236: x = 54.8708254180, f(x) = 5.3571223313
237: x = 55.0850054172, f(x) = 1.6566053288
238: x = 55.2938417841, f(x) = 3.9381723052
239: x = 55.5876211439, f(x) = 2.6032779010
240: x = 55.7771837662, f(x) = 0.8292869703
241: x = 55.9893821984, f(x) = 5.5894633707
242: x = 56.2030835972, f(x) = 1.3267770462
243: x = 56.4053333270, f(x) = 3.2966472039
244: x = 56.6921436299, f(x) = 3.3050179972
245: x = 56.8942866234, f(x) = 1.3349106763
246: x = 57.1080828666, f(x) = 5.6450276329
247: x = 57.3201764632, f(x) = 0.8406792173
248: x = 57.5098489115, f(x) = 2.6479062777
249: x = 57.8036334607, f(x) = 4.0265575641
250: x = 58.0123704361, f(x) = 1.7000538009
251: x = 58.2266407920, f(x) = 5.5185082954
252: x = 58.4353667915, f(x) = 0.2263505499
253: x = 58.6041388827, f(x) = 2.1285174877
254: x = 58.9188291072, f(x) = 4.6902506424
255: x = 59.1309299229, f(x) = 1.8993045021
256: x = 59.3447220274, f(x) = 5.2151299089
257: x = 59.6903102787, f(x) = 1.9314872079
258: x = 60.0359245262, f(x) = 5.2454129652
259: x = 60.2496314900, f(x) = 1.9171868524
260: x = 60.4618204956, f(x) = 4.7512689378
261: x = 60.7764920436, f(x) = 2.1676087740
262: x = 60.9451614448, f(x) = 0.2311603180
263: x = 61.1540038135, f(x) = 5.6555297467
264: x = 61.3681886066, f(x) = 1.7485337360
265: x = 61.5770198018, f(x) = 4.1559057632
266: x = 61.8707995082, f(x) = 2.7464669487
267: x = 62.0603678060, f(x) = 0.8747495501
268: x = 62.2725609062, f(x) = 5.8947537050
269: x = 62.4862671495, f(x) = 1.3989757116
270: x = 62.6885115246, f(x) = 3.4754127730
271: x = 62.9753219037, f(x) = 3.4833540050
272: x = 63.1774702161, f(x) = 1.4066919442
273: x = 63.3912618174, f(x) = 5.9474664786
274: x = 63.6033605670, f(x) = 0.8855571371
275: x = 63.7930277025, f(x) = 2.7888048608
276: x = 64.0868120738, f(x) = 4.2397544159
277: x = 64.2955538240, f(x) = 1.7897519218
278: x = 64.5098199232, f(x) = 5.8086307372
279: x = 64.7185517320, f(x) = 0.2382089436
280: x = 64.8873187609, f(x) = 2.2397163057
281: x = 65.2020081317, f(x) = 4.9340040960
282: x = 65.4141132631, f(x) = 1.9976664661
283: x = 65.6279012805, f(x) = 5.4842640985
284: x = 65.9734908373, f(x) = 2.0306013315
285: x = 66.3191039137, f(x) = 5.5130690129
286: x = 66.5328148991, f(x) = 2.0146759410
287: x = 66.7449998185, f(x) = 4.9920438605
288: x = 67.0596722889, f(x) = 2.2768992612
289: x = 67.2283464134, f(x) = 0.2427838998
290: x = 67.4371834968, f(x) = 5.9389623753
291: x = 67.6513721893, f(x) = 1.8358647135
292: x = 67.8601991697, f(x) = 4.3627863240
293: x = 68.1539791524, f(x) = 2.8825519334
294: x = 68.3435520789, f(x) = 0.9179633289
295: x = 68.5557408237, f(x) = 6.1849932943
296: x = 68.7694510224, f(x) = 1.4676269138
297: x = 68.9716910179, f(x) = 3.6454225026
298: x = 69.2585014533, f(x) = 3.6529941705
299: x = 69.4606541190, f(x) = 1.4749840483
300: x = 69.6744419147, f(x) = 6.2352528716
301: x = 69.8865448871, f(x) = 0.9282679252
302: x = 70.0762076623, f(x) = 2.9229193488
303: x = 70.3699918821, f(x) = 4.4427321960
304: x = 70.5787375537, f(x) = 1.8751642502
305: x = 70.7930001508, f(x) = 6.0849362004
306: x = 71.0017367373, f(x) = 0.2495043691
307: x = 71.1704995979, f(x) = 2.3456495052
308: x = 71.4851882605, f(x) = 5.1662696364
309: x = 71.6972969481, f(x) = 2.0914074299
310: x = 71.9110815917, f(x) = 5.7407948649
311: x = 72.2566722217, f(x) = 2.1250978158
312: x = 72.6022843256, f(x) = 5.7683189009
313: x = 72.8159986358, f(x) = 2.1076605223
314: x = 73.0281801714, f(x) = 5.2217284001
315: x = 73.3428534013, f(x) = 2.3811788524
316: x = 73.5115314399, f(x) = 0.2538758582
317: x = 73.7203641387, f(x) = 6.2094711309
318: x = 73.9345560652, f(x) = 1.9192259569
319: x = 74.1433795443, f(x) = 4.5602912678
320: x = 74.4371597524, f(x) = 3.0124957519
321: x = 74.6267365259, f(x) = 0.9592322826
322: x = 74.8389216461, f(x) = 6.4622103889
323: x = 75.0526351355, f(x) = 1.5332072523
324: x = 75.2548714822, f(x) = 3.8078493400
325: x = 75.5416819605, f(x) = 3.8150986602
326: x = 75.7438382549, f(x) = 1.5402511689
327: x = 75.9576228739, f(x) = 6.5103302121
328: x = 76.1697293701, f(x) = 0.9690981581
329: x = 76.3593885023, f(x) = 3.0511444641
330: x = 76.6531725917, f(x) = 4.6368331164
331: x = 76.8619215413, f(x) = 1.9568520628
332: x = 77.0761812066, f(x) = 6.3492287958
333: x = 77.2849217917, f(x) = 0.2603101220
334: x = 77.4536811602, f(x) = 2.4470010422
335: x = 77.7683692260, f(x) = 5.3885329775
336: x = 77.9804808945, f(x) = 2.1811232827
337: x = 78.1942627058, f(x) = 5.9863426934
338: x = 78.5398542338, f(x) = 2.2155675809
339: x = 78.8854655174, f(x) = 6.0127428119
340: x = 79.0991826220, f(x) = 2.1967126837
341: x = 79.3113613093, f(x) = 5.4417270429
342: x = 79.6260351756, f(x) = 2.4810794495
343: x = 79.7947165107, f(x) = 0.2645030828
344: x = 80.0035455134, f(x) = 6.4686775733
345: x = 80.2177401653, f(x) = 1.9991141305
346: x = 80.4265606898, f(x) = 4.7495903500
347: x = 80.7203410850, f(x) = 3.1370616358
348: x = 80.9099211065, f(x) = 0.9987975128
349: x = 81.1221031633, f(x) = 6.7280148904
350: x = 81.3358194331, f(x) = 1.5960952984
351: x = 81.5380526931, f(x) = 3.9636256183
352: x = 81.8248632047, f(x) = 3.9705905399
353: x = 82.0270225702, f(x) = 1.6028628721
354: x = 82.2408044974, f(x) = 6.7742468773
355: x = 82.4529139787, f(x) = 1.0082763211
356: x = 82.6425700218, f(x) = 3.1741940085
357: x = 82.9363539978, f(x) = 4.8231290084
358: x = 83.1451057283, f(x) = 2.0352638726
359: x = 83.3593629033, f(x) = 6.6029511741
360: x = 83.5681068841, f(x) = 0.2706848519
361: x = 83.7368632846, f(x) = 2.5443184984
362: x = 84.0515508405, f(x) = 5.6019847789
363: x = 84.2636650438, f(x) = 2.2672918906
364: x = 84.4774444437, f(x) = 6.2222079801
365: x = 84.8230367340, f(x) = 2.3024853310
366: x = 85.1686473163, f(x) = 6.2476114989
367: x = 85.3823668026, f(x) = 2.2822927854
