if if-else if-elif-else

Transcription

1 1 Python SageMath ( ) I Python 1 1 Python (Hello World) Python Python (print) Python (dictionary) (set) e π

2 if if-else if-elif-else if BMI For For for for for (for for ) for 2 if for 3: break ( ) n while while while lambda def

3 Set Set II SageMath SageMath Sage Sage Sage Sage Web Sage SageMathCloud Windows Sage Mac Sage Sage Sage n Sage Sage Sage Sage Sage

4 Sage plot (x, y) ( ) density plot contour plot x == solve find root Sage Taylor Taylor Taylor

5 Euler Riccati, Clairaut, Lagrange (ics) (animate)

6 Python SageMath Sage Sage 100 *1 Python 2 3 Sage Python Python Sage *2 Sage Python2 Python2 Python Sage Windows Mac Linux * * SageMath ver.8.1 6

7 1 PYTHON I Python 1 Python Python 3 (1) Python (2) (3) Sage (3) Sage 1.1 (Hello World) (1) T;GV6U1M"s e{>wn"iôö Ñ c/2y ÿ ***.py ƒžûœé œal Wÿ python ***.py 1 Python (1) Python Emacs (hello.py) Emacs 1 user@debian :~$ cd ~/ Desktop / dataproc1 / # 2 user@debian :~/ Desktop / dataproc1$ emacs hello.py & # E m a c s Emacs C-x C-s dataproc1 hello.py Emacs Python hello.py hello.py 1 print Hello World (C-x C-s) *3 Python *3 Alt+Tab 1

8 1.2 Python 1 PYTHON 1 user@ debian :~/ Desktop / dataproc1$ python hello. py 2 Hello World 3 user@ debian :~/ Desktop / dataproc1$ Hello World Python C 1.2 Python Python utf-8 1 # -*- coding : utf -8 -*- 1 # coding : utf -8 Emacs Python hellojp.py 1 print python hellojp.py sys:1: DeprecationWarning: Non-ASCII character xe3 in file test.py on line 1, but no encoding declared; see for details hellojp.py 1 # -*- coding : utf -8 -*- 2 print python hellojp.py 1.3 Python # *4 1 print Hello World # *4 LATEX % 2

9 1.4 Python Python Python python [ENTER] 1 user@ debian :~ $ python 2 Python ( default, Jun , 13: 08: 31) 3 [ GCC ] on linux2 4 Type " help ", " copyright ", " credits " or " license " for more information. 5 >>> 1 >>> >>> >>> 4* >>> 9/4 # >>> 9%4 # >>> 9.0/4 # # 13 >>> 2**10 # Python2 (/) 2^10 Python (XOR) **10 Python3 / Sage ^ 5/3 5 3 CRTL+d exit() 1 >>> exit () # ctrl +d 2 user@debian :~$ 2 1 >>> a = 6 # a 6 2 >>> a # a >>> a = 8 # a 8 5 >>> a 3

10 >>> a = a + 5 # a 5 8 >>> a >>> a += 1 # a 1 11 >>> a a=a+5 a a+5 = 1 >>> b 2 Traceback ( most recent call last ): # 3 File "<stdin >", line 1, in? # 4 NameError : name b is not defined # 5 >>> b = -4 6 >>> a+b 7 10 a z, A Z, 0 9, _ 3 Python Python print and for if elif else del is raise assert import from lambda return break global not try class except or while continue exec pass yield def finally in 4 Python 65, 3, 9.23 Hello >>> x = hello world 2 >>> x 3 hello world 4

11 4.2 4 x hello world >>> aa = Alice # a a A l i c e 2 >>> aa 3 Alice 4 >>> aa + Bob 5 AliceBob 6 >>> aa + and Bob 7 Alice and Bob " " 1 >>> a = " This is a pen " 2 >>> print a 3 This is a pen # \n 1 >>> a = aaa \ nbbb \ nccc 2 >>> a 3 aaa \ nbbb \ nccc 4 >>> print a 5 aaa 6 bbb 7 ccc \n """ """ 1 >>> a = aaa 2... bbb 3... ccc 4 >>> a 5 aaa \ nbbb \ nccc 6 >>> print a 7 aaa 8 bbb 9 ccc \,, \, ", \ \\ \ \" " \ \n \t 4.2 str (string) 5

12 >>> type ( hello ) # type ( hello ) 2 <type str > # s t r (int) (float) 1 >>> type (123) # <type int > # 3 >>> type (3.14) # <type float > # float 5 >>> a = 3.14 # a >>> type ( a) 7 <type float > # a (3.14) float 4.3 int (integer) float (floating point number) Python 1 >>> aa = 10 # int 2 >>> bb = 3.14 # float 3 >>> cc = aa + bb 4 >>> cc >>> type (cc) 7 <type float > int (PC ) long 1 >>> dd = 2**62 # d d >>> dd >>> type (dd) 5 <type int > # i n t 6 >>> ee = 2**65 # e e >>> ee L # L L o n g 9 >>> type (ee) 10 <type long > # long int long CPU Python3 int python2 long Python3 6

13 >>> Alice Traceback ( most recent call last ): # 3 File "<stdin >", line 1, in? # 4 TypeError : cannot concatenate str and int objects # str int str() str(123)= 123 int() int( 123 )=123) float() float( )= >>> Alice + str (1999) # Alice 2 Alice >>> aa = 123 # >>> int (aa) # >>> float (aa) # Python (operator) + (Addition) (Subtraction) * (Multiplication) 4*5 20 / (Division) 14/ / % (Modulus) 8%5 3 ** (Power) 2**3 2 3 = 8 5*4/3**2 7

14 6 (PRINT) 1 5*4/3**2 = 5*4/(3**2) # 2 = 5*4/9 3 = (5*4)/9 # 4 = 20/9 5 = 2 # 1 3**3**3 = L 2 (3**3)**3 = **(3**3) = L ** Python / 1 5.0*4/3**2 = 5.0*4/9 2 = 20.0/9 3 = (print) Python print print01.py 1 a = 6 2 print 6 3 b = 4 4 a + b print 1 print 4 2 print print, 1 print 4, 2 print print 8

15 7 7 1 >>> a = [ Alice, Bob, 2,3,8] # a 2 >>> a # a 3 [ Alice, Bob, 2,3,8] 4 >>> type (a) # a 5 <type list > # a 1 >>> a [0] # a 1 2 Alice 3 >>> a[ -1] # a >>> a [3] # a >>> a[ -2] # a >>> a [8] 2 Traceback ( most recent call last ): 3 File " <stdin >", line 1, in? 4 IndexError : list index out of range a[:n] n, a[-n:] n, a[n:] n, a[:-n] n, a[n:m] n m 1 >>> a = [ a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p ] 2 >>> print a [:5] 3 [ a, b, c, d, e ] 4 >>> print a [ -5:] 5 [ l, m, n, o, p ] 1 >>> a = [1,2,3] 2 >>> b = [ a, b, c ] 3 >>> a + b 4 [1, 2, 3, a, b, c ] append() 9

16 8 PYTHON 1 >>> a. append ( Alice ) 2 >>> a 3 [1,2,3, Alice ] 1 >>> a = [1,2,3] 2 >>> a [0] = Alice #a [0] A l i c e 3 >>> a 4 [ Alice, 2, 3] pop(), remove() 1 >>> aa = [ a, b, c, d, e, b, 23, 8, 13] 2 >>> aa.pop () # a a >>> aa 5 [ a, b, c, d, e, b, 23, 8, 12] 6 >>> aa.pop (3) # a a 3 7 d 8 >>> aa 9 [ a, b, c, e, b, 23, 8] 10 >>> aa. remove ( b ) # a a b 11 >>> aa 12 [ a, c, e, b, 23, 8] # b 1 >>> range (5) 2 [0, 1, 2, 3, 4] 3 >>> range (3,6) 4 [3, 4, 5] 5 >>> range ( -3,3) 6 [-3, -2, -1, 0, 1, 2] 8 Python 8.1 (list) (tuple) 1 >>> a = (3,7, abc ) # ( tuple ) 2 >>> print a 3 (3, 7, abc ) 4 >>> type ( a) 5 <type tuple > # a Immutable( ) Immutable 10

17 8.2 (dictionary) 8 PYTHON 8.2 (dictionary) (key) (value) 1 >>> a = { birth :1534, type : A, name : Nobunaga, death :1582} 1 >>> print a[ birth ] keys() values() 1 >>> b = a. keys () # a b 2 >>> print b 3 [ death, type, name, birth ] 4 >>> c = a. values () # a c 5 >>> print c 6 [1582, A, Nobunaga, 1534] del 1 >>> del a[ type ] # t y p e 2 >>> print a 3 { death : 1582, name : Nobunaga, birth : 1534} 1 >>> a[ hobby ] = tea 2 >>> print a 3 { hobby : tea, death : 1582, name : Nobunaga, birth : 1534} immutable 1 >>> a = {(1,1,0): police, (1,1,9): fire, (1,7,7): weather } 2 print a 3 {(1, 1, 0): police, (1, 1, 9): fire, (1, 7, 7): weather } 1 >>> a = [(1, One ), (2, Two ), (3, Three )] 2 >>> b = dict ( a) 3 >>> print b 8.3 (set) set immutable set 1 >>> a = [1,4,3,2,2,2] 2 >>> b = set ( a) 3 >>> print b 4 set ([1, 2, 3, 4]) 5 >>> b. add (5) # b 5 11

18 9 6 >>> print (b) 7 set ([1, 2, 3, 4, 5]) e π 20 pp = ee = ee pp 20 (print) Python problem01.py python problem01.py e π 20 12

