from sympy import symbols, cos, sin, diff, simplify
x = symbols('x', real=True)
f = -3*cos(x)
result = diff(f, x)
expected = 3*sin(x)
print("f(x) =", f)
print("f'(x) =", result)
print("Kontroll =", simplify(result - expected))
assert simplify(result - expected) == 0

Utskrift
f(x) = -3*cos(x)
f'(x) = 3*sin(x)
Kontroll = 0

from sympy import symbols, sin, cos, diff, simplify, trigsimp
x = symbols('x', real=True)
g = sin(x)**2
result = diff(g, x)
print("g(x) =", g)
print("g'(x) =", result)
print("Trigonometrisk forenkling =", trigsimp(result, method='fu'))
print("Kontroll mot sin(2x) =", simplify(result - sin(2*x)))
assert simplify(result - sin(2*x)) == 0

Utskrift
g(x) = sin(x)**2
g'(x) = 2*sin(x)*cos(x)
Trigonometrisk forenkling = sin(2*x)
Kontroll mot sin(2x) = 0

from sympy import symbols, exp, diff, factor, simplify
x = symbols('x', real=True)
h = x**3*exp(-x)
result = diff(h, x)
expected = x**2*exp(-x)*(3-x)
print("h(x) =", h)
print("h'(x) =", result)
print("Faktorisert =", factor(result))
print("Kontroll =", simplify(result - expected))
assert simplify(result - expected) == 0

Utskrift
h(x) = x**3*exp(-x)
h'(x) = -x**3*exp(-x) + 3*x**2*exp(-x)
Faktorisert = -x**2*(x - 3)*exp(-x)
Kontroll = 0

from sympy import symbols, integrate, Rational, simplify
x = symbols('x', real=True)
expr = x**2 + 2*x - 3
F = integrate(expr, x)
value = integrate(expr, (x, 1, 2))
print("Integrand =", expr)
print("Antiderivert =", F)
print("F(2) =", F.subs(x, 2))
print("F(1) =", F.subs(x, 1))
print("Bestemt integral =", value)
assert simplify(value - Rational(7,3)) == 0

Utskrift
Integrand = x**2 + 2*x - 3
Antiderivert = x**3/3 + x**2 - 3*x
F(2) = 2/3
F(1) = -5/3
Bestemt integral = 7/3

from sympy import symbols, apart, integrate, diff, simplify
x = symbols('x', real=True)
expr = 3*x/(x**2 - x - 2)
partial = apart(expr, x)
F = integrate(expr, x)
print("Integrand =", expr)
print("Delbrokoppspalting =", partial)
print("Antiderivert =", F)
print("Derivert tilbake =", simplify(diff(F, x)))
print("Kontroll =", simplify(diff(F, x) - expr))
assert simplify(diff(F, x) - expr) == 0

Utskrift
Integrand = 3*x/(x**2 - x - 2)
Delbrokoppspalting = 1/(x + 1) + 2/(x - 2)
Antiderivert = 2*log(x - 2) + log(x + 1)
Derivert tilbake = 3*x/((x - 2)*(x + 1))
Kontroll = 0

from sympy import symbols, log, integrate, diff, simplify
x = symbols('x', positive=True)
expr = x*log(x)
F = integrate(expr, x)
expected = x**2*log(x)/2 - x**2/4
print("Integrand =", expr)
print("Antiderivert =", F)
print("Forventet form =", expected)
print("Derivert tilbake =", simplify(diff(expected, x)))
print("Kontroll =", simplify(F - expected))
assert simplify(F - expected) == 0

Utskrift
Integrand = x*log(x)
Antiderivert = x**2*log(x)/2 - x**2/4
Forventet form = x**2*log(x)/2 - x**2/4
Derivert tilbake = x*log(x)
Kontroll = 0

from sympy import symbols, exp, integrate, diff, simplify
x = symbols('x', real=True)
expr = x*exp(x**2)
F = exp(x**2)/2
print("Integrand =", expr)
print("Foreslatt antiderivert =", F)
print("Derivert av antiderivert =", diff(F, x))
print("Kontroll =", simplify(diff(F, x) - expr))
assert simplify(diff(F, x) - expr) == 0

Utskrift
Integrand = x*exp(x**2)
Foreslatt antiderivert = exp(x**2)/2
Derivert av antiderivert = x*exp(x**2)
Kontroll = 0

from sympy import symbols, Function, Eq, dsolve, diff, exp, simplify
x = symbols('x', real=True)
y = Function('y')
ode = Eq(diff(y(x), x) + 2*x*y(x), 4*x)
solution = dsolve(ode, ics={y(0): 8})
print("Differensiallikning:", ode)
print("Losning med initialbetingelse:", solution)
rhs = solution.rhs
print("y(0) =", rhs.subs(x, 0))
print("Restledd =", simplify(diff(rhs, x) + 2*x*rhs - 4*x))
assert rhs.subs(x, 0) == 8
assert simplify(diff(rhs, x) + 2*x*rhs - 4*x) == 0

