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calculus_doctest.py
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calculus_doctest.py
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"""
This file (./calculus_doctest.sage) was *autogenerated* from ./calculus.tex,
with sagetex.sty version 2011/05/27 v2.3.1.
It contains the contents of all the sageexample environments from this file.
You should be able to doctest this file with:
sage -t ./calculus_doctest.sage
It is always safe to delete this file; it is not used in typesetting your
document.
Sage example in ./calculus.tex, line 76::
sage: bool(arctan(1+abs(x)) == pi/2 - arctan(1/(1+abs(x))))
False
Sage example in ./calculus.tex, line 121::
sage: a, x = var('a, x'); y = cos(x+a) * (x+1); y
(x + 1)*cos(a + x)
sage: y.subs(a=-x); y.subs(x=pi/2, a=pi/3); y.subs(x=0.5, a=2.3)
x + 1
-1/4*sqrt(3)*(pi + 2)
-1.41333351100299
sage: y(a=-x); y(x=pi/2, a=pi/3); y(x=0.5, a=2.3)
x + 1
-1/4*sqrt(3)*(pi + 2)
-1.41333351100299
Sage example in ./calculus.tex, line 143::
sage: x, y, z = var('x, y, z') ; q = x*y + y*z + z*x
sage: bool(q(x=y, y=z, z=x) == q), bool(q(z=y)(y=x) == 3*x^2)
(True, True)
Sage example in ./calculus.tex, line 155::
sage: y, z = var('y, z'); f = x^3 + y^2 + z
sage: f.substitute(x^3 == y^2, z==1)
2*y^2 + 1
Sage example in ./calculus.tex, line 176::
sage: f(x)=(2*x+1)^3 ; f(-3)
-125
sage: f.expand()
x |--> 8*x^3 + 12*x^2 + 6*x + 1
Sage example in ./calculus.tex, line 193::
sage: y = var('y'); u = sin(x) + x*cos(y)
sage: v = u.function(x, y); v
(x, y) |--> x*cos(y) + sin(x)
sage: w(x, y) = u; w
(x, y) |--> x*cos(y) + sin(x)
Sage example in ./calculus.tex, line 240::
sage: x, y = SR.var('x,y')
sage: p = (x+y)*(x+1)^2
sage: p2 = p.expand(); p2
x^3 + x^2*y + 2*x^2 + 2*x*y + x + y
Sage example in ./calculus.tex, line 251::
sage: p2.collect(x)
x^3 + x^2*(y + 2) + x*(2*y + 1) + y
Sage example in ./calculus.tex, line 260::
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 example in ./calculus.tex, line 416::
sage: (x^x/x).simplify()
x^(x - 1)
Sage example in ./calculus.tex, line 426::
sage: f = (e^x-1) / (1+e^(x/2)); f.canonicalize_radical()
e^(1/2*x) - 1
Sage example in ./calculus.tex, line 435::
sage: f = cos(x)^6 + sin(x)^6 + 3 * sin(x)^2 * cos(x)^2
sage: f.simplify_trig()
1
Sage example in ./calculus.tex, line 447::
sage: f = cos(x)^6; 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); f.expand_trig()
5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5
Sage example in ./calculus.tex, line 482::
sage: n = var('n'); f = factorial(n+1)/factorial(n)
sage: f.simplify_factorial()
n + 1
Sage example in ./calculus.tex, line 502::
sage: f = sqrt(abs(x)^2); f.canonicalize_radical()
abs(x)
sage: f = log(x*y); f.canonicalize_radical()
log(x) + log(y)
Sage example in ./calculus.tex, line 592::
sage: assume(x > 0); bool(sqrt(x^2) == x)
True
sage: forget(x > 0); bool(sqrt(x^2) == x)
False
sage: n = var('n'); assume(n, 'integer'); sin(n*pi)
0
Sage example in ./calculus.tex, line 600::
sage: forget(n, 'integer');
Sage example in ./calculus.tex, line 690::
sage: a = var('a')
sage: c = (a+1)^2 - (a^2+2*a+1)
Sage example in ./calculus.tex, line 700::
sage: eq = c * x == 0
Sage example in ./calculus.tex, line 707::
sage: eq2 = eq / c; eq2
x == 0
sage: solve(eq2, x)
[x == 0]
Sage example in ./calculus.tex, line 715::
sage: solve(eq, x)
[x == x]
Sage example in ./calculus.tex, line 725::
sage: expand(c)
0
Sage example in ./calculus.tex, line 738::
sage: c = cos(a)^2 + sin(a)^2 - 1
sage: eq = c*x == 0
sage: solve(eq, x)
[x == 0]