368: x = 85.5945430593, f(x) = 5.6531707076
369: x = 85.9092174667, f(x) = 2.5771103512
370: x = 86.0779016159, f(x) = 0.2747195119
371: x = 86.2867274608, f(x) = 6.7178901091
372: x = 86.5009244407, f(x) = 2.0759302359
373: x = 86.7097424385, f(x) = 4.9316285974
374: x = 87.0035229915, f(x) = 3.2568667140
375: x = 87.1931057918, f(x) = 1.0368540745
376: x = 87.4052852254, f(x) = 6.9837100291
377: x = 87.6190038755, f(x) = 1.6565976962
378: x = 87.8212344902, f(x) = 4.1135069642
379: x = 88.1080450283, f(x) = 4.1202185307
380: x = 88.3102070266, f(x) = 1.6631190914
381: x = 88.5239866439, f(x) = 7.0282602530
382: x = 88.7360986862, f(x) = 1.0459880688
383: x = 88.9257520765, f(x) = 3.2926482805
384: x = 89.2195359533, f(x) = 5.0024919381
385: x = 89.4282900727, f(x) = 2.1107648069
386: x = 89.6425451061, f(x) = 6.8472784554
387: x = 89.8512920066, f(x) = 0.2806763590
388: x = 90.0200458534, f(x) = 2.6380483619
389: x = 90.3347329686, f(x) = 5.8075966952
390: x = 90.5468493538, f(x) = 2.3503034410
391: x = 90.7606266759, f(x) = 6.4494531214
392: x = 91.1062196214, f(x) = 2.3862392389
393: x = 91.4518295972, f(x) = 6.4739650038
394: x = 91.6655511377, f(x) = 2.3647778134
395: x = 91.8777252959, f(x) = 5.8569859979
396: x = 92.1924001688, f(x) = 2.6696891777
397: x = 92.3610867486, f(x) = 0.2845693934
398: x = 92.5699098642, f(x) = 6.9581826370
399: x = 92.7841088559, f(x) = 2.1500035780
400: x = 92.9929246681, f(x) = 5.1071824738
401: x = 93.2867053561, f(x) = 3.3724184063
402: x = 93.4762905608, f(x) = 1.0735624213
403: x = 93.6884677227, f(x) = 7.2303684068
404: x = 93.9021884336, f(x) = 1.7149669542
405: x = 94.1044167561, f(x) = 4.2581159086
406: x = 94.3912273155, f(x) = 4.2645998980
407: x = 94.5933915961, f(x) = 1.7212672191
408: x = 94.8071692094, f(x) = 7.2734079860
409: x = 95.0192834730, f(x) = 1.0823866874
410: x = 95.2089345607, f(x) = 3.4069866187
411: x = 95.5027183497, f(x) = 5.1756427439
412: x = 95.7114745435, f(x) = 2.1836568211
413: x = 95.9257277156, f(x) = 7.0831829280
414: x = 96.1344771532, f(x) = 0.2903242117
415: x = 96.3032287795, f(x) = 2.7285603735
416: x = 96.6179155102, f(x) = 6.0061739370
417: x = 96.8300337932, f(x) = 2.4304814371
418: x = 97.0438093062, f(x) = 6.6689593697
419: x = 97.3894028210, f(x) = 2.4671515400
420: x = 97.7350122672, f(x) = 6.6926673598
421: x = 97.9487355975, f(x) = 2.4444811022
422: x = 98.1609079257, f(x) = 6.0539434242
423: x = 98.4755832033, f(x) = 2.7591634352
424: x = 98.6442719036, f(x) = 0.2940895597
425: x = 98.8530926368, f(x) = 7.1904494850
426: x = 99.0672933843, f(x) = 2.2216085229
427: x = 99.2761072873, f(x) = 5.2768991911
428: x = 99.5698880920, f(x) = 3.4841399289
429: x = 99.7594753975, f(x) = 1.1090564339
430: x = 99.9716505732, f(x) = 7.4688853870
【極小値】
個数 = 429
1: x = 0.2774363337, f(x) = 0.0000000000
2: x = 0.4087983937, f(x) = 0.0000000000
3: x = 0.7192029060, f(x) = 0.0000000000
4: x = 0.8288364482, f(x) = 0.0000000000
5: x = 1.3908460479, f(x) = 0.0000000000
6: x = 1.5331314548, f(x) = 0.0000000000
7: x = 1.8576063345, f(x) = 0.0000000000
8: x = 1.9177990078, f(x) = 0.0000000000
9: x = 2.5081618019, f(x) = 0.0000000000
10: x = 2.6551361780, f(x) = 0.0000000000
11: x = 3.6280491292, f(x) = 0.0000000000
12: x = 3.7750235053, f(x) = 0.0000000000
13: x = 4.3653862993, f(x) = 0.0000000000
14: x = 4.4255789727, f(x) = 0.0000000000
15: x = 4.7500538524, f(x) = 0.0000000000
16: x = 4.8923392593, f(x) = 0.0000000000
17: x = 5.4543488590, f(x) = 0.0000000000
18: x = 5.5639824012, f(x) = 0.0000000000
19: x = 5.8743869135, f(x) = 0.0000000000
20: x = 6.0057489735, f(x) = 0.0000000000
21: x = 6.5606216409, f(x) = 0.0000000000
22: x = 6.6919837009, f(x) = 0.0000000000
23: x = 7.0023882132, f(x) = 0.0000000000
24: x = 7.1120217553, f(x) = 0.0000000000
25: x = 7.6740313551, f(x) = 0.0000000000
26: x = 7.8163167620, f(x) = 0.0000000000
27: x = 8.1407916417, f(x) = 0.0000000000
28: x = 8.2009843150, f(x) = 0.0000000000
29: x = 8.7913471091, f(x) = 0.0000000000
30: x = 8.9383214851, f(x) = 0.0000000000
31: x = 9.9112344364, f(x) = 0.0000000000
32: x = 10.0582088124, f(x) = 0.0000000000
33: x = 10.6485716065, f(x) = 0.0000000000
34: x = 10.7087642799, f(x) = 0.0000000000
35: x = 11.0332391596, f(x) = 0.0000000000
36: x = 11.1755245665, f(x) = 0.0000000000
37: x = 11.7375341662, f(x) = 0.0000000000
38: x = 11.8471677084, f(x) = 0.0000000000
39: x = 12.1575722206, f(x) = 0.0000000000
40: x = 12.2889342806, f(x) = 0.0000000000
41: x = 12.8438069481, f(x) = 0.0000000000
42: x = 12.9751690081, f(x) = 0.0000000000
43: x = 13.2855735203, f(x) = 0.0000000000
44: x = 13.3952070625, f(x) = 0.0000000000
45: x = 13.9572166623, f(x) = 0.0000000000
46: x = 14.0995020692, f(x) = 0.0000000000
47: x = 14.4239769489, f(x) = 0.0000000000
48: x = 14.4841696222, f(x) = 0.0000000000
49: x = 15.0745324163, f(x) = 0.0000000000
50: x = 15.2215067923, f(x) = 0.0000000000
51: x = 16.1944197436, f(x) = 0.0000000000
52: x = 16.3413941196, f(x) = 0.0000000000
53: x = 16.9317569137, f(x) = 0.0000000000
54: x = 16.9919495870, f(x) = 0.0000000000
55: x = 17.3164244667, f(x) = 0.0000000000
56: x = 17.4587098736, f(x) = 0.0000000000
57: x = 18.0207194734, f(x) = 0.0000000000
58: x = 18.1303530156, f(x) = 0.0000000000
59: x = 18.4407575278, f(x) = 0.0000000000
60: x = 18.5721195878, f(x) = 0.0000000000
61: x = 19.1269922553, f(x) = 0.0000000000
62: x = 19.2583543153, f(x) = 0.0000000000
63: x = 19.5687588275, f(x) = 0.0000000000
64: x = 19.6783923697, f(x) = 0.0000000000
65: x = 20.2404019694, f(x) = 0.0000000000
66: x = 20.3826873763, f(x) = 0.0000000000
67: x = 20.7071622560, f(x) = 0.0000000000
68: x = 20.7673549294, f(x) = 0.0000000000
69: x = 21.3577177235, f(x) = 0.0000000000
70: x = 21.5046920995, f(x) = 0.0000000000
71: x = 22.4776050508, f(x) = 0.0000000000
72: x = 22.6245794268, f(x) = 0.0000000000
