# 기초수학실습-0114-서기영

## 2399 days 전, sky 작성

[-9,-7..30]
 [-9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29] [-9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29]
[-18,-15..100]
 [-18, -15, -12, -9, -6, -3, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99] [-18, -15, -12, -9, -6, -3, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99]
[1..100]
 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100] [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100]
[a.is_prime() for a in [1..100]]
 [False, True, True, False, True, False, True, False, False, False, True, False, True, False, False, False, True, False, True, False, False, False, True, False, False, False, False, False, True, False, True, False, False, False, False, False, True, False, False, False, True, False, True, False, False, False, True, False, False, False, False, False, True, False, False, False, False, False, True, False, True, False, False, False, False, False, True, False, False, False, True, False, True, False, False, False, False, False, True, False, False, False, True, False, False, False, False, False, True, False, False, False, False, False, False, False, True, False, False, False] [False, True, True, False, True, False, True, False, False, False, True, False, True, False, False, False, True, False, True, False, False, False, True, False, False, False, False, False, True, False, True, False, False, False, False, False, True, False, False, False, True, False, True, False, False, False, True, False, False, False, False, False, True, False, False, False, False, False, True, False, True, False, False, False, False, False, True, False, False, False, True, False, True, False, False, False, False, False, True, False, False, False, True, False, False, False, False, False, True, False, False, False, False, False, False, False, True, False, False, False]
b=[] for a in [1..100]: if a.is_prime(): b.append(a) print b
 [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] [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]
P(x)=(x-2)^3*(x^2-4*x+3)*2 P(x)
 2*(x^2 - 4*x + 3)*(x - 2)^3 2*(x^2 - 4*x + 3)*(x - 2)^3
expand(P(x))
 2*x^5 - 20*x^4 + 78*x^3 - 148*x^2 + 136*x - 48 2*x^5 - 20*x^4 + 78*x^3 - 148*x^2 + 136*x - 48
factor(2*x^5 - 20*x^4 + 78*x^3 - 148*x^2 + 136*x - 48)
 2*(x - 1)*(x - 2)^3*(x - 3) 2*(x - 1)*(x - 2)^3*(x - 3)
(P(x)/(x-1))
 2*(x^2 - 4*x + 3)*(x - 2)^3/(x - 1) 2*(x^2 - 4*x + 3)*(x - 2)^3/(x - 1)
(P(x)/(x-1)).simplify_full()
 2*x^4 - 18*x^3 + 60*x^2 - 88*x + 48 2*x^4 - 18*x^3 + 60*x^2 - 88*x + 48
W.<x>=QQ[] W(P(x)).quo_rem(x-5)
 (2*x^4 - 10*x^3 + 28*x^2 - 8*x + 96, 432) (2*x^4 - 10*x^3 + 28*x^2 - 8*x + 96, 432)
W.<x>=QQ[] W(P(x)).quo_rem(x^2-5*x+1)