Sage example in ./calculus.tex, line 750::
sage: c.simplify_trig()
0
sage: c.is_zero()
True
Sage example in ./calculus.tex, line 839::
sage: z, phi = var('z, phi')
sage: eq = z**2 - 2/cos(phi)*z + 5/cos(phi)**2 - 4 == 0; eq
z^2 - 2*z/cos(phi) + 5/cos(phi)^2 - 4 == 0
Sage example in ./calculus.tex, line 852::
sage: eq.lhs()
z^2 - 2*z/cos(phi) + 5/cos(phi)^2 - 4
sage: eq.rhs()
0
Sage example in ./calculus.tex, line 861::
sage: solve(eq, z)
[z == -(2*sqrt(cos(phi)^2 - 1) - 1)/cos(phi),
z == (2*sqrt(cos(phi)^2 - 1) + 1)/cos(phi)]
Sage example in ./calculus.tex, line 871::
sage: y = var('y'); solve(y^7==y, y)
[y == 1/2*I*sqrt(3) + 1/2, y == 1/2*I*sqrt(3) - 1/2, y == -1,
y == -1/2*I*sqrt(3) - 1/2, y == -1/2*I*sqrt(3) + 1/2, y == 1, y == 0]
Sage example in ./calculus.tex, line 880::
sage: solve(x^2-1, x, solution_dict=True)
[{x: -1}, {x: 1}]
Sage example in ./calculus.tex, line 894::
sage: solve([x+y == 3, 2*x+2*y == 6], x, y)
[[x == -r1 + 3, y == r1]]
Sage example in ./calculus.tex, line 910::
sage: solve([cos(x)*sin(x) == 1/2, x+y == 0], x, y)
[[x == 1/4*pi + pi*z..., y == -1/4*pi - pi*z...]]
Sage example in ./calculus.tex, line 920::
sage: solve(x^2+x-1 > 0, x)
[[x < -1/2*sqrt(5) - 1/2], [x > 1/2*sqrt(5) - 1/2]]
Sage example in ./calculus.tex, line 943::
sage: x, y, z = var('x, y, z')
sage: solve([x^2 * y * z == 18, x * y^3 * z == 24,\
....: x * y * z^4 == 6], x, y, z)
[[x == 3, y == 2, z == 1],
[x == (1.3372150673296... - 2.685489874065...*I),
y == (-1.7004342714592... + 1.0528643257547...*I),
z == (0.93247222940435... - 0.36124166618715...*I)], ...]
Sage example in ./calculus.tex, line 975::
sage: expr = sin(x) + sin(2 * x) + sin(3 * x)
sage: solve(expr, x)
[sin(3*x) == -sin(2*x) - sin(x)]
Sage example in ./calculus.tex, line 983::
sage: find_root(expr, 0.1, pi) # abs tol 1e-12
2.0943951023931957
Sage example in ./calculus.tex, line 989::
sage: f = expr.simplify_trig(); f
2*(2*cos(x)^2 + cos(x))*sin(x)
sage: solve(f, x)
[x == 0, x == 2/3*pi, x == 1/2*pi]
Sage example in ./calculus.tex, line 1022::
sage: (x^3+2*x+1).roots(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), 1),
(-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), 1),
((1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)
- 2/3/(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3), 1)]
Sage example in ./calculus.tex, line 1058::
sage: (x^3+2*x+1).roots(x, ring=RR)
[(-0.453397651516404, 1)]
Sage example in ./calculus.tex, line 1062::
sage: (x^3+2*x+1).roots(x, ring=CC)
[(-0.453397651516404, 1),
(0.226698825758202 - 1.46771150871022*I, 1),
(0.226698825758202 + 1.46771150871022*I, 1)]
Sage example in ./calculus.tex, line 1086::
sage: solve(x^(1/x)==(1/x)^x, x)
[(1/x)^x == x^(1/x)]
Sage example in ./calculus.tex, line 1124::
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 example in ./calculus.tex, line 1171::
sage: k, n = var('k, n')
sage: sum(k, k, 1, n).factor()
1/2*(n + 1)*n
Sage example in ./calculus.tex, line 1179::
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 example in ./calculus.tex, line 1189::
sage: k, n = var('k, n')
sage: sum(binomial(n,k), k, 0, n),\
....: sum(k * binomial(n, k), k, 0, n),\
....: sum((-1)^k*binomial(n,k), k, 0, n)
(2^n, 2^(n - 1)*n, 0)
Sage example in ./calculus.tex, line 1199::
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 example in ./calculus.tex, line 1212::
sage: assume(abs(q) < 1)
sage: sum(a*q^k, k, 0, infinity)
-a/(q - 1)
Sage example in ./calculus.tex, line 1218::
sage: forget(); assume(q > 1); sum(a*q^k, k, 0, infinity)
Traceback (most recent call last):
...