73: x = 23.2149422209, f(x) = 0.0000000000
74: x = 23.2751348942, f(x) = 0.0000000000
75: x = 23.5996097739, f(x) = 0.0000000000
76: x = 23.7418951808, f(x) = 0.0000000000
77: x = 24.3039047806, f(x) = 0.0000000000
78: x = 24.4135383227, f(x) = 0.0000000000
79: x = 24.7239428350, f(x) = 0.0000000000
80: x = 24.8553048950, f(x) = 0.0000000000
81: x = 25.4101775624, f(x) = 0.0000000000
82: x = 25.5415396224, f(x) = 0.0000000000
83: x = 25.8519441347, f(x) = 0.0000000000
84: x = 25.9615776769, f(x) = 0.0000000000
85: x = 26.5235872766, f(x) = 0.0000000000
86: x = 26.6658726835, f(x) = 0.0000000000
87: x = 26.9903475632, f(x) = 0.0000000000
88: x = 27.0505402365, f(x) = 0.0000000000
89: x = 27.6409030306, f(x) = 0.0000000000
90: x = 27.7878774067, f(x) = 0.0000000000
91: x = 28.7607903579, f(x) = 0.0000000000
92: x = 28.9077647340, f(x) = 0.0000000000
93: x = 29.4981275281, f(x) = 0.0000000000
94: x = 29.5583202014, f(x) = 0.0000000000
95: x = 29.8827950811, f(x) = 0.0000000000
96: x = 30.0250804880, f(x) = 0.0000000000
97: x = 30.5870900877, f(x) = 0.0000000000
98: x = 30.6967236299, f(x) = 0.0000000000
99: x = 31.0071281422, f(x) = 0.0000000000
100: x = 31.1384902022, f(x) = 0.0000000000
101: x = 31.6933628696, f(x) = 0.0000000000
102: x = 31.8247249296, f(x) = 0.0000000000
103: x = 32.1351294419, f(x) = 0.0000000000
104: x = 32.2447629841, f(x) = 0.0000000000
105: x = 32.8067725838, f(x) = 0.0000000000
106: x = 32.9490579907, f(x) = 0.0000000000
107: x = 33.2735328704, f(x) = 0.0000000000
108: x = 33.3337255437, f(x) = 0.0000000000
109: x = 33.9240883378, f(x) = 0.0000000000
110: x = 34.0710627139, f(x) = 0.0000000000
111: x = 35.0439756651, f(x) = 0.0000000000
112: x = 35.1909500412, f(x) = 0.0000000000
113: x = 35.7813128352, f(x) = 0.0000000000
114: x = 35.8415055086, f(x) = 0.0000000000
115: x = 36.1659803883, f(x) = 0.0000000000
116: x = 36.3082657952, f(x) = 0.0000000000
117: x = 36.8702753949, f(x) = 0.0000000000
118: x = 36.9799089371, f(x) = 0.0000000000
119: x = 37.2903134494, f(x) = 0.0000000000
120: x = 37.4216755094, f(x) = 0.0000000000
121: x = 37.9765481768, f(x) = 0.0000000000
122: x = 38.1079102368, f(x) = 0.0000000000
123: x = 38.4183147490, f(x) = 0.0000000000
124: x = 38.5279482912, f(x) = 0.0000000000
125: x = 39.0899578910, f(x) = 0.0000000000
126: x = 39.2322432979, f(x) = 0.0000000000
127: x = 39.5567181776, f(x) = 0.0000000000
128: x = 39.6169108509, f(x) = 0.0000000000
129: x = 40.2072736450, f(x) = 0.0000000000
130: x = 40.3542480210, f(x) = 0.0000000000
131: x = 41.3271609723, f(x) = 0.0000000000
132: x = 41.4741353483, f(x) = 0.0000000000
133: x = 42.0644981424, f(x) = 0.0000000000
134: x = 42.1246908158, f(x) = 0.0000000000
135: x = 42.4491656955, f(x) = 0.0000000000
136: x = 42.5914511024, f(x) = 0.0000000000
137: x = 43.1534607021, f(x) = 0.0000000000
138: x = 43.2630942443, f(x) = 0.0000000000
139: x = 43.5734987565, f(x) = 0.0000000000
140: x = 43.7048608165, f(x) = 0.0000000000
141: x = 44.2597334840, f(x) = 0.0000000000
142: x = 44.3910955440, f(x) = 0.0000000000
143: x = 44.7015000562, f(x) = 0.0000000000
144: x = 44.8111335984, f(x) = 0.0000000000
145: x = 45.3731431982, f(x) = 0.0000000000
146: x = 45.5154286051, f(x) = 0.0000000000
147: x = 45.8399034848, f(x) = 0.0000000000
148: x = 45.9000961581, f(x) = 0.0000000000
149: x = 46.4904589522, f(x) = 0.0000000000
150: x = 46.6374333282, f(x) = 0.0000000000
151: x = 47.6103462795, f(x) = 0.0000000000
152: x = 47.7573206555, f(x) = 0.0000000000
153: x = 48.3476834496, f(x) = 0.0000000000
154: x = 48.4078761229, f(x) = 0.0000000000
155: x = 48.7323510026, f(x) = 0.0000000000
156: x = 48.8746364095, f(x) = 0.0000000000
157: x = 49.4366460093, f(x) = 0.0000000000
158: x = 49.5462795515, f(x) = 0.0000000000
159: x = 49.8566840637, f(x) = 0.0000000000
160: x = 49.9880461237, f(x) = 0.0000000000
161: x = 50.5429187912, f(x) = 0.0000000000
162: x = 50.6742808512, f(x) = 0.0000000000
163: x = 50.9846853634, f(x) = 0.0000000000
164: x = 51.0943189056, f(x) = 0.0000000000
165: x = 51.6563285053, f(x) = 0.0000000000
166: x = 51.7986139122, f(x) = 0.0000000000
167: x = 52.1230887919, f(x) = 0.0000000000
168: x = 52.1832814653, f(x) = 0.0000000000
169: x = 52.7736442594, f(x) = 0.0000000000
170: x = 52.9206186354, f(x) = 0.0000000000
171: x = 53.8935315867, f(x) = 0.0000000000
172: x = 54.0405059627, f(x) = 0.0000000000
173: x = 54.6308687568, f(x) = 0.0000000000
174: x = 54.6910614301, f(x) = 0.0000000000
175: x = 55.0155363098, f(x) = 0.0000000000
176: x = 55.1578217167, f(x) = 0.0000000000
177: x = 55.7198313165, f(x) = 0.0000000000
178: x = 55.8294648586, f(x) = 0.0000000000
179: x = 56.1398693709, f(x) = 0.0000000000
180: x = 56.2712314309, f(x) = 0.0000000000
181: x = 56.8261040983, f(x) = 0.0000000000
182: x = 56.9574661583, f(x) = 0.0000000000
183: x = 57.2678706706, f(x) = 0.0000000000
184: x = 57.3775042128, f(x) = 0.0000000000
185: x = 57.9395138125, f(x) = 0.0000000000
186: x = 58.0817992194, f(x) = 0.0000000000
187: x = 58.4062740991, f(x) = 0.0000000000
188: x = 58.4664667724, f(x) = 0.0000000000
189: x = 59.0568295665, f(x) = 0.0000000000
190: x = 59.2038039426, f(x) = 0.0000000000
191: x = 60.1767168938, f(x) = 0.0000000000
192: x = 60.3236912699, f(x) = 0.0000000000
193: x = 60.9140540640, f(x) = 0.0000000000
194: x = 60.9742467373, f(x) = 0.0000000000
195: x = 61.2987216170, f(x) = 0.0000000000
196: x = 61.4410070239, f(x) = 0.0000000000
197: x = 62.0030166236, f(x) = 0.0000000000
198: x = 62.1126501658, f(x) = 0.0000000000
199: x = 62.4230546781, f(x) = 0.0000000000
200: x = 62.5544167381, f(x) = 0.0000000000
201: x = 63.1092894055, f(x) = 0.0000000000
202: x = 63.2406514655, f(x) = 0.0000000000
203: x = 63.5510559778, f(x) = 0.0000000000
204: x = 63.6606895200, f(x) = 0.0000000000
205: x = 64.2226991197, f(x) = 0.0000000000