19 10 10 raw_input() input() *5 input1.py 1 # -*- coding : utf -8 -*- 2 3 namae = raw_input ( ) 4 print, namae, 1 user@ debian :~/ Desktop / dataproc1$ python input1. py 2 Enter namae 3. namae 4. raw_input() namae input() input2.py 1 # -*- coding : utf -8 -*- 2 3 num = input ( ) 4 print, num **2, 1 user@ debian :~/ Desktop / dataproc1$ python input2. py BMI bmi1.py 1 # -*- coding : utf -8 -*- 2 3 namae = raw_input ( ) 4 shintyo = input ( ) *5 Python3 raw input() input() 13

20 11 5 weight = input ( ) 6 bmi = 10000* weight /( shintyo **2) 7 print namae,, bmi, BMI 11 Python Python True False 1 >>> a = 3 # a 3 2 >>> a == 3 # a 3 3 True #a ==3 4 >>> a == 2 # a 2 5 False #a ==2 6 >>> a > 2 7 True 8 >>> 3!= 2 #3 2 9 True #3 2 ==, >,!= ==!= < > <= >= =,, <, >,, less than or equal!= not equal True False and, or, not 1 >>> a = True # a T r u e 2 >>> b = False # b F a l s e 3 >>> a and b # a b 4 False 5 >>> a or b # a b 6 True 7 >>> not a # a 8 False ==!= in ==!= in 14

21 12 1 >>> a = [3,6,5,1] # a [3,6,5,1] 2 >>> 5 in a #5 a 3 True 4 >>> 7 in a #7 a 5 False type 1 >>> a = [3,5,6,1] 2 >>> type ( a) 3 <type list > 4 >>> type ( a) == list 5 True 6 >>> type ( a) == tuple 7 False if (True False) if if A A X A Python 15

22 12.2 if-else 12 Python if 1 if A: 2 [ X ] 3 [ X ] 4 [ X ] A True X False X if X X X X if A: X if Python (1) a (2) a a is even a (3) a if1.py 1 # -*- coding : utf -8 -*- 2 a = input ( Input integer a: ) 3 4 if a%2 == 0: # a 5 print a is even 6 a = a/2 # a 7 8 print a # a 4 6 if 12.2 if-else X Y 16

23 12.2 if-else 12 if A: A A X A else: X Y Y if 1 if A: # A 2 X # X 3 if not A: # A 4 Y # Y if-else Python if-else 1 if A: 2 X 3 else : 4 Y A True X False Y X, Y if-else (1) a (2) a a (3) a a 3*a+1 (4) a if2.py 1 # -*- coding : utf -8 -*- 2 a = input ( a?: ) 3 4 if a%2 == 0: # a 5 a = a/2 # a 6 else : # 17

24 12.3 if-elif-else 12 7 a = 3*a + 1 # a 3 a print a # a 12.3 if-elif-else if-elif-else elif else if if-elif-else 1 if A: 2 X 3 elif B: 4 Y 5 else : 6 Z A X A B True Y A B Z C D elif if A: A A X A B B elif B: B Y X Y Z else: Z elif if-elif-elif-elif-else 12.4 if if-elif-else

25 Python uruuq.py 1 # -*- coding : utf -8 -*- 2 year = input ( : ) # y e a r 3 4 if year % 400 == 0: # y e a r print year, 6 elif year % 100 == 0: # y e a r print year, 8 elif year % 4 == 0: # y e a r 4 9 print year, 10 else : # 11 print year, BMI BMI (1) namae (2) shintyo, weight (3) bmi (4) bmi<18.5 (5) bmi<25.0 (6) bmi<30 (7) ***** bmi2.py 1 # -*- coding : utf -8 -*- 2 3 namae = raw_input ( ) 4 shintyo = input ( ) 5 weight = input ( ) 6 7 bmi = * weight /( shintyo **2) 19

26 13.1 BMI 13 8 bmi = round (bmi,1) # b m i print namae,, bmi,, if **********: 13 ***** 14 elif **********: 15 ***** 16 ***************: 17 ***** 18 ****: 19 ***** 20

27 14 FOR 14 For 14.1 For for hello 1 >>> print hello 2 hello 3 >>> print hello 4 hello 5 >>> print hello 6 hello 7 >>> print hello 8 hello 9 >>> print hello 10 hello for 1 for j in range (5): 2 print hello Python >>> for j in range(5):... print hello... hello hello hello hello hello range(5) [0,1,2,3,4] *6 1 >>> range (5) 2 [0,1,2,3,4] for for j in range(5): j 0 4 j 1 for j in range (5): 2 print j j 1 for x in range (5): 2 print x *6 Python3.x range list(range(5)) [0,1,2,3,4] 21

28 14.2 for 14 FOR for For 10 1 for j in range (10): < > 6 for for for j in range(10): X j for Start j=0 j=1 j=2 j=3 j=4 X X X X X End 3 for (5 ) 14.2 for for for1.py 1 for j in range (5): # 5 2 print wanwan, # 22

29 14.2 for 14 FOR 3 print nyannyan # wanwan nyannyan wanwan nyannyan wanwan nyannyan wanwan nyannyan wanwan nyannyan for for2.py 1 for j in range (5): # 5 2 print wanwan, # 3 print nyannyan # 4 5 print gaogao # wanwan nyannyan wanwan nyannyan wanwan nyannyan wanwan nyannyan wanwan nyannyan gaogao for for3.py 1 for j in range (10): # j print j, j **2 # j j 3 4 print owari # owari for4.py 1 # -*- coding : utf -8 -*- 2 for j in range (1,101): 3 print + str (j) for5.py 1 aa = [ Alice, falls, down, a, rabbit, hole. ] # a a 23

30 14.3 for 14 FOR 2 3 for i in aa: # a a i 4 print i, # i Alice falls down a rabbit hole for Python if-else for for6.py 1 for j in range (5): 2 print wanwan, 3 print nyannyan File "for6.py", line 3 print nyannyan ^ IndentationError: unindent does not match any outer indentation level for7.py 1 for j in range (5): 2 print wanwan, 3 print nyannyan File "for7.py", line 3 print nyannyan ^ IndentationError: unindent does not match any outer indentation level 14.4 for (for for ) for for for8.py 1 for i in range (5): # i = 0 ~ 5 2 for j in [ a, b, c ]: # j = a, b, c 3 print i, j, # i j 0 a 0 b 0 c 1 a 1 b 1 c 2 a 2 b 2 c 3 a 3 b 3 c 4 a 4 b 4 c for9.py 1 for i in range (1,10): # i =1,2,...,9 2 for j in range (1,10): # j =1,2,...,9 3 print i*j, # i* j 4 print # 24

31 14.5 for 2 if 14 FOR for 2 if if for j j is even j is odd for10.py 1 for j in range (1,10): 2 if j%2 == 0: 3 print j, is even 4 else : 5 print j, is odd 1 is odd 2 is even 3 is odd 4 is even 5 is odd 6 is even 7 is odd 8 is even 9 is odd 14.6 for 3: break for break for break for 2 2j j 1, 2, 3,, j for for11.py 1 for j in range (11): 2 a = 2**2** j 3 if a < : 4 print a 5 else : 6 print Too big!! 7 break # f o r Too big!! 25

32 15 15 Python 15.1 ( ) Python print 1 user@ debian :~/ Desktop / dataproc1 /06 $ 1 $ python. py >. txt 1 $ python for4. py > for4result. txt for4result.txt for4.py >> 1 $ python for3. py >> for4result. txt for4result.txt for3.py Python write1.py 1 # -*- coding : utf -8 -*- 2 abc = hogehoge # a b c 3 f = open ( test. txt, a ) # ( a) 4 f. write ( abc ) # abc text. t x t ( ) 5 f. close () # 1 $ python write1. py test.txt hogehoge 3 1 f = open ( test. txt, w ) # ( w) CountSheep.py 1 # -*- coding : utf -8 -*- 2 3 abc = # a b c 26

33 for i in range (10): 6 abc = abc + + str (i)+ \n # \ n 7 8 f = open ( hitsuji. txt, w ) # ( hitsuji. txt ) w r i t e 9 f. write ( abc ) 10 f. close () hitsuji.txt 15.3 read readlines 1 $ python for11. py > hoge. txt hoge.txt for11.py read read1.py 1 # -*- coding : utf -8 -*- 2 3 f = open ( hoge. txt, r ) # 4 aaa = f. read () # hoge. t x t a a a 5 print aaa # a a a 6 7 f. close () # hoge. t x t Too big!! readlines read2.py 1 # -*- coding : utf -8 -*- 2 3 f = open ( hoge. txt, r ) # 4 aaa = f. readlines () # hoge. t x t a a a 5 print aaa # a a a 6 7 f. close () # hoge. t x t [ \n, 4\n, 16\n, 256\n, 65536\n, \n, Too big!!\n,] aaa hoge.txt \n 27

34 n + 1 3n + 1 n n n n n 3n Collatz a 1. a 2. maxiter for maxiter 4. a 1 break for 5. a a 6. a a 3a a Python **** collatz.py 1 a = input ( a?: ) 2 print a, 3 4 maxiter = for i in range ( maxiter ): 6 *********** 7 break 8 ************** 9 ******* 10 ***** 11 *********** 12 print a, a 19 a?: while (1000 ) while 28

35 x x 8 1 import random # r a n d o m 2 x = random. randint (1,10) # x 3 print x (i) x (ii) x (iii) a (iv) a = x (v) a < x a > x (vi) (iii) (v) 8 Game Over (1) random (2) x (3) x x 8 (4) for j 0 7 (5) (8) (5) input 8 j x a (6) a = x break for (7) a < x (8) (9) x a (10),(11) (10) x (11) Game Over Python find number.py 29

36 17 WHILE 17 while 3n + 1 n n n n n 3n + 1 n n = 1 for 1000 n = 1 n = 1 for while 17.1 while for while while 1 while A: 2 [ X ] 3 [ X ] A True X A False X while A True X X A A True X False while A True A False while break, continue X break while X continue X A Start while A: A X True False 4 while X End 5 while 30