 (2*x^3 - 10*x^2 + 26*x - 8, 70*x - 40) (2*x^3 - 10*x^2 + 26*x - 8, 70*x - 40)
var ('x, y') #두개이상일때 변수선언 한번 선언하면 더이상 할 필요없음 solve( [3*x - y == 2, -2*x -y == 1 ], x,y)
 [[x == (1/5), y == (-7/5)]] [[x == (1/5), y == (-7/5)]]
solve( [ 2*x + y == -1 , -4*x - 2*y == 2],x,y)
 [[x == -1/2*r2 - 1/2, y == r2]] [[x == -1/2*r2 - 1/2, y == r2]]
solve( [ 2*x - y == -1 , 2*x - y == 2],x,y)
 [] []
var ('z') solve([ 2*x + 3*y + 5*z == 1, 4*x + 6*y + 10*z == 2, 6*x + 9*y + 15*z == 3], x,y,z)
 [[x == -5/2*r3 - 3/2*r4 + 1/2, y == r4, z == r3]] [[x == -5/2*r3 - 3/2*r4 + 1/2, y == r4, z == r3]]
G(X)=x^3-x-7*x^2+7 G(X)
 x^3 - 7*x^2 - x + 7 x^3 - 7*x^2 - x + 7
solve(G(X), x)
 [x == 7, x == -1, x == 1] [x == 7, x == -1, x == 1]
var('t') solve(abs(t-7)>=3, t) #ineq=부등식
 #0: solve_rat_ineq(ineq=abs(t-7) >= 3) [[t == 10], [t == 4], [t < 4], [10 < t]] #0: solve_rat_ineq(ineq=abs(t-7) >= 3) [[t == 10], [t == 4], [t < 4], [10 < t]]
solve([2*x+y==17, x-3*y==-16], x,y)
 [[x == 5, y == 7]] [[x == 5, y == 7]]
solve(x^3-x==7*x^2-7, x)
 [x == 7, x == -1, x == 1] [x == 7, x == -1, x == 1]
f(x) = sin(x) #sin/cos p = plot(f(x), (x, -pi/2, pi/2)) #범위 p.show() #보여주기
f(x)=cos(x) p = plot(f(x), (x,-pi/2, pi/2), axes_labels=['x','cos(x)'], color='orange') p.show()
f(x) = sin(x) g(x) = cos(x) p = plot(f(x),(x,-pi/2,pi/2), color='black') q = plot(g(x), (x,-pi/2, pi/2), color='red') r = p + q r.show()
sage: p = plot(f(x), (x,-pi/2, pi/2), linestyle='--', thickness=0.7) sage: p.show()
sage: f(x) = (x^3 + x^2 + x)/(x^2 - x -2 ) #유리함수 sage: p = plot(f(x), (x, -5,5)) sage: p.show()
p.show(xmin=-2, xmax=4, ymin=-20, ymax=20) #최소범위, 최대범위
m(x)=sin(pi*x-pi) p = plot(m(x), (x,-1, 1), linestyle='--', thickness=17, color='red') p.show()
l(x)=cos(pi*x-pi) p = plot(l(x), (x,-1, 1), linestyle='--', thickness=17, color='blue') p.show()
p = plot(m(x),(x,-1,1), color='red') q = plot(l(x), (x,-1,1), color='blue') r = p + q r.show()
plot(1/x, (x, -1, 1), ymin=-10, ymax=10) #x, -1, 1은 범위
P(x)=2*x^3+3*x^2-5*x-6 solve(P(x)==0, x) # x절편 구하기
 [x == -2, x == -1, x == (3/2)] [x == -2, x == -1, x == (3/2)]
plot(P(x), (x, -4, 4), ymax=20, ymin=-20)
P(x)=4*x^4++4*x^3-9*x^2-x+2 solve(P(x)==0, x)
 [x == (1/2), x == -2, x == 1, x == (-1/2)] [x == (1/2), x == -2, x == 1, x == (-1/2)]
plot(P(x), (x, -3, 3), ymax=20, ymin=-20)
f(x)=2/(x-5) plot(f(x), (x, 2, 7), ymax=10, ymin=-10)
f(x)=-4/((x-4)^2) plot(f(x), (x, 2, 7), ymax=10, ymin=-20)
f(x)=(2*x^2+1)/(x^2-4*x+3) plot(f(x), (x, -2, 6), ymax=20, ymin=-20)
f(x)=(2*x^2-1)/(x^2+1) plot(f(x), (x, -6, 6), ymax=3)+plot(2, (x, -6, 6), color='red')
f(x)=(x^2-4)/(x-2) plot(f(x), (x, -6, 6))
f(x)=(x+2)/(x^2-x-6) plot(f(x), (x, -4, 6), ymax=20, ymin=-20)
x,y = var("x y") f(x,y) = x^2 - y^2 p = plot3d(f(x,y), (x,-10,10), (y,-10,10)) p.show()
 Sleeping...