206: x = 64.3649845266, f(x) = 0.0000000000
207: x = 64.6894594063, f(x) = 0.0000000000
208: x = 64.7496520796, f(x) = 0.0000000000
209: x = 65.3400148737, f(x) = 0.0000000000
210: x = 65.4869892498, f(x) = 0.0000000000
211: x = 66.4599022010, f(x) = 0.0000000000
212: x = 66.6068765771, f(x) = 0.0000000000
213: x = 67.1972393711, f(x) = 0.0000000000
214: x = 67.2574320445, f(x) = 0.0000000000
215: x = 67.5819069242, f(x) = 0.0000000000
216: x = 67.7241923311, f(x) = 0.0000000000
217: x = 68.2862019308, f(x) = 0.0000000000
218: x = 68.3958354730, f(x) = 0.0000000000
219: x = 68.7062399853, f(x) = 0.0000000000
220: x = 68.8376020453, f(x) = 0.0000000000
221: x = 69.3924747127, f(x) = 0.0000000000
222: x = 69.5238367727, f(x) = 0.0000000000
223: x = 69.8342412849, f(x) = 0.0000000000
224: x = 69.9438748271, f(x) = 0.0000000000
225: x = 70.5058844269, f(x) = 0.0000000000
226: x = 70.6481698338, f(x) = 0.0000000000
227: x = 70.9726447135, f(x) = 0.0000000000
228: x = 71.0328373868, f(x) = 0.0000000000
229: x = 71.6232001809, f(x) = 0.0000000000
230: x = 71.7701745569, f(x) = 0.0000000000
231: x = 72.7430875082, f(x) = 0.0000000000
232: x = 72.8900618842, f(x) = 0.0000000000
233: x = 73.4804246783, f(x) = 0.0000000000
234: x = 73.5406173517, f(x) = 0.0000000000
235: x = 73.8650922314, f(x) = 0.0000000000
236: x = 74.0073776383, f(x) = 0.0000000000
237: x = 74.5693872380, f(x) = 0.0000000000
238: x = 74.6790207802, f(x) = 0.0000000000
239: x = 74.9894252924, f(x) = 0.0000000000
240: x = 75.1207873524, f(x) = 0.0000000000
241: x = 75.6756600199, f(x) = 0.0000000000
242: x = 75.8070220799, f(x) = 0.0000000000
243: x = 76.1174265921, f(x) = 0.0000000000
244: x = 76.2270601343, f(x) = 0.0000000000
245: x = 76.7890697341, f(x) = 0.0000000000
246: x = 76.9313551410, f(x) = 0.0000000000
247: x = 77.2558300207, f(x) = 0.0000000000
248: x = 77.3160226940, f(x) = 0.0000000000
249: x = 77.9063854881, f(x) = 0.0000000000
250: x = 78.0533598641, f(x) = 0.0000000000
251: x = 79.0262728154, f(x) = 0.0000000000
252: x = 79.1732471914, f(x) = 0.0000000000
253: x = 79.7636099855, f(x) = 0.0000000000
254: x = 79.8238026588, f(x) = 0.0000000000
255: x = 80.1482775385, f(x) = 0.0000000000
256: x = 80.2905629454, f(x) = 0.0000000000
257: x = 80.8525725452, f(x) = 0.0000000000
258: x = 80.9622060874, f(x) = 0.0000000000
259: x = 81.2726105996, f(x) = 0.0000000000
260: x = 81.4039726596, f(x) = 0.0000000000
261: x = 81.9588453271, f(x) = 0.0000000000
262: x = 82.0902073871, f(x) = 0.0000000000
263: x = 82.4006118993, f(x) = 0.0000000000
264: x = 82.5102454415, f(x) = 0.0000000000
265: x = 83.0722550412, f(x) = 0.0000000000
266: x = 83.2145404481, f(x) = 0.0000000000
267: x = 83.5390153278, f(x) = 0.0000000000
268: x = 83.5992080012, f(x) = 0.0000000000
269: x = 84.1895707953, f(x) = 0.0000000000
270: x = 84.3365451713, f(x) = 0.0000000000
271: x = 85.3094581226, f(x) = 0.0000000000
272: x = 85.4564324986, f(x) = 0.0000000000
273: x = 86.0467952927, f(x) = 0.0000000000
274: x = 86.1069879660, f(x) = 0.0000000000
275: x = 86.4314628457, f(x) = 0.0000000000
276: x = 86.5737482526, f(x) = 0.0000000000
277: x = 87.1357578524, f(x) = 0.0000000000
278: x = 87.2453913945, f(x) = 0.0000000000
279: x = 87.5557959068, f(x) = 0.0000000000
280: x = 87.6871579668, f(x) = 0.0000000000
281: x = 88.2420306342, f(x) = 0.0000000000
282: x = 88.3733926942, f(x) = 0.0000000000
283: x = 88.6837972065, f(x) = 0.0000000000
284: x = 88.7934307487, f(x) = 0.0000000000
285: x = 89.3554403484, f(x) = 0.0000000000
286: x = 89.4977257553, f(x) = 0.0000000000
287: x = 89.8222006350, f(x) = 0.0000000000
288: x = 89.8823933083, f(x) = 0.0000000000
289: x = 90.4727561024, f(x) = 0.0000000000
290: x = 90.6197304785, f(x) = 0.0000000000
291: x = 91.5926434297, f(x) = 0.0000000000
292: x = 91.7396178058, f(x) = 0.0000000000
293: x = 92.3299805999, f(x) = 0.0000000000
294: x = 92.3901732732, f(x) = 0.0000000000
295: x = 92.7146481529, f(x) = 0.0000000000
296: x = 92.8569335598, f(x) = 0.0000000000
297: x = 93.4189431595, f(x) = 0.0000000000
298: x = 93.5285767017, f(x) = 0.0000000000
299: x = 93.8389812140, f(x) = 0.0000000000
300: x = 93.9703432740, f(x) = 0.0000000000
301: x = 94.5252159414, f(x) = 0.0000000000
302: x = 94.6565780014, f(x) = 0.0000000000
303: x = 94.9669825137, f(x) = 0.0000000000
304: x = 95.0766160558, f(x) = 0.0000000000
305: x = 95.6386256556, f(x) = 0.0000000000
306: x = 95.7809110625, f(x) = 0.0000000000
307: x = 96.1053859422, f(x) = 0.0000000000
308: x = 96.1655786155, f(x) = 0.0000000000
309: x = 96.7559414096, f(x) = 0.0000000000
310: x = 96.9029157856, f(x) = 0.0000000000
311: x = 97.8758287369, f(x) = 0.0000000000
312: x = 98.0228031129, f(x) = 0.0000000000
313: x = 98.6131659070, f(x) = 0.0000000000
314: x = 98.6733585804, f(x) = 0.0000000000
315: x = 98.9978334601, f(x) = 0.0000000000
316: x = 99.1401188670, f(x) = 0.0000000000
317: x = 99.7021284667, f(x) = 0.0000000000
318: x = 99.8117620089, f(x) = 0.0000000000
319: x = 1.0837370622, f(x) = 0.2311929249
320: x = 2.1797213246, f(x) = 0.2227092270
321: x = 2.9980019264, f(x) = 0.1057125421
322: x = 3.2848285644, f(x) = 0.1106536661
323: x = 4.1023079566, f(x) = 0.3055094914
324: x = 5.1965039326, f(x) = 0.5059801427
325: x = 6.2827115798, f(x) = 0.6266452571
326: x = 7.3690066788, f(x) = 0.6025277425
327: x = 8.4634686918, f(x) = 0.4388101797
328: x = 9.2813126992, f(x) = 0.1859984197
329: x = 9.5681251699, f(x) = 0.1888504113
330: x = 10.3857349160, f(x) = 0.4860940149
331: x = 11.4799647897, f(x) = 0.7520379530
332: x = 12.5661337702, f(x) = 0.8862227497
333: x = 13.6523560350, f(x) = 0.8201110319
334: x = 14.7467368131, f(x) = 0.5792269903
335: x = 15.5645221876, f(x) = 0.2408640568
336: x = 15.8513335273, f(x) = 0.2430731525
337: x = 16.6689797728, f(x) = 0.6158219384