37 17.2 while 17 WHILE while while1.py 1 a = 1 2 while a <10: # a print a*a, # a 4 a = a+1 # a 5 6 print owari # w h i l e owari A a<10 X print a*a; a = a+1, while1.py 1. a 1 2. a<10 a=1 a<10 3. a*a=1 a 1 (a=2 ) 4. a<10 a<10 5. a*a=4 a 1 (a=3 ) a*a=9 a 1 (a=10 ) 8. a<10 a<10 while 9. owari while for for while1.py 1 for a in range (1,10): 2 print a* a 3 4 print owari 17.2 while n + 1 while n 3n + 1 while2.py 1 a = input ( Input an integer : ) # a 2 print a, # a 3 4 while a!= 1: # a 1 5 if a%2 == 0: # a 6 a = a/2 7 else : # a 8 a = 3*a+1 9 print a, 31

38 17.2 while 17 WHILE Input an integer: a= a = 1 Input an integer: while while3.py 1 # -*- coding : utf -8 -*- 2 j=1 3 while True : # 4 print + str (j)+ 5 j = j while True CTRL + C while break, else, continue while break while4.py 1 # -*- coding :utf -8 -*- 2 j=1 3 while True : 4 print + str (j)+ 5 j=j +1 6 if j > : # w h i l e 7 break 8 9 print " Too much sheeps! I m already asleep... zzz " Too much sheeps! I m already asleep...zzz. while else while False while else 32

39 17.2 while 17 WHILE while else.py 1 a=0 2 while a <5: 3 a = a+1 4 print a, 5 else : 6 print hogehoge 7 8 print END hogehoge END break while-else while continue ( ) Shinshu ns 1 >>> ns in Shinshu 2 True 3 >>> sn in Shinshu 4 False while continue.py 1 # -*- coding : utf -8 -*- 2 a=0 3 while a <20: 4 a=a +1 5 if 4 in str (a): # a 4 6 continue # p r i n t w h i l e 7 else : # 8 print + str (a)+ # 9 10 print zzz zzz... 33

40 17.2 while 17 WHILE for 1 # -*- coding : utf -8 -*- 2 for i in range (1,21): 3 if not 4 in str (i): # i 4 4 print + str (i)+ 5 6 print zzz , 1, 2, 3, 5, 8, 13, 21, 34, a 1 = 1, a 2 = 1, a n+1 = a n + a n 1, n = 2, 3, 4, (1) n a n while 1. maxvalue = a=1 b=1 3. while a<maxvalue a 5. a, b b, a+b 6. a<maxvalue while while5.py 1 maxvalue = # m a x v a l u e 2 a = 1 # a = 1 3 b = 1 # b = 1 4 while a < maxvalue : 5 print a # 6 a, b = b, a+b # 7 8 print The next value is, a # the next value i s a The next value is ( 2) n 2 n n n 1. n ( n 2 ) 2. i = 2 34

41 18 3. i 2 n n i n 3 5. i i n while primeq.py 1 # -*- coding :utf -8 -*- 2 n = input ( Input an integer ( >1): ) # n 3 i = 2 # i = 2 4 while i*i <= n: # i* i n 5 if n % i == 0: # n i 6 print n, is not a prime. # n 7 break # w h i l e else : 9 i += 1 # i 10 else : # i*i>n 11 print n, is prime. # n print Input an integer: n = while primeq.pyn = a, b b = 0 a b 0 a b b r r a b Python 1. a, b 2. b 0 3,4,5 while b = a b r 4. a b 5. b r 6. b = 0 a a 35

42 gcd.py 1 a = input ( Input an integer : ) 2 b = input ( Input an integer : ) 3 while *******: 4 ********** 5 ********** 6 ********** 7 8 print greatest common divisor is, a ***** 36

43 19 19 [3,6,3,9] 1 >>> a = [1,3,2,4] 2 >>> a 3 [1,3,2,4] append 1 >>> a. append (7) 2 >>> a 3 [1,3,2,4,7] for 1 aa = [] # a a 2 for i in range (10): # i aa. append (i*i) # i* i a 4 5 print aa # a a [0, 1, 4, 9, 16, 25, 36, 49, 64, 81] i 2 (i = 0, 1, 2,, 9) 1, 9, 25, 49, 81, for 1 bb = [] # b b 2 for i in range (20): # i if i%2 == 1: # i 4 bb. append (i*i) # i* i a 5 6 print bb # b b [1, 9, 25, 49, 81, 121, 169, 225, 289, 361] Python 1 cc = [i*i for i in range (10)] # 2 print cc [0, 1, 4, 9, 16, 25, 36, 49, 64, 81] bb 37

44 20 1 dd = [ i* i for i in range (20) if i %2==1] 2 print dd [1, 9, 25, 49, 81, 121, 169, 225, 289, 361] 1 lis = [ a, b, c ] 2 ee = [ (i, j) for i in range (1,5) for j in lis ] 3 print ee [(1, a ), (1, b ), (1, c ), (2, a ), (2, b ), (2, c ), (3, a ), (3, b ), (3, c ), (4, a ), (4, b ), (4, c )] 20 for while INPUT x FUNCTION f: OUTPUT f(x) 1 def ( ) : 2 [ X ] # 3 return [ ] # def def : func1.py 38

45 def greeting (): # 2 print "I m fine." # 3 4 print How are you? # How are you? 5 greeting () # How are you? I m fine. greeting() print "I m fine" return func2.py 1 def sanjou (a): # s a n j o u a 2 return a*a*a # a 3 4 print sanjou (2) # 8 5 print sanjou (3) # b = sanjou (1)+ sanjou (2)+ sanjou (3)+ sanjou (4) 7 print b sanjou( ) a odd even func3.py 1 def guuki (a): 2 if a%2 == 0: # a 3 return even # e v e n 4 else : # 5 return odd # o d d 6 7 print guuki (1232), guuki (99) even odd n True False uruuq func4.py 1 def uruuq (n): 2 if n %400 == 0: 3 return True 4 elif n %100 == 0: 5 return False 6 elif n%4 == 0: 7 return True 8 else : 9 return False 39

46 for i in range (1000,2018): 12 if uruuq (i): # i 13 print i, # i func5.py 1 # -*- coding : utf -8 -*- 2 def CountSheeps ( a): 3 for i in range (1,a +1): # i 1 a 4 print + str (i) + # i 5 6 CountSheeps (45) # def nobu (): 2 print De aruka 3 4 oda = nobu # n o b u o d a 5 oda () # o d a De aruka a, b ***** def gcd.py 1 def gcd (a,b): 2 ************* 3 ******* 4 ***** 5 ***** 6 ******** 7 8 print gcd ( , )

47 n n True False primeq(n) primelist.py 41

48 20.3 lambda lambda lambda Python lambda lambda 1 lambda x, y : ( x y ) n n 2 f lambda 1 >>> f = lambda a : a*a # f a a * a 2 >>> f (3) 3 9 lambda 1 a = [ lambda x, y : x+y, lambda x,y: x-y, lambda x,y: x*y] 2 3 for f in a: 4 print f (3,5) N N = 100 S = k 2 N N N S = = k 2 = j 2 = l 2 (2) k=1 k=1 j=1 l=1 Σ k, j, l *7 (2) k, j, l Σ S N S N k c s daikei.py 1 def daikei (a,b,h): # a b h *7 42

49 c = a+b # c 3 s = 1.0* c*h / 2 # s 4 return s 5 6 S = daikei (3,5,8) # S 7 print S 8 9 print c # c 32.0 Traceback (most recent call last): File "daikei.py", line 8, in? print c NameError: name c is not defined Python def """ """ gcd(a,b) 1 # -*- coding : utf -8 -*- 2 3 def gcd (a,b): 4 """ a b 5 6 """ 7 while b!= 0: 8 a, b = b, a%b 9 return a print gcd (1428,2618) 238 4,5, def (while primeq.py) n True False primeq.py 1 # -*- coding :utf -8 -*- 43

50 def primeq (n): 3 """ n T r u e F a l s e """ 4 if n < 2: 5 return False 6 else : 7 i=2 8 while i **2 <= n: 9 if n%i == 0: # n i 10 return False 11 i=i return True # print [ k for k in range (1,101) if primeq ( k)] [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97] AKS Miller-Rabin Sage is_prime 20.7 n! = (n 1) n for kaijou1.py 1 def kaijou (n): 2 """ n n! f o r """ 3 a=1 4 for i in range (1,n +1): 5 a = a* i 6 return a 7 8 for i in range (1,10): #i =1,...,9 9 print kaijou (i), # kaijou (i) f(n) f(0) = 1, f(1) = 1, f(n) = n f(n 1) n f(n) = n! (recursive definition) Python 44

51 n! kaijou2.py 1 def kaijou (n): 2 """ n! """ 3 if n ==0 or n ==1: 4 return 1 5 else : 6 return n* kaijou (n -1) # kaijou (n -1) 7 8 for i in range (10): #i=0,1,2,,9 9 print kaijou (i), # kaijou (i) gcd(a,b) gcd rec.py 1 def gcd (a,b): 2 if b == 0: # b=0 3 return a # a 4 else : # 5 return gcd (b,a%b) # gcd (b,a%b) 6 7 print gcd (1428,2618) a=1428, b=2618 gcd(a,b) 2. b = 0 gcd(2618,1428%2618) gcd(2618,1428) gcd(1428,2618%1428) %1428=1190 gcd(1428,1190) gcd(1190,238) 238=1428% gcd(238,1190%238) %238= print

