% This file was *autogenerated* from P3b.sagetex.sage with % sagetex.py version 2015/08/26 v3.0-92d9f7a sage: a = 5 sage: type(a) sage: a = 5/3 sage: type(a) sage: a = 'hello' sage: type(a) sage: phi = var('phi') sage: find_root(cos(phi)==sin(phi),0,pi/2) 0.785398163397 sage: solve(x^2+x-1 > 0, x) [[x < -1/2*sqrt(5) - 1/2], [x > 1/2*sqrt(5) - 1/2]] sage: diff(sin(x^2), x, 4) 16*x^4*sin(x^2) - 48*x^2*cos(x^2) - 12*sin(x^2) sage: x, y = var('x,y') sage: f = x^2 + 17*y^2 sage: f.diff(y) 34*y sage: integral(x*sin(x^2), x) -1/2*cos(x^2) sage: integral(x/(x^2+1), x, 0, 1) 1/2*log(2) sage: f = 1/((1+x)*(x-1)) sage: f.partial_fraction(x) -1/2/(x + 1) + 1/2/(x - 1) sage: simplify(arccos(sin(pi/3))) 1/6*pi sage: simplify(exp(i*pi/6)) 1/2*sqrt(3) + 1/2*I sage: a = var('a') sage: y = cos(x+a) * (x+1) sage: y.subs(a=-x) x + 1 sage: y.subs(x=pi/2, a=pi/3) -1/4*sqrt(3)*(pi + 2) sage: y, z = var('y, z') sage: f = x^3 + y^2 + z sage: f.subs_expr(x^3 == y^2, z==1) 2*y^2 + 1 sage: f(x)=(2*x+1)^3 sage: f(-3) -125 sage: f.expand() x |--> 8*x^3 + 12*x^2 + 6*x + 1 sage: ((x+y+sin(x))^2).expand().collect(sin(x)) x^2 + 2*x*y + y^2 + 2*(x + y)*sin(x) + sin(x)^2 sage: u = sin(x) + x*cos(y) sage: v = u.function(x, y) sage: v (x, y) |--> x*cos(y) + sin(x) sage: f = (e^x-1) / (1+e^(x/2)) sage: f.simplify_exp() e^(1/2*x) - 1 sage: f = cos(x)^6 + sin(x)^6 + 3 * sin(x)^2 * cos(x)^2 sage: f.simplify_trig() 1 sage: f = cos(x)^6 sage: f.reduce_trig() 1/32*cos(6*x) + 3/16*cos(4*x) + 15/32*cos(2*x) + 5/16 sage: f = sin(5 * x) sage: f.expand_trig() 5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5 sage: n = var('n') sage: f = factorial(n+1)/factorial(n) sage: f.simplify_factorial() n + 1 sage: f = sqrt(abs(x)^2) sage: f.simplify_radical() abs(x) sage: assume(x > 0) None sage: bool(sqrt(x^2) == x) True sage: forget(x > 0) None sage: bool(sqrt(x^2) == x) False sage: n = var('n') sage: assume(n, 'integer') None sage: sin(n*pi).simplify() 0 sage: t = var('t') sage: x = function('x',t) sage: DE = diff(x, t) + x - 1 sage: desolve(DE, [x,t]) (_C + e^t)*e^(-t) sage: x = var('x') sage: y = function('y', x) sage: desolve(diff(y,x,x) + x*diff(y,x) + y == 0, y, [0,0,1]) -1/2*I*sqrt(2)*sqrt(pi)*erf(1/2*I*sqrt(2)*x)*e^(-1/2*x^2) sage: k, n = var('k, n') sage: sum(k, k, 1, n).factor() 1/2*(n + 1)*n sage: sum(k * binomial(n, k), k, 0, n) 2^(n - 1)*n sage: n, k, y = var('n, k, y') sage: sum(binomial(n,k) * x^k * y^(n-k), k, 0, n) (x + y)^n sage: a, q, k, n = var('a, q, k, n') sage: sum(a*q^k, k, 0, n) (a*q^(n + 1) - a)/(q - 1) sage: a, q, k, n = var('a, q, k, n') sage: sum(a*q^k, k, 0, n) (a*q^(n + 1) - a)/(q - 1) sage: assume(abs(q) < 1) None sage: sum(a*q^k, k, 0, infinity) -a/(q - 1) sage: limit((x**(1/3) - 2) / ((x + 19)**(1/3) - 3), x = 8) 9/4 sage: f(x) = (cos(pi/4-x)-tan(x))/(1-sin(pi/4 + x)) sage: limit(f(x), x = pi/4) Infinity sage: limit(f(x), x = pi/4, dir='minus') +Infinity sage: limit(f(x), x = pi/4, dir='plus') -Infinity sage: u(n) = n^100 / 100^n sage: limit(u(n), n=infinity) 0 sage: taylor((1+arctan(x))**(1/x), x, 0, 3) 1/16*x^3*e + 1/8*x^2*e - 1/2*x*e + e sage: (ln(2*sin(x))).series(x==pi/6, 3) (sqrt(3))*(-1/6*pi + x) + (-2)*(-1/6*pi + x)^2 + Order(-1/216*(pi - 6*x)^3) sage: (ln(2*sin(x))).series(x==pi/6, 3).truncate() -1/18*(pi - 6*x)^2 - 1/6*sqrt(3)*(pi - 6*x) sage: f = arctan(x).series(x, 10) sage: f 1*x + (-1/3)*x^3 + 1/5*x^5 + (-1/7)*x^7 + 1/9*x^9 + Order(x^10) sage: (16*f.subs(x==1/5) - 4*f.subs(x==1/239)).n() 3.14159268240440 sage: 1+1 2 sage: factor(x^2 + 2*x + 1) (x + 1)^2 %0c95e22d5dc91a28a96308e5af3d31aa% md5sum of corresponding .sage file (minus "goboom", "current_tex_line", and pause/unpause lines)