ValueError: Sum is divergent.
Sage example in ./calculus.tex, line 1300::
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 example in ./calculus.tex, line 1317::
sage: limit(f(x), x = pi/4, dir='minus')
+Infinity
sage: limit(f(x), x = pi/4, dir='plus')
-Infinity
Sage example in ./calculus.tex, line 1368::
sage: u(n) = n^100 / 100^n
sage: u(1.);u(2.);u(3.);u(4.);u(5.);u(6.);u(7.);u(8.);u(9.);u(10.)
0.0100000000000000
1.26765060022823e26
5.15377520732011e41
1.60693804425899e52
7.88860905221012e59
6.53318623500071e65
3.23447650962476e70
2.03703597633449e74
2.65613988875875e77
1.00000000000000e80
Sage example in ./calculus.tex, line 1389::
sage: plot(u(x), x, 1, 40)
Graphics object consisting of 1 graphics primitive
Sage example in ./calculus.tex, line 1407::
sage: v(x) = diff(u(x), x); sol = solve(v(x) == 0, x); sol
[x == 50/log(10), x == 0]
sage: floor(sol[0].rhs())
21
Sage example in ./calculus.tex, line 1420::
sage: limit(u(n), n=infinity)
0
sage: n0 = find_root(u(n) - 1e-8 == 0, 22, 1000); n0
105.07496210187252
Sage example in ./calculus.tex, line 1502::
sage: ((1+arctan(x))^(1/x)).series(x==0, 3)
(e) + (-1/2*e)*x + (1/8*e)*x^2 + Order(x^3)
Sage example in ./calculus.tex, line 1507::
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 example in ./calculus.tex, line 1520::
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 example in ./calculus.tex, line 1537::
sage: taylor((x**3+x)**(1/3) - (x**3-x)**(1/3), x, infinity, 2)
2/3/x
Sage example in ./calculus.tex, line 1577::
sage: tan(4*arctan(1/5)).simplify_trig()
120/119
sage: tan(pi/4+arctan(1/239)).simplify_trig()
120/119
Sage example in ./calculus.tex, line 1591::
sage: f = arctan(x).series(x, 10); 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(); pi.n()
3.14159268240440
3.14159265358979
Sage example in ./calculus.tex, line 1662::
sage: k = var('k')
sage: sum(1/k^2, k, 1, infinity),\
....: sum(1/k^4, k, 1, infinity),\
....: sum(1/k^5, k, 1, infinity)
(1/6*pi^2, 1/90*pi^4, zeta(5))
Sage example in ./calculus.tex, line 1689::
sage: s = 2*sqrt(2)/9801*(sum((factorial(4*k)) * (1103+26390*k) /
....: ((factorial(k)) ^ 4 * 396 ^ (4 * k)) for k in (0..11)))
sage: (1/s).n(digits=100)
3.141592653589793238462643383279502884197169399375105820974...