52 mc n m n mc n = m! (m n)!n! 0 if m < n or n < 0 mc n = 1 if m = 0 or m = n m 1C n + m 1 C n 1 (3) (4) n < 0 m < n m C n 0 (3), (4) m C n C(m,n), D(m,n) combinat1.py 1 def kaijou (n): # kaijou (n)=n! 2 if n == 0 or n ==1: 3 return 1 4 else : 5 return n* kaijou (n -1) 6 7 def C(m,n): # (1) 8 return kaijou (m )/( kaijou (m-n)* kaijou (n)) 9 10 def D(m,n): # (2) 11 if n <0 or m<n: 12 return 0 13 elif m == 0 or m == n: 14 return 1 15 else : 16 return D(m -1,n) + D(m -1,n -1) for i in range (7): #i=0,1,2,3,4,5,6 19 print C(6,i), #C(6,i) print # for i in range (7): #i=0,1,2,3,4,5,6 24 print D(6,i), #C(6,i) combinat2.py 1 # -*- coding :utf -8 -*- 2 from time import time # t i m e t i m e 3 4 def kaijou (n): 5 if n == 0 or n ==1: 6 return 1 7 else : 46

53 return n* kaijou (n -1) 9 10 def C(m,n): # (1) 11 return kaijou (m )/( kaijou (m-n)* kaijou (n)) def D(m,n): # (2) 14 if n <0 or m<n: 15 return 0 16 elif m == 0 or m == n: 17 return 1 18 else : 19 return D(m -1,n) + D(m -1,n -1) time1 = time () # t i m e 1 22 print C (23,12) # C (23,12) 23 time2 = time () # t i m e 2 24 print, time2 - time1, 25 print D (23,12) # D (23,12) 26 time3 = time () # t i m e 3 27 print, time3 - time2, (3) (4) 1.78 (4) (4) C(5,2) C(5, 2) = C(4, 2) + C(4, 1) = C(3, 2) + C(3, 1) + C(3, 1) + C(3, 0) = C(2, 2) + C(2, 1) + C(2, 1) + C(2, 0) + C(2, 1) + C(2, 0) + C(2, 0) + C(2, 1) = 1 + C(1, 1) + C(1, 0) + C(1, 1) + C(1, 0) + C(1, 0) + C(1, 1) + C(1, 1) + C(1, 0) + C(1, 0) + C(1, 1) + C(1, 0) + C(1, 1) + 0 = C(0, 0) + C(0, 1) C(0, 0) + C(0, 1) + C(0, 0) + C(0, 1) C(0, 0) + C(0, 1) + C(0, 0) + C(0, 1) C(0, 0) + C(0, 1) + 0 = = 10 C(54,24)= m C n = m!/(m n)!n! 20.9 n n j=1 1 j 2 (5) Basel sum(n) Basel sum.py 1 def Basel_sum (n): 47

54 if n <2: 3 return 1 4 else : 5 ****** ************************ 6 7 print Basel_ sum (10) 8 print Basel_ sum (100) 48

55 21 21 Sage Python (expression) (statement) ans = 2^2^2 ans ans a=1 a= eval exec eval exec eval (expression) exec (statement) 1 >>> eval ( 1+3 ) # >>> exec ( a=3 ) #a=3 4 >>> print a a=3 1+3 a=3 eval 1 def func1 (): 2 return 2**2**2 3 4 def func2 (): 5 return hogehoge 6 7 def func3 (): 8 return type (1.4) 9 10 def func4 (): 11 return [1,2,3,4] print func1 () 14 print func2 () 15 print func3 () 16 print func4 () 16 hogehoge <type float > [1,2,3,4] func1, func2, func3, func4 4 4 eval 1 def func1 (): 2 return 2**2**2 49

56 22 SET 3 4 def func2 (): 5 return hogehoge 6 7 def func3 (): 8 return type (1.4) 9 10 def func4 (): 11 return [1,2,3,4] for i in range (1,5): 14 print eval ( func + str (i)+ () ) str 22 Set 22.1 Python (set) set([1,3,4]) union intersection, & Python 1 >>> a = set ([1,3,4]) # a 2 >>> b = set ([3,7]) # b 3 >>> c = a b # c a b 4 >>> c 5 set ([1, 3, 4, 7]) 6 >>> d = a & b # d a b 7 >>> d 8 set ([3]) union() intersection() = 1 >>> a. union (b) #a. union (b) 2 set ([1, 3, 4, 7]) # a b 3 >>> a. intersection (b) #a. intersection (b) 4 set ([3]) # a b 5 >>> a 6 set ([1, 3, 4]) # a 7 >>> b 8 set ([3, 7]) # b 9 >>> a = b # a b 10 >>> a # a 11 set ([1, 3, 4, 7]) # b add remove 50

57 22.2 Set SET 1 >>> a. add (8) # a 8 2 >>> a # a 3 set ([8, 1, 3, 4]) # a 8 4 >>> a. remove (1) # a 1 5 >>> a 6 set ([8, 3, 4]) # a Set (2 + 9) (3/3) = Python Python set a, b a + b, a b, b a, a b, a/b, b/a nikou(a,b) set 0 nikou.py 1 def nikou (a,b): 2 if a!=0 and b!=0: # a b 0 3 return set ([a+b,a-b,b-a,a*b,1.0* a/b, 1.0* b/a]) # 4 elif b == 0: # b=0 5 return set ([a, -a, 0]) 6 else : # a=0 7 return set ([b, -b, 0]) 8 9 print nikou (3,4) #3,4 set([0.75, 1, 7, 12, , -1]) 1.0* / a,b,c sankou(a,b,c) 3 3,, (a b) c : a, b c (a c) d : a, c d 51

58 22.2 Set SET (b c) a : b, c a 3 3 a,b,c 2 sankou(a,b,c) sankou1.py 1 def nikou (a,b): 2 if a!=0 and b!=0: 3 return set ([a+b,a-b,b-a,a*b, 1.0* a/b,1.0* b/a]) 4 elif b == 0: 5 return set ([a,-a,0]) 6 else : 7 return set ([b, -b, 0]) 8 9 def sankou (a,b,c): # a,b, c 10 ResultSet = set ([]) # R e s u l t S e t 11 for i in nikou (a,b): # a, b i 12 ResultSet = nikou (i,c) # i, c R e s u l t S e t 13 for i in nikou (b,c): 14 ResultSet = nikou (i, a) 15 for i in nikou (a,c): 16 ResultSet = nikou (i, b) 17 return ResultSet # R e s u l t S e t print sankou (4,4,9) # 4,4,9 20 print # 21 print 10 in sankou (4,4,9) # sankou (4,4,9) 1 0 set([ , , , 1, , 9, , , 144, 17, 20, 25, , , 0, 32, , 40, 7, , 52, , , , 72, , -7, -32, , , -20, , -9, , , -1, ]) True a b c 9 10 sankou2.py 1 # -*- coding :utf -8 -*- 2 def nikou (a,b): 3 if a!=0 and b!=0: 4 return set ([a+b,a-b,b-a,a*b, 1.0* a/b,1.0* b/a]) 5 elif b == 0: 6 return set ([a,-a,0]) 7 else : 8 return set ([b, -b, 0]) 9 10 def sankou (a,b,c): # a,b, c 11 ResultSet = set ([]) # R e s u l t S e t 12 for i in nikou (a,b): # a, b i 13 ResultSet = nikou (i,c) # i, c R e s u l t S e t 52

59 22.2 Set SET 14 for i in nikou (b,c): 15 ResultSet = nikou (i, a) 16 for i in nikou (a,c): 17 ResultSet = nikou (i, b) 18 return ResultSet # R e s u l t S e t number = 0 # n u m b e r for i in range (10): 23 for j in range (i,10): 24 for k in range (j,10): 25 if 10 in sankou (i,j,k): # 1 0 sankou (i,j,k) 26 print i,j,k #i,j, k 27 number = number +1 # n u m b e r print 10, number, a,b,c,d 2 (A) 3 (B) 2 (a+d*c)*b (a*d)+(b/c) (A) (B) (A) yonkoua(a,b,c,d) 1 def yonkoua (a,b,c,d): 2 ResultSet = set ([]) 3 for i in sankou (a,b,c): # a,b, c i 4 ResultSet = nikou (i,d) # i d R e s u l t S e t 5 for i in sankou (a,b,d): 6 ResultSet = nikou (i, c) 7 for i in sankou (a,c,d): 8 ResultSet = nikou (i, b) 9 for i in sankou (b,c,d): 10 ResultSet = nikou (i, a) 11 return ResultSet (A) 10 4 yonkou1.py 1 # -*- coding :utf -8 -*- 53

60 22.2 Set SET 2 def nikou (a,b): 3 if a!=0 and b!=0: 4 return set ([a+b,a-b,b-a,a*b, 1.0* a/b,1.0* b/a]) 5 elif b == 0: 6 return set ([a,-a,0]) 7 else : 8 return set ([b, -b, 0]) 9 10 def sankou (a,b,c): # a,b, c 11 ResultSet = set ([]) # R e s u l t S e t 12 for i in nikou (a,b): # a, b i 13 ResultSet = nikou (i,c) # i, c R e s u l t S e t 14 for i in nikou (b,c): 15 ResultSet = nikou (i, a) 16 for i in nikou (a,c): 17 ResultSet = nikou (i, b) 18 return ResultSet # R e s u l t S e t def yonkoua (a,b,c,d): 21 ResultSet = set ([]) 22 for i in sankou (a,b,c): # a,b, c i 23 ResultSet = nikou (i,d) # i d R e s u l t S e t 24 for i in sankou (a,b,d): 25 ResultSet = nikou (i, c) 26 for i in sankou (a,c,d): 27 ResultSet = nikou (i, b) 28 for i in sankou (b,c,d): 29 ResultSet = nikou (i, a) 30 return ResultSet number = for i in range (10): 35 for j in range (i,10): 36 for k in range (j,10): 37 for l in range (k,10): 38 if 10 in yonkoua (i,j,k,l): 39 print i,j,k,l 40 number = number print 10, number, (A) 10 4 B) (1+1)*(1+4)=