338: x = 17.7632307912, f(x) = 0.9354685761
339: x = 18.8493980278, f(x) = 1.0853996089
340: x = 19.9356019358, f(x) = 0.9910206771
341: x = 21.0299554303, f(x) = 0.6917028813
342: x = 21.8477177689, f(x) = 0.2853691463
343: x = 22.1345287971, f(x) = 0.2872361619
344: x = 22.9521920362, f(x) = 0.7226240775
345: x = 24.0464546365, f(x) = 1.0884120508
346: x = 25.1326228091, f(x) = 1.2533126610
347: x = 26.2188187850, f(x) = 1.1365127122
348: x = 27.3131587177, f(x) = 0.7882903471
349: x = 28.1309087610, f(x) = 0.3238141395
350: x = 28.4177196610, f(x) = 0.3254606884
351: x = 29.2353927158, f(x) = 0.8155576762
352: x = 30.3296625185, f(x) = 1.2223664484
353: x = 31.4158318004, f(x) = 1.4012467477
354: x = 32.5020234357, f(x) = 1.2653853196
355: x = 33.5963552783, f(x) = 0.8742713481
356: x = 34.4140976774, f(x) = 0.3581558883
357: x = 34.7009085124, f(x) = 0.3596452460
358: x = 35.5185879577, f(x) = 0.8989343842
359: x = 36.6128626540, f(x) = 1.3430258692
360: x = 37.6990328969, f(x) = 1.5349892583
361: x = 38.7852218178, f(x) = 1.3822947280
362: x = 39.8795482921, f(x) = 0.9525224767
363: x = 40.6972854794, f(x) = 0.3894812982
364: x = 40.9840962771, f(x) = 0.3908513070
365: x = 41.8017802137, f(x) = 0.9752085403
366: x = 42.8960584464, f(x) = 1.4537046225
367: x = 43.9822294822, f(x) = 1.6579781233
368: x = 45.0684165535, f(x) = 1.4900595065
369: x = 46.1627392077, f(x) = 1.0248159461
370: x = 46.9804726141, f(x) = 0.4184683043
371: x = 47.2672833884, f(x) = 0.4197437126
372: x = 48.0849706539, f(x) = 1.0459351540
373: x = 49.1792515600, f(x) = 1.5565332337
374: x = 50.2654232479, f(x) = 1.7724533289
375: x = 51.3516089816, f(x) = 1.5905394978
376: x = 52.4459287792, f(x) = 1.0923352795
377: x = 53.2636593176, f(x) = 0.4455735192
378: x = 53.5504700763, f(x) = 0.4467715563
379: x = 54.3681599079, f(x) = 1.1121730900
380: x = 55.4624429052, f(x) = 1.6529773870
381: x = 56.5486151339, f(x) = 1.8799707685
382: x = 57.6347998568, f(x) = 1.6850384022
383: x = 58.7291174382, f(x) = 1.1559173810
384: x = 59.5468457265, f(x) = 0.4711218523
385: x = 59.8336564742, f(x) = 0.4722550817
386: x = 60.6513483443, f(x) = 1.1746819180
387: x = 61.7456330216, f(x) = 1.7440965416
388: x = 62.8318057042, f(x) = 1.9816632753
389: x = 63.9179896372, f(x) = 1.7745120211
390: x = 65.0123054493, f(x) = 1.2161799264
391: x = 65.8300319251, f(x) = 0.4953542551
392: x = 66.1168426649, f(x) = 0.4964321711
393: x = 66.9345361932, f(x) = 1.2340284552
394: x = 68.0288222498, f(x) = 1.8306859995
395: x = 69.1149953175, f(x) = 2.0783860452
396: x = 70.2011786168, f(x) = 1.8596858308
397: x = 71.2954929837, f(x) = 1.2735942170
398: x = 72.1132179684, f(x) = 0.5184552787
399: x = 72.4000287021, f(x) = 0.5194852630
400: x = 73.2177236059, f(x) = 1.2906490008
401: x = 74.3120108150, f(x) = 1.9133608375
402: x = 75.3981842132, f(x) = 2.1708034796
403: x = 76.4843669929, f(x) = 1.9411259234
404: x = 77.5786801573, f(x) = 1.3285295788
405: x = 78.3964038937, f(x) = 0.5405699898
406: x = 78.6832146228, f(x) = 0.5415579159
407: x = 79.5009106859, f(x) = 1.3448878965
408: x = 80.5951988723, f(x) = 1.9926083780
409: x = 81.6813725567, f(x) = 2.2594439412
410: x = 82.7675549031, f(x) = 2.0192841059
411: x = 83.8618670512, f(x) = 1.3812818143
412: x = 84.6795897272, f(x) = 0.5618148730
413: x = 84.9664004527, f(x) = 0.5627655053
414: x = 85.7840975062, f(x) = 1.3970225704
415: x = 86.8783865318, f(x) = 2.0688225134
416: x = 87.9645604665, f(x) = 2.3447358245
417: x = 89.0507424458, f(x) = 2.0945278087
418: x = 90.1450537239, f(x) = 1.4320921957
419: x = 90.9627754881, f(x) = 0.5822851441
420: x = 91.2495862106, f(x) = 0.5832024104
421: x = 92.0672841199, f(x) = 1.4472804386
422: x = 93.1615738741, f(x) = 2.1423270123
423: x = 94.2477480293, f(x) = 2.4270321874
424: x = 95.3339296939, f(x) = 2.1671606251
425: x = 96.4282402186, f(x) = 1.4811605792
426: x = 97.2459611903, f(x) = 0.6020598196
427: x = 97.5327719105, f(x) = 0.6029470034
428: x = 98.3504705668, f(x) = 1.4958506850
429: x = 99.4447609592, f(x) = 2.2133918445
2)
原点、関数$$y = \sqrt{(\sqrt{x} × sin(7x) × sin(3x) × cos(7x) + \frac{\sqrt{x}}{4})^2}$$の 0 < x < 100 なる x の変域における y が極大値になる点、そして (100, 0) を x の昇順に直線で繋いだ線と x 軸で囲まれる領域の面積をできるだけ正しく求めるプログラムを作成し、プログラムと実行結果を提出せよ。
import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import find_peaks
# 関数定義
def f(x):
return np.abs(
np.sqrt(x) * np.sin(7 * x) * np.sin(3 * x) * np.cos(7 * x)
+ np.sqrt(x) / 4
)
# 0 < x < 100 を高密度にサンプリング
x = np.linspace(1e-8, 100, 2_000_000)
y = f(x)
# 極大点を検出
peaks, _ = find_peaks(y)
x_peak = x[peaks]
y_peak = y[peaks]
# 極大点数
print(f"極大点の個数: {len(x_peak)}")
# 原点 → 極大点列 → (100,0)
x_poly = np.concatenate(([0], x_peak, [100]))
y_poly = np.concatenate(([0], y_peak, [0]))
# 台形公式を手実装
area = 0.0
for i in range(len(x_poly) - 1):
width = x_poly[i + 1] - x_poly[i]
average_height = (y_poly[i] + y_poly[i + 1]) / 2
area += width * average_height
print(f"面積: {area:.15f}")
# 極大点一覧を出力
print("\n極大点一覧")
print("-" * 40)
for i, (xp, yp) in enumerate(zip(x_peak, y_peak), start=1):
print(f"{i:3d}: x = {xp:.10f}, y = {yp:.10f}")
# グラフ表示
plt.figure(figsize=(15, 6))
# Original function
plt.plot(
x,
y,
color="lightgray",
linewidth=0.5,
alpha=0.7,
label="Original Function"
)
# Polyline connecting local maxima
plt.plot(
x_poly,
y_poly,
color="green",
linewidth=0.5,
label="Polyline Connecting Local Maxima"
)
# Target area
plt.fill_between(
x_poly,
y_poly,
0,
color="green",
alpha=0.25,
label="Target Area"
)
# Local maxima
plt.scatter(
x_peak,
y_peak,
color="red",
s=4,
alpha=0.7,
label="Local Maxima"
)
plt.title(f"Area = {area:.6f}")
plt.xlabel("x")
plt.ylabel("y")
plt.xlim(0, 100)
plt.grid(True, alpha=0.3)
plt.legend()
plt.show()
注: 問1で導出した極大点の情報を使わずScipyのfind_peaks関数を利用して極大値の情報を取得
実行結果
極大点の個数: 430
面積: 267.814982927899564
極大点一覧
----------------------------------------