sage: (pi-1/s).n(digits=100).n()
-4.36415445739398e-96
Sage example in ./calculus.tex, line 1722::
sage: n = var('n'); u = sin(pi*(sqrt(4*n^2+1)-2*n))
sage: taylor(u, n, infinity, 3)
1/4*pi/n - 1/384*(6*pi + pi^3)/n^3
Sage example in ./calculus.tex, line 1762::
sage: diff(sin(x^2), x)
2*x*cos(x^2)
sage: function('f')(x); function('g')(x); diff(f(g(x)), x)
f(x)
g(x)
D[0](f)(g(x))*diff(g(x), x)
sage: diff(ln(f(x)), x)
diff(f(x), x)/f(x)
Sage example in ./calculus.tex, line 1780::
sage: f(x,y) = x*y + sin(x^2) + e^(-x); derivative(f, x)
(x, y) |--> 2*x*cos(x^2) + y - e^(-x)
sage: derivative(f, y)
(x, y) |--> x
Sage example in ./calculus.tex, line 1803::
sage: x, y = var('x, y'); f = ln(x**2+y**2) / 2
sage: delta = diff(f,x,2) + diff(f,y,2)
sage: delta.simplify_rational()
0
Sage example in ./calculus.tex, line 1854::
sage: sin(x).integral(x, 0, pi/2)
1
sage: integrate(1/(1+x^2), x)
arctan(x)
sage: integrate(1/(1+x^2), x, -infinity, infinity)
pi
sage: integrate(exp(-x**2), x, 0, infinity)
1/2*sqrt(pi)
Sage example in ./calculus.tex, line 1864::
sage: integrate(exp(-x), x, -infinity, infinity)
Traceback (most recent call last):
...
ValueError: Integral is divergent.
Sage example in ./calculus.tex, line 1878::
sage: u = var('u'); f = x * cos(u) / (u^2 + x^2)
sage: assume(x>0); f.integrate(u, 0, infinity)
1/2*pi*e^(-x)
sage: forget(); assume(x<0); f.integrate(u, 0, infinity)
-1/2*pi*e^x
Sage example in ./calculus.tex, line 1904::
sage: integral_numerical(sin(x)/x, 0, 1) # abs tol 1e-12
(0.946083070367183, 1.0503632079297087e-14)
sage: g = integrate(exp(-x**2), x, 0, infinity)
sage: g, g.n() # abs tol 1e-12
(1/2*sqrt(pi), 0.886226925452758)
sage: approx = integral_numerical(exp(-x**2), 0, infinity)
sage: approx # abs tol 1e-12
(0.8862269254527568, 1.714774436012769e-08)
sage: approx[0]-g.n() # abs tol 1e-12
-1.11022302462516e-15
Sage example in ./calculus.tex, line 2228::
sage: A = matrix(QQ, [[1,2],[3,4]]); A
[1 2]
[3 4]
Sage example in ./calculus.tex, line 2468::
sage: A = matrix(QQ, [[2,4,3],[-4,-6,-3],[3,3,1]])
sage: A.characteristic_polynomial()
x^3 + 3*x^2 - 4
sage: A.eigenvalues()
[1, -2, -2]
sage: A.minimal_polynomial().factor()
(x - 1) * (x + 2)^2
Sage example in ./calculus.tex, line 2487::
sage: A.eigenvectors_right()
[(1, [
(1, -1, 1)
], 1), (-2, [
(1, -1, 0)
], 2)]
Sage example in ./calculus.tex, line 2499::
sage: A.jordan_form(transformation=True)
(
[ 1| 0 0]
[--+-----] [ 1 1 1]
[ 0|-2 1] [-1 -1 0]
[ 0| 0 -2], [ 1 0 -1]
)
Sage example in ./calculus.tex, line 2533::
sage: A = matrix(QQ, [[1,-1/2],[-1/2,-1]])
sage: A.jordan_form()
Traceback (most recent call last):
...
RuntimeError: Some eigenvalue does not exist in Rational Field.
Sage example in ./calculus.tex, line 2543::
sage: A = matrix(QQ, [[1,-1/2],[-1/2,-1]])
sage: A.minimal_polynomial()
x^2 - 5/4
Sage example in ./calculus.tex, line 2557::
sage: R = QQ[sqrt(5)]
sage: A = A.change_ring(R)
sage: A.jordan_form(transformation=True, subdivide=False)
(
[ 1/2*sqrt5 0] [ 1 1]
[ 0 -1/2*sqrt5], [-sqrt5 + 2 sqrt5 + 2]
)
Sage example in ./calculus.tex, line 2597::
sage: K.<sqrt2> = NumberField(x^2 - 2)
sage: L.<sqrt3> = K.extension(x^2 - 3)
sage: A = matrix(L, [[2, sqrt2*sqrt3, sqrt2], \
....: [sqrt2*sqrt3, 3, sqrt3], \
....: [sqrt2, sqrt3, 1]])
sage: A.jordan_form(transformation=True)
(
[6|0|0]
[-+-+-]
[0|0|0] [ 1 1 0]
[-+-+-] [1/2*sqrt2*sqrt3 0 1]
[0|0|0], [ 1/2*sqrt2 -sqrt2 -sqrt3]
)
"""