61 22.2 Set SET (B) yonkoub(a,b,c,d) 4 yonkoub(a,b,c,d) 1 def yonkoub (a,b,c,d): 2 ResultSet = set ([]) # R e s u l t S e t 3 for i in nikou (a,b): # a b i 4 for j in nikou (c,d): # c d j 5 ResultSet = nikou (i,j) # i j R e s u l t S e t 6 for i in nikou (a,c): 7 for j in nikou (b,d): 8 ResultSet = nikou (i, j) 9 for i in nikou (a,d): 10 for j in nikou (b,c): 11 ResultSet = nikou (i, j) 12 return ResultSet # R e s u l t S e t a, b, c, d a b c d 10 make10.py 1 # -*- coding :utf -8 -*- 2 def nikou (a,b): 3 if a!=0 and b!=0: 4 return set ([a+b,a-b,b-a,a*b, 1.0* a/b,1.0* b/a]) 5 elif b == 0: 6 return set ([a,-a,0]) 7 else : 8 return set ([b, -b, 0]) 9 10 def sankou (a,b,c): # a,b, c 11 ResultSet = set ([]) # R e s u l t S e t 12 for i in nikou (a,b): # a, b i 13 ResultSet = nikou (i,c) # i, c R e s u l t S e t 14 for i in nikou (b,c): 15 ResultSet = nikou (i, a) 16 for i in nikou (a,c): 17 ResultSet = nikou (i, b) 18 return ResultSet # R e s u l t S e t def yonkoua (a,b,c,d): 21 ResultSet = set ([]) 22 for i in sankou (a,b,c): # a,b, c i 23 ResultSet = nikou (i,d) # i d R e s u l t S e t 24 for i in sankou (a,b,d): 25 ResultSet = nikou (i, c) 26 for i in sankou (a,c,d): 27 ResultSet = nikou (i, b) 28 for i in sankou (b,c,d): 29 ResultSet = nikou (i, a) 30 return ResultSet 31 55

62 SET 32 def yonkoub (a,b,c,d): 33 ResultSet = set ([]) # R e s u l t S e t 34 for i in nikou (a,b): # a b i 35 for j in nikou (c,d): # c d j 36 ResultSet = nikou (i,j) # i j R e s u l t S e t 37 for i in nikou (a,c): 38 for j in nikou (b,d): 39 ResultSet = nikou (i, j) 40 for i in nikou (a,d): 41 for j in nikou (b,c): 42 ResultSet = nikou (i, j) 43 return ResultSet # R e s u l t S e t def yonkou (a,b,c,d): 46 ResultSet = set ([]) 47 ResultSet = yonkoua (a,b,c, d) 48 ResultSet = yonkoub (a,b,c, d) 49 return ResultSet counter = for i in range (10): 54 for j in range (i,10): 55 for k in range (j,10): 56 for l in range (k,10): 57 if 10 in yonkou (i,j,k,l): 58 print i,j,k,l 59 counter = counter print 10 4, counter, >>> == 1 2 True 3 >>> == 1 4 False a, b, c, d, e(0 a b c d e 9) make100.py 56

63 24 SAGE II SageMath 23 SageMath SageMath( Sage ) Sage Python Sage Python Sage Sage Sage Linux, Mac, Windows 24 Sage Sage 4 1. ( sage CUI) 2. Sage ( sage.sage ) 3. Sage ( notebook() ) 4. Sage (sage:load(.sage )) 24.1 Sage Sage PC sage Sage 1 user@debian :~$ sage # S a g e M a t h $ 1 2 SageMath version 8.2, Release Date : Type " notebook ()" for the browser - based notebook interface. 4 Type " help ()" for help. 5 6 sage : Sage Python Sage 1+2 factor(2010) 1 sage : sage : factor ( 2010) 4 2 * 3 * 5 * 67 57

64 24.2 Sage 25 SAGE WEB factor Sage Sage exit quit 1 sage : exit 2 Exiting Sage ( CPU time 0m0.06s, Wall time 2m8.71 s) Sage Python Sage 1. Sage.sage 2. sage.sage * 8 testfactor.sage 1 a = factor (2014) # 2014 a 2 print a # a 1 $ sage testfactor. sage 2 2 * 19 * 53 3 $ Python print Python 1 $ sage testfactor. sage > testfactor. txt 2 $ testfactor.txt 25 Sage Web Sage Try Sage Online : Sage Documentation : Sage Python *8 58

65 26 SAGE Download : Sage ( OS ) Sage Feature Tour : 26 Sage 26.1 SageMathCloud Sage Online Sage Web Try Sage Online Sage 26.2 Windows Sage SageMath8.0 Windows 1. SageMath SageMath-**.exe (xx Mac *9 PC Max OS X 1. Sage Web download sage dmg mac CPU intel intel OS bit sage-xx-x86_64-darwin-osx-10.10_x86_64-app.dmg -app Mac -app Unix command-line -app 2. dmg 3. sage 4. finder sage sage 5. Select to run it with Terminal : Choose Applications, then select All Applications in the Enable: drop down. Change the Applications drop down to Utilities. On the left, scroll and select Terminal. Click Open, then in the next dialog select Update. 6. Sage Window 7. Sage notebook() firefox safari URL *9 Mac 59

66 28 SAGE 27 Sage Sage Sage notebook() Sage 1 sage : notebook () firefox Sage * 10 worksheet New Worksheet Sage ( ) shift+enter evaluate Sage notebook() ctrl+c Sage exit Sage 28 Sage 28.1 Sage Python / a, b Python2 a/b Sage a/b a/b Python a b a**b Sage a^b a + b a+b a b a-b a b a*b a b a/b ab a^b *10 username = admin, password = sagesage 60

67 SAGE 28.2 Sage π pi e e i i 1 pi pi n() 1 n(pi) n( pi, digits =30) n(e) (square root) 1 sqrt (3) sqrt(3) 1 sqrt (3)* sqrt (3) 3 e pi 3 n(sqrt(3)) 28.3 Sage n Sage n n(a) a n 28.4 Sage i I 5 + 3i 5+3*I 61

68 SAGE 1 sage : (5+3* I )^2 2 30* I sage : a = 5+3* I; a. conjugate () # 4-3* I sage : exp (pi*i) 6-1 # E u l e r 28.5 Sage e π Sage (golden ratio) ϕ golden ratio ( 2 n ) 1 (Euler s constant) γ euler_gamma lim n k ln(n) k=1 ( 1) n (Catalan s constant) K catalan (2n + 1) 2. (twin prime) C 2 twinprime 28.6 k=1 p 3; p(p 2) (p 1) 2 Sage Sage Sage ZZ Integer Ring QQ Rational Field QQbar Algebraic Field Q SR Symbolic Ring (53bit) RR Real Field with 53 bits of precision 53 (53bit) RDF Real Double Field (53bit) CC Complex Field with 53 bits of precision 53 (400bit) RealField(400) 400 Z/pZ Q[ 5] 1 sage : ZZ 2 Integer Ring # QQ,RR, S R 3 sage : 5 in ZZ # 5 4 True 5 sage : 1.5 in ZZ 6 False 7 sage : 1.5 in QQ # True 9 sage : sqrt (2) in QQ # 2 10 False 62

69 31 11 sage : sqrt (2) in QQbar # 2 12 True 13 sage : x = var ( x ) # x 14 sage : x in SR # 15 True 1 print pi 2 print RDF (pi) 3 print RR(pi) 4 print RealField (200)( pi) pi sage : a = RR(pi ); exp (a*i) e -16* I e iπ = e-16*I i (plot) 29 Sage notebook a=3 a 3 a=5 a 5 Sage Action... Restart worksheet 30 Sage Tab Sage Sage dif Tab diff differences 2 diffe Tab differences diff differences 31 Typeset Typeset 1 integral ( sin (x)*e^x,x) Typeset 63

70 32 SAGE 8 Typeset 32 Sage Sage Sage worksheet sws Web * 11 * 12 Sage worksheet * Sage [File...] [Save worksheet to a file...] 2. OK 3. Web OK 4. sage01.sws dataproc1 sws worksheet 1. Sage notebook upload 2. Browse your computer to select a file to upload: *11 firefox *12 ( ) *13 64

71 34 SAGE 3. [upload] 33 Sage? 1 sum? sum sum 1 sum?? 34 Sage 34.1 Sage Word Sage New Worksheet Worksheet Edit Edit 1 2 {{{ id =1 3 4 /// 5 }}} Worksheet Worksheet 1 integrate ( sin (x),x) Edit 1 # 2 {{{ id =1 # 3 integrate ( sin (x),x) # 65

72 SAGE 4 /// # 5 -cos (x) # 6 }}} # 7 # 8 {{{ id =2 # 2 9 # 10 /// # 11 }}} # 2 {{{id=1 1 1 $\ sin (x)$ Save changes L A TEX L A TEX $ $ \[ \] \begin{equation}...\end{equation} html html 1 <h1 >1. </ h1 > 66

73 34.2 Sage 34 SAGE 2 <h2 >1.1 </h2 > 3 $\ int \ sin (x) dx$ 4 5 {{{ id =1 6 integrate ( sin (x),x) 7 /// 8 -cos (x) 9 }}} <h2 > 1.2 </h2 > 12 $\ int_0 ^\ pi \ sin (x) dx $ {{{ id =2 15 integrate ( sin (x),x,0, pi) 16 /// }}} \ begin { equation } 21 \ int_0 ^\ infty \ frac {1}{1+ x ^4} dx = \ frac {\ pi }{2\ sqrt {2}} 22 \ end { equation } {{{ id =3 25 integrate (1/(1+ x^4),x,0, oo) 26 /// 27 1/4* pi* sqrt (2) 28 }}} {{{ id = /// 33 }}} <h1> 1</h1> 1 2 <h2>... </h2> Worksheet html html 34.2 Sage 67