1: x = 0.1653500927, y = 0.1728509649
2: x = 0.3484001842, y = 0.1042525941
3: x = 0.5658502929, y = 0.5602949303
4: x = 0.7724003961, y = 0.0975598096
5: x = 0.9651504925, y = 0.3426650892
6: x = 1.2580006389, y = 0.5936497425
7: x = 1.4644507421, y = 0.2700745047
8: x = 1.6801008499, y = 0.9371035634
9: x = 1.8868009532, y = 0.0406725333
10: x = 2.0570010383, y = 0.3987012233
11: x = 2.3717011956, y = 0.9408624253
12: x = 2.5827013011, y = 0.3969222348
13: x = 2.7973514084, y = 1.1321246115
14: x = 3.1425515810, y = 0.4431468538
15: x = 3.4882517538, y = 1.2642871117
16: x = 3.7012518603, y = 0.4751744513
17: x = 3.9140519666, y = 1.2088050278
18: x = 4.2285521239, y = 0.5717271082
19: x = 4.3965522078, y = 0.0620866019
20: x = 4.6060523126, y = 1.5520568893
21: x = 4.8197524194, y = 0.4900143515
22: x = 5.0290525240, y = 1.1876364190
23: x = 5.3228026709, y = 0.8055387836
24: x = 5.5118027654, y = 0.2606882093
25: x = 5.7244528717, y = 1.7871995711
26: x = 5.9377529783, y = 0.4312467175
27: x = 6.1404530796, y = 1.0876782740
28: x = 6.4272032230, y = 1.1127903084
29: x = 6.6289533238, y = 0.4556562443
30: x = 6.8430534308, y = 1.9540444598
31: x = 7.0548035367, y = 0.2949284945
32: x = 7.2448036317, y = 0.9398049064
33: x = 7.5386037785, y = 1.4541006066
34: x = 7.7470038827, y = 0.6212516175
35: x = 7.9615539900, y = 2.0405802374
36: x = 8.1699040941, y = 0.0846353668
37: x = 8.3390041787, y = 0.8029063492
38: x = 8.6537043360, y = 1.7974842256
39: x = 8.8655544419, y = 0.7354256529
40: x = 9.0796045489, y = 2.0398693850
41: x = 9.4251047216, y = 0.7675014518
42: x = 9.7707548944, y = 2.1160902125
43: x = 9.9842550011, y = 0.7804474897
44: x = 10.1966551073, y = 1.9511645265
45: x = 10.5112552646, y = 0.9014414053
46: x = 10.6797053488, y = 0.0967660174
47: x = 10.8888054533, y = 2.3864238408
48: x = 11.1028055603, y = 0.7437336602
49: x = 11.3118056648, y = 1.7812275424
50: x = 11.6056058116, y = 1.1894933627
51: x = 11.7949559063, y = 0.3813499510
52: x = 12.0073060125, y = 2.5884379683
53: x = 12.2208561192, y = 0.6186825780
54: x = 12.4233062204, y = 1.5471341272
55: x = 12.7101063638, y = 1.5648912065
56: x = 12.9120564647, y = 0.6359380840
57: x = 13.1260065717, y = 2.7063334448
58: x = 13.3379066776, y = 0.4055273680
59: x = 13.5277567725, y = 1.2842288330
60: x = 13.8215569194, y = 1.9689407979
61: x = 14.0301570237, y = 0.8360507560
62: x = 14.2445571309, y = 2.7295004569
63: x = 14.4530572351, y = 0.1125702158
64: x = 14.6220073195, y = 1.0632002191
65: x = 14.9367074769, y = 2.3615378888
66: x = 15.1487075828, y = 0.9613339012
67: x = 15.3626076898, y = 2.6534133854
68: x = 15.7081578625, y = 0.9908348100
69: x = 16.0538080353, y = 2.7124486074
70: x = 16.2674081421, y = 0.9961980144
71: x = 16.4797082482, y = 2.4805165768
72: x = 16.7943584055, y = 1.1394448911
73: x = 16.9628584897, y = 0.1219531973
74: x = 17.1718585942, y = 2.9968725670
75: x = 17.3859587012, y = 0.9306813620
76: x = 17.5948588057, y = 2.2215099443
77: x = 17.8886589525, y = 1.4767914002
78: x = 18.0781090472, y = 0.4721203197
79: x = 18.2904091534, y = 3.1946837860
80: x = 18.5040092602, y = 0.7612906491
81: x = 18.7063593613, y = 1.8984797888
82: x = 18.9931595047, y = 1.9129783924
83: x = 19.1952096057, y = 0.7753790106
84: x = 19.4091097126, y = 3.2909326285
85: x = 19.6211098186, y = 0.4918558569
86: x = 19.8108599134, y = 1.5541118257
87: x = 20.1046600603, y = 2.3746721700
88: x = 20.3133101646, y = 1.0059867561
89: x = 20.5276602718, y = 3.2766462791
90: x = 20.7362603761, y = 0.1348369285
91: x = 20.9051104605, y = 1.2712728514
92: x = 21.2198106178, y = 2.8147456668
93: x = 21.4318607238, y = 1.1434491677
94: x = 21.6457108307, y = 3.1496286048
95: x = 21.9912610034, y = 1.1723697770
96: x = 22.3369111762, y = 3.1995212979
97: x = 22.5505612830, y = 1.1729125064
98: x = 22.7628113891, y = 2.9152868034
99: x = 23.0774615464, y = 1.3356928281
100: x = 23.2460616307, y = 0.1427638482
101: x = 23.4550117352, y = 3.5024974124
102: x = 23.6691118422, y = 1.0859081765
103: x = 23.8780119466, y = 2.5879434181
104: x = 24.1717620935, y = 1.7166637998
105: x = 24.3612621882, y = 0.5480578724
106: x = 24.5735622943, y = 3.7029733394
107: x = 24.7871624011, y = 0.8811122917
108: x = 24.9895125023, y = 2.1942725478
109: x = 25.2763126456, y = 2.2068287568
110: x = 25.4783627466, y = 0.8933129414
111: x = 25.6922128535, y = 3.7863235940
112: x = 25.9042629595, y = 0.5651478713
113: x = 26.0940130544, y = 1.7836159880
114: x = 26.3877632012, y = 2.7205538928
115: x = 26.5964633056, y = 1.1511030274
116: x = 26.8108134127, y = 3.7446839403
117: x = 27.0194635170, y = 0.1539152582
118: x = 27.1882636014, y = 1.4497847843
119: x = 27.5029637587, y = 3.2044862673
120: x = 27.7150138647, y = 1.3003035402
121: x = 27.9288639716, y = 3.5776704208
122: x = 28.2744141444, y = 1.3293415556
123: x = 28.6200643172, y = 3.6216711983
124: x = 28.8337144240, y = 1.3262869573
125: x = 29.0459645301, y = 3.2931499863
126: x = 29.3606146874, y = 1.5065911142
127: x = 29.5292147717, y = 0.1609049812
128: x = 29.7381648761, y = 3.9438227575
129: x = 29.9522649831, y = 1.2215667053
130: x = 30.1611650876, y = 2.9085733462
131: x = 30.4549152344, y = 1.9269037022
132: x = 30.6444653292, y = 0.6146847027
133: x = 30.8567154353, y = 4.1494608485
134: x = 31.0703655421, y = 0.9864859629
135: x = 31.2726656432, y = 2.4546773133
136: x = 31.5594657866, y = 2.4659079401
137: x = 31.7615658876, y = 0.9973984900
138: x = 31.9754159945, y = 4.2240099900
139: x = 32.1874661005, y = 0.6299697288
140: x = 32.3771661953, y = 1.9867837288
141: x = 32.6709663422, y = 3.0271706466
142: x = 32.8796664465, y = 1.2798697950
143: x = 33.0939665537, y = 4.1603978376
144: x = 33.3026166580, y = 0.1708766968
145: x = 33.4714167424, y = 1.6086073287
146: x = 33.7861168997, y = 3.5517141644