74 35 Sage Shift + L A TEX Save Changes Sage Worksheet 35 Sage Sage sagews01.sws Sage Upload 68

75 36 36 Python Python Python 1 def fib (n): # f i b n 2 a, b = 0, 1 # 3 for i in range (n): # 4 a, b = b, a+b # 5 return a # a 6 7 for i in range (1,10): #i=1,,9 8 print fib (i) # fib (i) 9 10 print type ( fib ) # f i b <type function > fib n 2 4 a (return) Python 4x + 3, x 2, sin(x), 1 x (6) x (variable) (indeterminate) x Python x x Sage 1 var ( x ) # x 2 f = x^2 # x^2 f 1 x x 2 f = x 2 1 print f # f 2 print f(x =5) # x=5 f 3 print type (f) # f 4 print type ( x) 69

76 36.1 Sage 37 x^2 25 <type sage.symbolic.expression.expression > <type sage.symbolic.expression.expression > f x 2 3,4 f x sage.symbolic.expression.expression fib function 2 1 var ( x ) 2 f(x) = x^2 # ( x) 3 4 print f 5 print f (5) # 6 print type ( f) x --> x^2 25 <type sage.symbolic.expression.expression > 2 x,y 1 var ( x y ) #x, y 2 g = (x+y )^2 # g 3 print g 4 print g(x=3, y =6) #x=3, y=6 g 5 print g(y =3) #y=3 g (x + y)^2 81 (x + 3)^ Sage Sage Sage sin, cos, tan, csc, sec, cot,arcsin, arccos log, ln, exp sinh, cosh, tanh arcsinh gamma(z) zeta 37 Sage Sage 37.1 plot f(x) plot 70

77 37.1 plot 37 plot f(x) 1 plot (f(x), (x,a,b)) f(x) x [a, b] sin(x) [ 6, 6] 1 plot ( sin (x), (x, -6,6)) sin(x) 1/x 2 1 plot (1/x, (x, -2,2)) /x 1/x 71

78 37.1 plot 37 f(x) [c, d] 1 plot (f(x), (x,a,b), ymin =c, ymax =d) f(x) x [a, b] y ymin=c, ymax=d 1/x [ 3, 4] 1 plot (1/x, (x, -2,2), ymin =-3, ymax =4) /x [ 3, 4] f(x), g(x), h(x) 1 plot ( (f(x),g(x),h(x)), (x,a,b) ) 3 sin(x), cos(x), tan(x) [ 7, 7] [ 5, 5] 1 plot ( ( sin (x), cos (x), tan (x)), (x, -7,7), ymin =-5, ymax =5) sin(x), cos(x), tan(x) 72

79 plot plot 1 plot ( f(x), (x,a,b), color =, 2 axes_labels =[, ], legend_label = ) red, yellow, blue, green RGB #3F4A46 axes_labels axes=false $ $ L A TEX \sin(x) sin(x) \\tan(x) tan(x) \\frac{a}{b} a b 1 plot ( sin (x ^2),(x,0,5), color = red, axes_labels =[ time, amplitude ], 2 legend_label = $\ sin (x ^2) $ ) amplitude 1 sin(x 2 ) time sin(x 2 ) 37.2 red, blue colors 1 for i in sorted ( colors ): 2 print i aliceblue antiquewhite aqua yellow yellowgreen 73

80 show() show() show 1 p1 = plot ( sin (x), (x, -6,6) ) 2 show (p1) 1 sin(x) p1 2 p1 2 p1.show() show() show 1 p1 = plot ( tan (x), (x, -6,6) ) 2 p1. show ( ymin =-5, ymax =5) 1 p1 = plot ( sin (x), (x, -5,5), legend_label = sin, color =" red ") 2 p2 = plot ( cos (x), (x, -5,5), legend_label = cos, color =" blue ") 3 p3 = plot ( tan (x), (x, -5,5), legend_label = $\\ tan (x)$, color =" green ") 4 p4 = p1+p2+p3 # p 4 p 1,p2, p 3 5 p4. show ( ymin =-3, ymax =3) # p 4 s h o w sin, cos, tan p1,p2,p3 4 p4 show p sin cos(x) tan(x) sin(x), cos(x), tan(x) graphics_array 74

81 graphics array 1 p1 = plot ( sin (x), (x, -4,4) ) 2 p2 = plot ( cos (x), (x, -4,4) ) 3 p3 = graphics_array ([p1,p2 ]) 4 p3. show ( aspect_ratio =1) aspect ratio aspect ratio sin(x), cos(x) fill 1 p1 = plot ( sin (x), (x, -5,5), fill = axis ) # 2 p2 = plot ( sin (x), (x, -5,5), fill = min ) # 3 p3 = plot ( sin (x), (x, -5,5), fill = max ) # 4 p4 = plot ( sin (x), (x, -5,5), fill = 0.5) # graphics_array ([[ p1, p2], [p3, p4 ]]). show () graphics_array([[a,b],[c,d]]) a c b d fill 75

82 sin(x) x plot ( sin (x), (x, -2,2), fill = x ^2-1) sin(x) x 2 1 (thickness) (dashed) plot plot 1 plot? 37.4 f(x, y) = 0 x, y implicit_plot implicit plot f(x, y) = 0 1 var ("x y") 2 implicit_plot (f(x,y), (x,a,b), (y,c,d)) 2 f(x, y) (x, y) [a, b] [c, d] x,y = var( x y ) x, y (x 3 + y 3 3xy) x 3 + y 3 3xy = var ("x y") 2 implicit_plot (x ^3+ y ^3-3* x*y, (x, -2,2), (y, -2,2)) 76

83 (x(t), y(t)) parametric_plot (x(t), y(t)) 11 1 t = var ( t ) 2 parametric_plot ([ cos (t) + 2* cos (t/4), sin (t) -2* sin (t /4)], 3 (t,0, 8* pi), fill = true ) x(t) = cos(t) + 2 cos(t/4), y(t) = sin(t) 2 sin(t/4) t 8π fill parametric plot(with filling) 37.5 list_plot 1 a = 2 list_plot (a) [3,2,6,3,12,-2,2] 2 [(1,2),(4,1),(3,4),(4,2)]

84 a = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10] 2 list_plot (a) plotjoined=true 13 1 a = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10] 2 list_plot (a, plotjoined = True ) (plotjoined=true) a = [( sqrt (i)* cos (i), sqrt (i)* sin (i)) for i in range (20)]; 2 list_plot (a, plotjoined = True ) { i(cos(i), sin(i)) i = 0, 1,, 19} 78

85 37.6 (x, y) (x, y) 2 f(x, y) f(x, y) > 0 regin_plot region plot f(x, y) > 0 (x, y) 1 x,y = var ( x,y ) 2 region_plot (f(x,y)>0, (x,a,b), (y,c,d)) sin(x 2 y 3 ) > 0 x, y 14 1 x,y = var ( x,y ) 2 region_plot ( sin (x^2 -y^3) >0, (x,-3, 3), (y,-3, 3)) sin(x 2 y 3 ) > 0 (x, y) [f(x,y)>0,g(x,y)>0,h(x,y)>0] 15 1 x,y = var ( x,y ) 2 region_plot ([x >0, y >0, x ^2+ y ^2 >0.5], (x, -1,1), (y, -1,1) ) x > 0, y > 0, x 2 + y 2 > 0.5 x, y region plot 100 plot_points=300 region_plot 79

86 37.7 ( ) region_ plot? 37.7 ( ) Sage 2 plot circle((a,b),r) (a, b), r ellipse((a,b),c,d) (a, b) c, d arrow((a,b),(c,d)) (a, b), (c, d) arc((a,b),r,sector=(c,d)) (a, b), r, (c, d) (c, d) 0 90 (0, π/2) line([(a,b),(c,d)]) (a, b) (c, d) linestype= -- marker= o point(a,b,size=r) (a, b) r text(, (a,b), fontsize=r) (a, b) r 16 1 s1 = plot (x^2,(x, -1,1)) 2 s2 = point ((0,0), size =100, color = black ) 3 s3 = arrow (( -1/2,1/4),(1,1), color = red ) 4 s4 = arc ((0,0),0.5, sector =(0, pi), color = green ) 5 s5 = s1 + s2 + s3 + s4 6 s5. show () f(x, y) f(x, y) 3 plot3d 80

87 38.2 density plot contour plot 38 2 plot3d f(x, y) 3 1 var ( x y ) 2 plot3d (f(x,y), (x,a,b),(y,c,d)) plot_points : () opacity : ( ) aspect_ratio : [1,2,1] 17 1 var ( x y ) 2 plot3d ( sin (x-y)*y* cos (x), (x, -3,3), (y, -3,3), 3 opacity =0.5, aspect_ ratio =[1,1,1]) 26 y sin(x y) cos(x) 38.2 density plot contour plot plot3d density plot contour plot 18 1 x, y = var ( x y ) 2 density_plot ( sin (x*y), (x, -3,3), (y, -3,3)) 81

88 sin(xy) 19 1 x, y = var ( x y ) 2 contour_plot ( cos (x)+ cos (y), (x, -8, 8), (y, -8, 8), contours =20) cos(x) + cos(y) implicit_plot3d parametric_plot3d plot_vector_field3d 38.4 Sage notebook png png pdf,eps,ps 82

89 39 notebook 1 p1 = plot ( sin (x), (x, -4,4)) 2 p1. save (". pdf ") eps pdf eps 3 eps, pdf, ps png Sage (**.sage ) plot.sage 1 p1 = plot ( sin (x), (x, -4,4)) 2 p1. save (" plot. pdf ") # p 1 p d f Emacs 1 user@debian :~$ sage plot. sage # p d f ( 2 user@debian :~$ 39 e z n z nz n k=0 z k k! n (nz) n k=0 k! (7) n ze z 1 = 1 roots x 5 + 3x + 1 == 0 1 eq = x ^5+3* x +1==0 2 eq. roots (x, ring =CC, multiplicities = False ) [ , *I, *I, *I, *I] multiplicities False 7 83

90 var ( z k ) #z, k 2 nn = 30 # n n f = sum (( nn*z)^k/ factorial (k), k,0, nn) # f 4 eq = f ==0 # f ==0 e q 5 lis = eq. roots (z, ring =CC, multiplicities = False ); # e q l i s 6 list_plot ( lis ) # l i s 29 n = 30 1 var ( z k ) #z, k 2 nn = 200 # n n f = sum (( nn*z)^k/ factorial (k), k,0, nn) # f 4 eq = f ==0 # f ==0 e q 5 lis = eq. roots (z, ring =CC, multiplicities = False ); # e q l i s 6 list_plot ( lis ) # l i s 30 n = 100? roots CC ComplexField(300) 1 var ( z k ) 2 nn = 200 # n n f = sum (( nn*z)^k/ factorial (k), k,0, nn) 4 eq = f ==0 5 lis = eq. roots (z, ring = ComplexField (300), multiplicities = False ) # 300 b i t 6 list_plot ( lis ) 84

91 n = 200 ze 1 z = 1 z = x + iy ze 1 z = x 2 + y 2 e 1 x 1 var ( x y ) 2 implicit_plot ( sqrt (x ^2+ y ^2)* exp (1 -x)-1, (x, -1,2), (y, -1,1), color = red ) 1 var ( z k ) 2 nn =200 3 f = sum (( nn*z)^k/ factorial (k), k,0, nn) # f 4 eq = f ==0 5 lis = eq. roots (z, ring = ComplexField (300), multiplicities = False ); 6 p1 = list_plot ( lis ) 7 var ( x y ) 8 p2 = implicit_plot ( sqrt (x ^2+ y ^2)* exp (1 -x)-1, (x, -1,2), (y, -1,1), color = red ) 9 (p1+p2 ). show () ze 1 z = 1 complex plot 40 k=1 (40z)k /k! 1 var ( k ) 2 nn =40 3 f = sum (( nn*x)^k/ factorial (k),k,0, nn) 4 complex_plot (f,( -1,2),( -1,1), plot_points =500, aspect_ratio =1) 85

92 k=1 (40z)k /k! 40 3 SageMath sagews02.sws 86

93 (expand) f f 1 f. expand () expand (f) (x + 1) 3 1 var ( x ); f = (x +1)^3 2 print f. expand () x^3 + 3*x^2 + 3*x + 1 f f (factor) 1 f. factor () factor (f) x 3 + 3x 2x 6 1 f = x^3 + 3*x^2-2*x factor (f) (x + 3)*(x^2-2) solve simplify f (simplify simplify full) f 1 f. simplify () simplify (f) 1 f. simplify_full () simplify_full (f) f = x 2 + 2x + 5 3x 1 f = x ^2 + 2* x + 5-3* x 2 simplify ( f) x^2 - x + 5 simplify simplify_full 1 sage : var ( x ); f = sin (x )^2 + cos (x )^2 87

94 41.2 x sage : f. simplify () # simplify sin ^2+ cos ^2=1 3 sin (x )^2 + cos (x )^2 # 4 sage : f. simplify_full () # 5 1 # x x var ( x ) 2 factor (x ^20-1) ( x 8 x 6 + x 4 x )( x 4 + x 3 + x 2 + x + 1 )( x 4 x 3 + x 2 x + 1 )( x ) (x + 1)(x 1) x n 1 ±1 1 var ( x ) 2 factor ( x ^105-1) ( x 48 + x 47 + x 46 x 43 x 42 2 x 41 x 40 x 39 + x 36 + x 35 + x 34 + x 33 + x 32 + x 31 x 28 x 26 x 24 x 22 x 20 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 x 9 x 8 2 x 7 x 6 x 5 + x 2 + x + 1 ) ( x 24 x 23 + x 19 x 18 + x 17 x 16 + x 14 x 13 + x 12 x 11 + x 10 x 8 + x 7 x 6 + x 5 x + 1 ) ( x 12 x 11 + x 9 x 8 + x 6 x 4 + x 3 x + 1)(x 8 x 7 + x 5 x 4 + x 3 x + 1 ) ( x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 )( x 4 + x 3 + x 2 + x + 1 ) (x 2 + x + 1 )( x 1 ) ± x 2 (x + 1) 2 = 2 x (x + 1) f 1 f. partial_ fraction () 1 sage : var ( x ); f = x ^2/( x +1)^2 2 sage : f. partial_ fraction () 3-2/( x + 1) + 1/( x + 1) ^2 +1 factor 88

95 sage : var ( x ); g = -2/(x + 1) + 1/( x + 1)^ sage : g. factor () 3 x ^2/( x + 1)^ x x a 0 = x x 1 = 1/(x a 0 ) a 1 = x 1 x 2 = 1/(x 1 a 1 ) x = a x 1 x = a a x 2 1 x = a a a a 3 + a 4 + x x = [a 0 : a 1, a 2, a 3, a 4, ]. x = = = = /29 = [11; 1, 8, 1, 2] a 1 continued_fraction (a) # 2 continued_fraction_list (a, ) # bits=40 1 sage : continued_ fraction ( 345/ 29) 2 [11, 1, 8, 1, 2] 3 sage : continued_fraction ( sqrt (2)) 4 [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,...] 89

96 = [1; 2, 2, 2, 2, ] continued fraction list 1 sage : a = continued_fraction_list ( sqrt (3), bits =40) #40 2 sage : a 3 [1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3] 1 sage : continued_fraction ( RealField (400)( e)) # 2 [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 3 12, 1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, 1, 1, 22, 4 1, 1, 24, 1, 1, 26, 1, 1, 28, 1, 1, 30, 1, 1, 32, 1, 5 1, 34, 1, 1, 36, 1, 1, 38, 1, 1, 40, 1, 1, 42, 1, 1, 6 44, 1, 1, 46, 1, 1, 48, 1, 1, 50, 1, 1, 52, 1, 1, 54, 7 1, 1, 56, 1, 1, 58, 1, 1, 60, 1, 1, 62, 1, 1, 64, 1, 1, 8 66, 1, 1, 68, 2] RealField(400)(e) e 41.5 Sage oo lim x a f(x) limit(...) f lim x a f(x) 1 f. limit (x = a) limit (f, x=a) 2 f. limit (x=a, dir = + ) # 3 f. limit (x=a, dir = - ) # ( e = lim ) x x x 1 sage : var ( x ) 2 sage : f = (1 + 1/x)^x 3 sage : f. limit (x=oo) 4 e # 5 sage : n(e) f.limit(x=oo) limit( f, x = oo) 41.6 Sage sum 90

97 b f(k) k=a 42 1 var ( k ) # k 2 sum (f(k),k,a,b) 1 k var ( k ); sum (k,k,1,50) 1275 sum m k = k=1 m(m + 1) 2 1 sage : var ( k m ) # k m 2 sage : ans = sum (k,k,1,m) # a n s 3 sage : print ans # a n s 4 1/2* m^2 + 1/2* m # 5 sage : factor ( ans ) # a n s 6 1/2*( m + 1)* m # k=0 1 k! = e, k=1 1 k 2 = π2 6, k=1 2 k k(k + 1) = log var ( k ) 2 print sum (1/ factorial (k),k,0, oo) 3 print sum (1/ k^2,k,1, oo) 4 print sum (2^( - k )/( k*(k+1)),k,1, oo) e 1/6*pi^2 -log(2) == Python = 1 >>> x = 3 2 >>> print x

98 42.2 solve 42 x x 3 == if while 1 >>> 2**3 == 8 2 True 3 >>> 8 > 9 4 False Sage == 1 var ( x y ) 2 x^2 + y^2 == x*y + 1 # 1 var ( x y ) # x, y ( = ) 2 f = x^2 + y^2 == x*y + 1 # f x ^2+ y ^2== x*y+1 f 1 print f # f 2 print f. rhs () # f 3 print f. lhs () # f x^2 + y^2 == x*y + 1 x*y + 1 x^2 + y^2 rhs, lhs right hand side left hand side 42.2 solve f(x) = g(x) x solve f(x) = g(x) 1 var ( x ) 2 solve (f(x )== g(x),x) # f(x)=g(x) x == 2x + 3 = 7 1 var ( x ) 2 solve (2* x +3==7, x) [x==2] x, y (solve) 1 var ( x y ) 2 solve ([ 1, 2], x,y) x + y = 6, x y = 4 92