147: x = 33.9982170057, y = 1.4401745703
148: x = 34.2120171126, y = 3.9597088157
149: x = 34.5576172854, y = 1.4696419934
150: x = 34.9032174581, y = 3.9995088128
151: x = 35.1169175649, y = 1.4636770841
152: x = 35.3291176710, y = 3.6319111865
153: x = 35.6437678283, y = 1.6599875020
154: x = 35.8124179126, y = 0.1771986851
155: x = 36.0213180171, y = 4.3405059469
156: x = 36.2354681241, y = 1.3435975011
157: x = 36.4443182285, y = 3.1972094664
158: x = 36.7381183754, y = 2.1163605305
159: x = 36.9276184701, y = 0.6747644962
160: x = 37.1398685762, y = 4.5523667322
161: x = 37.3535186830, y = 1.0816422988
162: x = 37.5558187842, y = 2.6899908301
163: x = 37.8426189275, y = 2.7002429169
164: x = 38.0447190286, y = 1.0916039494
165: x = 38.2585691355, y = 4.6204197352
166: x = 38.4706192415, y = 0.6887175422
167: x = 38.6603193363, y = 2.1710213539
168: x = 38.9541194832, y = 3.3054670551
169: x = 39.1628195875, y = 1.3968163643
170: x = 39.3771196946, y = 4.5381896322
171: x = 39.5858197990, y = 0.1863002407
172: x = 39.7546198833, y = 1.7531000374
173: x = 40.0693200407, y = 3.8678956544
174: x = 40.2813701467, y = 1.5676148656
175: x = 40.4952202536, y = 4.3079998646
176: x = 40.8407704263, y = 1.5976690169
177: x = 41.1864205991, y = 4.3446104522
178: x = 41.4000707059, y = 1.5892338304
179: x = 41.6122708120, y = 3.9416647949
180: x = 41.9269709693, y = 1.8003612828
181: x = 42.0956210536, y = 0.1921153003
182: x = 42.3044711580, y = 4.7038545002
183: x = 42.5186212651, y = 1.4554321978
184: x = 42.7274713695, y = 3.4618633907
185: x = 43.0212715163, y = 2.2901979084
186: x = 43.2108216111, y = 0.7299158065
187: x = 43.4230217172, y = 4.9224040036
188: x = 43.6367218240, y = 1.1690790910
189: x = 43.8389719251, y = 2.9063139773
190: x = 44.1257720685, y = 2.9158055054
191: x = 44.3279221695, y = 1.1783017912
192: x = 44.5417222764, y = 4.9854084266
193: x = 44.7538223824, y = 0.7428335556
194: x = 44.9434724772, y = 2.3408028045
195: x = 45.2372726241, y = 3.5620868999
196: x = 45.4460227285, y = 1.5047008445
197: x = 45.6602728356, y = 4.8868619749
198: x = 45.8689729399, y = 0.2005409166
199: x = 46.0377730243, y = 1.8865582256
200: x = 46.3524731816, y = 4.1601155466
201: x = 46.5645732876, y = 1.6854465641
202: x = 46.7783733945, y = 4.6301658223
203: x = 47.1239735673, y = 1.7161715754
204: x = 47.4695737400, y = 4.6642483519
205: x = 47.6832738469, y = 1.7055726200
206: x = 47.8954739529, y = 4.2287901932
207: x = 48.2101241102, y = 1.9305550962
208: x = 48.3787741945, y = 0.2059543744
209: x = 48.5876242990, y = 5.0410813353
210: x = 48.8018244060, y = 1.5592665634
211: x = 49.0106745104, y = 3.7076744225
212: x = 49.3044246573, y = 2.4517403522
213: x = 49.4940247521, y = 0.7811828968
214: x = 49.7062248581, y = 5.2665054161
215: x = 49.9199249650, y = 1.2504163899
216: x = 50.1221750661, y = 3.1076150672
217: x = 50.4089752094, y = 3.1164936849
218: x = 50.6111253105, y = 1.2590434553
219: x = 50.8249254174, y = 5.3254403892
220: x = 51.0369755234, y = 0.7932663609
221: x = 51.2266756182, y = 2.4990763981
222: x = 51.5204757651, y = 3.8014225856
223: x = 51.7291758694, y = 1.6053516029
224: x = 51.9434759765, y = 5.2122621151
225: x = 52.1521760809, y = 0.2138355410
226: x = 52.3209761653, y = 2.0111797923
227: x = 52.6356763226, y = 4.4331145341
228: x = 52.8477264286, y = 1.7955621186
229: x = 53.0615265355, y = 4.9313290853
230: x = 53.4071267082, y = 1.8270040849
231: x = 53.7527268810, y = 4.9633439795
232: x = 53.9664269878, y = 1.8144671642
233: x = 54.1786270939, y = 4.4976227901
234: x = 54.4933272512, y = 2.0525070677
235: x = 54.6619773355, y = 0.2189204495
236: x = 54.8708274399, y = 5.3571223298
237: x = 55.0850275470, y = 1.6566051508
238: x = 55.2938276514, y = 3.9381722554
239: x = 55.5876277983, y = 2.6032778935
240: x = 55.7771778930, y = 0.8292869601
241: x = 55.9893779991, y = 5.5894633640
242: x = 56.2030781059, y = 1.3267770360
243: x = 56.4053282070, y = 3.2966471985
244: x = 56.6921283504, y = 3.3050179489
245: x = 56.8942784514, y = 1.3349106536
246: x = 57.1080785583, y = 5.6450276257
247: x = 57.3201786644, y = 0.8406792158
248: x = 57.5098287592, y = 2.6479062070
249: x = 57.8036289060, y = 4.0265575588
250: x = 58.0123790104, y = 1.7000537735
251: x = 58.2266291175, y = 5.5185082445
252: x = 58.4353792218, y = 0.2263505103
253: x = 58.6041293062, y = 2.1285174729
254: x = 58.9188294635, y = 4.6902506424
255: x = 59.1309295696, y = 1.8993045020
256: x = 59.3447296764, y = 5.2151298886
257: x = 59.6903298492, y = 1.9314871458
258: x = 60.0359300220, y = 5.2454129546
259: x = 60.2496301288, y = 1.9171868517
260: x = 60.4618302349, y = 4.7512689085
261: x = 60.7764803922, y = 2.1676087516
262: x = 60.9451804765, y = 0.2311602232
263: x = 61.1539805809, y = 5.6555295399
264: x = 61.3681806880, y = 1.7485337120
265: x = 61.5770307924, y = 4.1559057314
266: x = 61.8707809392, y = 2.7464668864
267: x = 62.0603810340, y = 0.8747494953
268: x = 62.2725811401, y = 5.8947535403
269: x = 62.4862812469, y = 1.3989756408
270: x = 62.6885313480, y = 3.4754126876
271: x = 62.9753314914, y = 3.4833539850
272: x = 63.1774815924, y = 1.4066918978
273: x = 63.3912816993, y = 5.9474663181
274: x = 63.6033818053, y = 0.8855569941
275: x = 63.7930319001, y = 2.7888048576
276: x = 64.0868320470, y = 4.2397543087
277: x = 64.2955321513, y = 1.7897517373
278: x = 64.5098322585, y = 5.8086306774
279: x = 64.7185323628, y = 0.2382088424
280: x = 64.8873324472, y = 2.2397162737
281: x = 65.2020326045, y = 4.9340039037
282: x = 65.4141327105, y = 1.9976663101
283: x = 65.6278828174, y = 5.4842639740
284: x = 65.9734829901, y = 2.0306013210
285: x = 66.3190831629, y = 5.5130688547
286: x = 66.5328332698, y = 2.0146758006
287: x = 66.7449833758, y = 4.9920437727
288: x = 67.0596835331, y = 2.2768992392