99 42.2 solve 42 1 var ( x y ) 2 solve ([x+y==6, x-y ==4], x,y) [[x == 5, y == 1]] x x + 1 == 0 1 var ( x ); solve (x ^3+2* x +1==0) [x == -1/2*(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1) - 1/3*(I*sqrt(3) - 1)/(1/18 *sqrt(59)*sqrt(3) - 1/2)^(1/3), x == -1/2*(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)*(-I*sqrt(3) + 1) - 1/3*(-I*sqrt(3) - 1)/(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3), x == (1/18*sqrt(59) *sqrt(3) - 1/2)^(1/3) - 2/3/(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)] 1 a = solve (x ^3+2* x +1==0, x) 2 print a [0]; print "" 3 print a [1]; print "" 4 print a [2] x == -1/2*(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1) - 1/3*(I*sqrt(3) - 1)/(1/18 *sqrt(59)*sqrt(3) - 1/2)^(1/3) x == -1/2*(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)*(-I*sqrt(3) + 1) - 1/3*(-I*sqrt(3) - 1)/(1/ 18*sqrt(59)*sqrt(3) - 1/2)^(1/3) x == (1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3) - 2/3/(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3) rhs() 1 a = solve (x ^3+2* x +1==0, x) 2 a [0]. rhs () -1/2*(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1) - 1/3*(I*sqrt(3) - 1)/(1/18*sqrt(59)*sqrt(3) - 1/2) solve 1 solve (x ^5+3* x +1==0, x) [0=x^5+3x+1] 1 solve (x ^5+ x +1==0, x) [x == -1/2*(I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(23) - 25/54)^(1/3) - 1/18*(-I*sqrt(3) + 1)/(1/18*sqrt(3)*sqrt(23) - 25/54)^(1/3) + 1/3, x == -1/2*(-I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(23) - 25/54)^(1/3) - 1/18*(I*sqrt(3) + 1)/(1/18*sqrt(3)*sqrt(23) - 25/54)^(1/3) + 1/3, x == (1/18*sqrt(3)*sqrt(23) - 25/54)^(1/3) + 1/9/(1/18*sqrt(3)*sqrt(23) - 25/54)^(1/3) + 1/3, x == -1/2*I*sqrt(3) - 1/2, x == 1/2*I*sqrt(3) - 1/2] 1 solve ( sin (x )==1/2, x) 93

100 42.3 find root 42 [x == 1/6*pi] x = 1 6 π π 1 solve ( sin (x)* cos (x )==1/2, x) [sin(x) == 1/2/cos(x)] solve 42.3 find root find_root [a, b] 1 find_root (, a,b) find_root 1 x 2 + 2x 9 = 0 [ 5, 5] 1 sage : find_root (x ^2+2*x -9==0, -5,5) find_root (x ^2+2*x -9==0, -2,2) Traceback (click to the left of this block for traceback)... RuntimeError: f appears to have no zero on the interval f 0 find_root 1 sage : find_root ( tan (x )==8 - x ^3, -10,10) sage : find_root (x ^400, -5,5) x = 0 Newton 1 sage : limit (1/x, x=0, dir = + ) # 2 + Infinity # 3 sage : limit (1/x, x=0, dir = - ) # 4 - Infinity # 94

101 43 SAGE 1 sage : limit (1/x, x =0) 2 Infinity Infinity + lim x 0 xa (8) a assume a Restart worksheet 1 var ( a ) 2 limit (x^a, x =0) Traceback (click to the left of this block for traceback)... Is a positive, negative, or zero? #a a > 0 1 var ( a ) 2 assume (a >0) #a >0 3 limit (x^a, x =0) Traceback (click to the left of this block for traceback)... Is a an integer? # 1 var ( a ); assume (a >0) 2 limit (x^a, x=0, dir = + ) # 0 a 1 var ( a ); 2 assume (a, even ) 3 limit (x^a, x =0) 0 43 Sage f(x) 95

102 f(x) 43 SAGE 1 diff (f,x) derivative (f,x) f. derivative (x) f x 1 sage : diff ( sin (x),x) 2 cos (x) Sage f(x) f(x)dx 1 integral (f(x),x) integrate (f(x),x) 1 sage : integral ( sin (x),x) 2 -cos (x) 3 sage : integral (x ^4) 4 1/5* x^5 5 sage : integral (1/(1 - x ^3)) 6 1/3* sqrt (3)* arctan (1/3*(2* x + 1)* sqrt (3)) - 1/3* log (x - 1) 7 + 1/6* log (x^2 + x + 1) 1 sage : integral ( sin (x)/ log (x),x) 2 -( log (x)* integrate ( cos (x )/( x* log (x )^2), x) + cos (x ))/ log (x) 1 a = var ( a ) 2 integral ( cos (a*x),x) sin(a*x)/a x f f(x) b a f(x)dx 1 integral (f,x,a,b) integral (f,(x,a,b)) f. integral (x,a,b) integral integrate 1 sage : integral (x^2,x,0,2) 96

103 SAGE 2 8/ (1 x 4 ) 1/2 integral 1 0 (1 x4 ) 1/2 dx 1 var ( x ) 2 ans = integrate (1/ sqrt ((1 -x^4)),0,1) # a n s 3 print ans # a n s 4 print n( ans ) # a n s 1/4*beta(1/4, 1/2) beta β(x, y) = 1 0 t x 1 (1 t) y 1 dt Sage4* Maxima Sage integrate numerical_integral Sage f(x) b a f(x)dx 1 numerical_integral (f(x),a,b) ( ) max_points 1 sage : numerical_integral ( sin (x),0,pi) 2 ( , e - 14) sage : val = numerical_integral ( sin (x),0,pi) 2 sage : val [0]

104 SAGE numerical_integral sage : numerical_integral ( sin (1/ x),0,1) 2 ( , ) 3 sage : numerical_integral ( sin (1/ x),0,1, max_points =1000) 4 ( , e - 05) numerical_integral GNU Scientific Library(GSL) C maxima b a f(x)dx (maxima ) 1 f. nintegral (x,a,b, ) (,,, ) : 1 : 2 : 3 : 4 : 5 : 6 : 1 sage : f= sin (1/ x); f. nintegral (x,0,1) 2 ( , e -05, 8379, 1) 5 1 1/100 1 sage : f= sin (1/ x); f. nintegral (x,0,1,1/100) # 1/100 2 ( , , 567, 0)

105 44 TAYLOR 44 Taylor 44.1 Taylor f f Taylor ( Taylor ) f(x) = f(a) + f (a) 1! (x 1) + f (a) 2! (x a) f (n) (a) (x a) n + n! a = 0 Maclaurin Taylor Taylor f(a) + f (a) 1! (x 1) + f (a) 2! (x a) f (n) (a) n! (x a) n f(x) x=a x n Taylor 1 taylor (f(x),x,a,n) e x x = 0 3 Taylor 1 var ( x ) 2 taylor (e^x,x,0,3) 1/6*x^3 + 1/2*x^2 + x f(x, y) Taylor 2 f(x, y) (a, b) n Taylor 1 taylor (f(x,y),(x,a),(y,b),n) 1 var ( x y ) 2 taylor ( sin (x+y),(x,0),(y,0),4) 1 6 x3 1 2 x2 y 1 2 xy2 1 6 y3 + x + y 44.2 Taylor cos(x) Taylor x 10 1 var ( x ) # x 2 f = taylor ( cos (x),x,0,10) # f c o s (x) p1 = plot (f,(x, -9,9)) # f 4 p2 = plot ( cos (x),(x, -9,9), color = red ) # cos (x) 5 p = p1 + p2; # p 6 p. show ( ymin = -10, ymax =5) # p 99

106 Sage (1) lim h 0 (x + h) 3 x 3 h (2) lim x ( x x x! (3) lim x ) 1/x x! factorial(x) (4) 3x 37 (x + 1)(x 4) (5) 9 9x 2x 2 + 7x 4 (6) 4x 2 (x 1)(x 2) 2 (7) 8x 2 12 x(x 2 + 2x 6) 11 (8) = golden ratio (9) = pi x (10) sin(cx) c (11) x /(1 + x 2 )( x abs(x) (absolute value )) x (12) 1 a + x (13) a x (14) 1 a2 + x 2 (15) x 2 [0, 1] (16) sin(x), [0, π] (17) x cos(x) [0, π] x = 0 5 (18) x 2 e x (19) sin(x)/(1 + x 2 ) Sage answer.sage sage answer.sage 100

107 45 (print) answer.sage 1 # -*- coding : utf -8 -*- 2 print (1), 3 var ( x h ) 4 f1 = ((x+h)^3 -x ^3)/ h 5 print limit (f1,h =0) 101

108 dy dx = y x (9) y(x) = x Ce x C dy dx = y x, y(0) = 2 3 x = 0 y(0) = 1 + C = 2/3 C C = 1/3 (10) y(x) = x ex (11) (9) y = y(x) (x, y) y x (x, y) (1, y x) (slope field) m = (1 + (y x) 2 ) 1/2 V (x, y) = (m, m(y x)) 1 var ( x y ) 2 m = sqrt (1+(y-x )^2)^( -1) 3 plot_vector_field ( (m,m*(y-x)), (x, -3,3), (y, -3,3) ) (9) (0, y(0)) y = x + 1 x ( ) y = x + 1 ( ) (11)( ) 1 var ( x y ) 2 m = sqrt (1+(y-x )^2)^( -1) 3 vecf = plot_vector_field ( (m,m*(y-x)), (x, -4,4), (y, -4,4),color =" red ") 4 sol = plot (x +1 -(1/3)* e^x, (x, -4,4)) 5 asym = plot (x+1, (x, -4,4), color = black ) 6 ( vecf + sol + asym ). show ( ymin =-4, ymax =4) 102

109 46.2 Euler Euler Euler f(x, y) dy = f(x, y) (12) dx y(x 0 ) = y 0 (x 0, y 0 ) dx y(x + dx) = y(x) + f(x, y)dx (13) y x x + dx Euler y(x) x x 1 = x 0 + x y x 2 = x 1 + x y x n = x 0 + n x y y 1 = y 0 + f(x 0, y 0 ) x (14) y 2 = y 1 + f(x 1, y 1 ) x (15) y n = y n 1 + f(x n 1, y n 1 ) x (16) Euler (10) y(x) x n ((x 0, y 0 ), (x 1, y 1 ),, (x n, y n )) 1 x0, y0 = 0, 2/3 # 2 Dx= 0.5 # Delta x 3 var ( x y ) 4 f(x,y) = y-x # f(x,y) 5 SOL = [(x0,y0 )] # 6 for j in range (1,11): 103