289: x = 67.2283336174, y = 0.2427838548
290: x = 67.4371837218, y = 5.9389623753
291: x = 67.6513838289, y = 1.8358646589
292: x = 67.8601839333, y = 4.3627862599
293: x = 68.1539840802, y = 2.8825519288
294: x = 68.3435341749, y = 0.9179632235
295: x = 68.5557342810, y = 6.1849932762
296: x = 68.7694343878, y = 1.4676268103
297: x = 68.9716844889, y = 3.6454224929
298: x = 69.2584846323, y = 3.6529941058
299: x = 69.4606347334, y = 1.4749839070
300: x = 69.6744348402, y = 6.2352528502
301: x = 69.8865349463, y = 0.9282678924
302: x = 70.0761850411, y = 2.9229192505
303: x = 70.3699851880, y = 4.4427321834
304: x = 70.5787352923, y = 1.8751642481
305: x = 70.7929853994, y = 6.0849361108
306: x = 71.0017355038, y = 0.2495043687
307: x = 71.1704855881, y = 2.3456494701
308: x = 71.4851857454, y = 5.1662696343
309: x = 71.6972858515, y = 2.0914073768
310: x = 71.9110859584, y = 5.7407948576
311: x = 72.2566861311, y = 2.1250977813
312: x = 72.6022863039, y = 5.7683188994
313: x = 72.8159864107, y = 2.1076604572
314: x = 73.0281865168, y = 5.2217283864
315: x = 73.3428366741, y = 2.3811788017
316: x = 73.5115367584, y = 0.2538758501
317: x = 73.7203868628, y = 6.2094709137
318: x = 73.9345369699, y = 1.9192258034
319: x = 74.1433870743, y = 4.5602912514
320: x = 74.4371372211, y = 3.0124956513
321: x = 74.6267373159, y = 0.9592322823
322: x = 74.8389374220, y = 6.4622102791
323: x = 75.0526375288, y = 1.5332072500
324: x = 75.2548876299, y = 3.8078492780
325: x = 75.5416877733, y = 3.8150986522
326: x = 75.7438378743, y = 1.5402511688
327: x = 75.9576379812, y = 6.5103301107
328: x = 76.1697380873, y = 0.9690981317
329: x = 76.3593881821, y = 3.0511444641
330: x = 76.6531883289, y = 4.6368330437
331: x = 76.8619384333, y = 1.9568519403
332: x = 77.0761885404, y = 6.3492287727
333: x = 77.2849386447, y = 0.2603100383
334: x = 77.4536887291, y = 2.4470010315
335: x = 77.7683888864, y = 5.3885328419
336: x = 77.9804889924, y = 2.1811232532
337: x = 78.1942390993, y = 5.9863424711
338: x = 78.5398392721, y = 2.2155675392
339: x = 78.8854894449, y = 6.0127425825
340: x = 79.0991895517, y = 2.1967126619
341: x = 79.3113396577, y = 5.4417268770
342: x = 79.6260398151, y = 2.4810794455
343: x = 79.7947398994, y = 0.2645029189
344: x = 80.0035400038, y = 6.4686775600
345: x = 80.2177401108, y = 1.9991141305
346: x = 80.4265402152, y = 4.7495902239
347: x = 80.7203403621, y = 3.1370616357
348: x = 80.9099404569, y = 0.9987973789
349: x = 81.1220905629, y = 6.7280148174
350: x = 81.3358406698, y = 1.5960951149
351: x = 81.5380407709, y = 3.9636255831
352: x = 81.8248409142, y = 3.9705904166
353: x = 82.0270410153, y = 1.6028627331
354: x = 82.2407911222, y = 6.7742467946
355: x = 82.4528912282, y = 1.0082761343
356: x = 82.6425913230, y = 3.1741939138
357: x = 82.9363414699, y = 4.8231289605
358: x = 83.1450915742, y = 2.0352637831
359: x = 83.3593416813, y = 6.6029509728
360: x = 83.5680917857, y = 0.2706847820
361: x = 83.7368418700, y = 2.5443184097
362: x = 84.0515420274, y = 5.6019847506
363: x = 84.2636421334, y = 2.2672916449
364: x = 84.4774422403, y = 6.2222079781
365: x = 84.8230424130, y = 2.3024853247
366: x = 85.1686425858, y = 6.2476114896
367: x = 85.3823426926, y = 2.2822925115
368: x = 85.5945427987, y = 5.6531707076
369: x = 85.9091929560, y = 2.5771102333
370: x = 86.0778930403, y = 0.2747194891
371: x = 86.2867431447, y = 6.7178899972
372: x = 86.5009432518, y = 2.0759300747
373: x = 86.7097433562, y = 4.9316285972
374: x = 87.0035435031, y = 3.2568666239
375: x = 87.1930935978, y = 1.0368540193
376: x = 87.4052937039, y = 6.9837099949
377: x = 87.6189938107, y = 1.6565976534
378: x = 87.8212439118, y = 4.1135069413
379: x = 88.1080440552, y = 4.1202185304
380: x = 88.3101941563, y = 1.6631190212
381: x = 88.5239942631, y = 7.0282602252
382: x = 88.7360943692, y = 1.0459880618
383: x = 88.9257444640, y = 3.2926482679
384: x = 89.2195446109, y = 5.0024919144
385: x = 89.4282947152, y = 2.1107647969
386: x = 89.6425448223, y = 6.8472784554
387: x = 89.8512949267, y = 0.2806763562
388: x = 90.0200450110, y = 2.6380483618
389: x = 90.3347451683, y = 5.8075966389
390: x = 90.5468452744, y = 2.3503034329
391: x = 90.7606453812, y = 6.4494529711
392: x = 91.1061955540, y = 2.3862391228
393: x = 91.4518457268, y = 6.4739648915
394: x = 91.6655458336, y = 2.3647777997
395: x = 91.8777459397, y = 5.8569858356
396: x = 92.1923960970, y = 2.6696891744
397: x = 92.3610961813, y = 0.2845693648
398: x = 92.5698962857, y = 6.9581825501
399: x = 92.7840963928, y = 2.1500035047
400: x = 92.9929464972, y = 5.1071823198
401: x = 93.2866966440, y = 3.3724183894
402: x = 93.4762967388, y = 1.0735624067
403: x = 93.6884468449, y = 7.2303681917
404: x = 93.9021969517, y = 1.7149669225
405: x = 94.1043970528, y = 4.2581158053
406: x = 94.3912471962, y = 4.2645997925
407: x = 94.5933972972, y = 1.7212672049
408: x = 94.8071474041, y = 7.2734077499
409: x = 95.0192975101, y = 1.0823866110
410: x = 95.2089476050, y = 3.4069865807
411: x = 95.5026977518, y = 5.1756426048
412: x = 95.7114978562, y = 2.1836565607
413: x = 95.9257479633, y = 7.0831827313
414: x = 96.1344980676, y = 0.2903240679
415: x = 96.3032481520, y = 2.7285602956
416: x = 96.6178983093, y = 6.0061738213
417: x = 96.8300484153, y = 2.4304813298
418: x = 97.0437985222, y = 6.6689593181
419: x = 97.3893986950, y = 2.4671515365
420: x = 97.7349988677, y = 6.6926672797
421: x = 97.9487489746, y = 2.4444810118
422: x = 98.1608990806, y = 6.0539433934
423: x = 98.4755992380, y = 2.7591633811
424: x = 98.6442493223, y = 0.2940893899
425: x = 98.8530994267, y = 7.1904494626
426: x = 99.0672995337, y = 2.2216085045
427: x = 99.2760996381, y = 5.2768991716
428: x = 99.5698997850, y = 3.4841398975
429: x = 99.7594998798, y = 1.1090561959
430: x = 99.9716499858, y = 7.4688853869