Utskrift
Differensiallikning: Eq(2*x*y(x) + Derivative(y(x), x), 4*x)
Losning med initialbetingelse: Eq(y(x), 2 + 6*exp(-x**2))
y(0) = 8
Restledd = 0

from sympy import symbols, Abs, solve_univariate_inequality
x = symbols('x', real=True, nonzero=True)
k = 1/x
region = solve_univariate_inequality(Abs(k) < 1, x)
print("Kvotient k =", k)
print("Konvergenskrav |k| < 1 gir:", region)
print("Ved x=1 er |k| =", Abs(k).subs(x, 1))
print("Ved x=-1 er |k| =", Abs(k).subs(x, -1))
assert bool(Abs(k).subs(x, 2) < 1)
assert not bool(Abs(k).subs(x, 1) < 1)

Utskrift
Kvotient k = 1/x
Konvergenskrav |k| < 1 gir: (1 < x) | (x < -1)
Ved x=1 er |k| = 1
Ved x=-1 er |k| = 1

from sympy import symbols, Eq, solve, simplify
x = symbols('x', real=True)
S = 2/(1-1/x)
solutions = solve(Eq(S, 4), x)
print("Sumfunksjon S(x) =", simplify(S))
print("Losninger av S(x)=4:", solutions)
for value in solutions:
    print("S(", value, ") =", simplify(S.subs(x, value)), ", |x| =", abs(value))
assert solutions == [2]
assert simplify(S.subs(x, 2)) == 4

Utskrift
Sumfunksjon S(x) = 2*x/(x - 1)
Losninger av S(x)=4: [2]
S( 2 ) = 4 , |x| = 2

from sympy import Matrix, sqrt, Rational
AB = Matrix([-3, 4, 0])
AC = Matrix([-3, 0, 1])
cross = AB.cross(AC)
norm = sqrt(cross.dot(cross))
area = norm/2
print("AB =", AB.T)
print("AC =", AC.T)
print("AB x AC =", cross.T)
print("Lengde av kryssprodukt =", norm)
print("Areal av trekanten =", area)
assert cross == Matrix([4, 3, 12])
assert area == Rational(13,2)

Utskrift
AB = Matrix([[-3, 4, 0]])
AC = Matrix([[-3, 0, 1]])
AB x AC = Matrix([[4, 3, 12]])
Lengde av kryssprodukt = 13
Areal av trekanten = 13/2

from sympy import symbols, simplify
x, y, z = symbols('x y z', real=True)
plane = 4*x + 3*y + 12*z - 12
points = {'A':(3,0,0), 'B':(0,4,0), 'C':(0,0,1)}
print("Planuttrykk =", plane)
for name, point in points.items():
    value = simplify(plane.subs({x:point[0], y:point[1], z:point[2]}))
    print(name, point, "gir", value)
    assert value == 0

Utskrift
Planuttrykk = 4*x + 3*y + 12*z - 12
A (3, 0, 0) gir 0
B (0, 4, 0) gir 0
C (0, 0, 1) gir 0

from sympy import symbols, solve, Eq, Rational, simplify
t = symbols('t', real=True)
x = t
y = t**2/3
z = -t/4
equation = Eq(4*x + 3*y + 12*z, 12)
roots = solve(equation, t)
valid = [r for r in roots if r >= 0]
print("Likning for treff:", equation)
print("Alle rotter:", roots)
print("Gyldig tid:", valid)
t0 = valid[0]
point = (t0, simplify(t0**2/3), simplify(-t0/4))
print("Treffpunkt:", point)
assert t0 == 3
assert point == (3, 3, Rational(-3,4))

Utskrift
Likning for treff: Eq(t**2 + t, 12)
Alle rotter: [-4, 3]
Gyldig tid: [3]
Treffpunkt: (3, 3, -3/4)

from sympy import symbols, simplify, factor
k = symbols('k', integer=True, positive=True)
hypothesis = k*(k-3)/2
next_from_recursion = hypothesis + k - 1
target = (k+1)*((k+1)-3)/2
print("Induksjonshypotese a_k =", hypothesis)
print("a_(k+1) fra rekursjonen =", next_from_recursion)
print("Faktorisert =", factor(next_from_recursion))
print("Mal for k+1 =", target)
print("Differanse =", simplify(next_from_recursion - target))
assert simplify(next_from_recursion - target) == 0

Utskrift
Induksjonshypotese a_k = k*(k - 3)/2
a_(k+1) fra rekursjonen = k*(k - 3)/2 + k - 1
Faktorisert = (k - 2)*(k + 1)/2
Mal for k+1 = (k - 2)*(k + 1)/2
Differanse = 0

from sympy import symbols, cos, pi, solveset, Interval
x = symbols('x', real=True)
f = 3 - 3*cos(1-x**2)
domain = Interval.open(-pi/2, pi/2)
roots = solveset(f, x, domain=domain)
print("f(x) =", f)
print("Definisjonsomrade =", domain)
print("Nullpunkter i omradet =", roots)
print("f(-1) =", f.subs(x, -1))
print("f(1) =", f.subs(x, 1))
assert f.subs(x, -1) == 0 and f.subs(x, 1) == 0

Utskrift
f(x) = 3 - 3*cos(x**2 - 1)
Definisjonsomrade = Interval.open(-pi/2, pi/2)
Nullpunkter i omradet = Union(Intersection(ImageSet(Lambda(_n, -sqrt(2*_n*pi + 1)), Integers), Interval.open(-pi/2, pi/2)), Intersection(ImageSet(Lambda(_n, sqrt(2*_n*pi + 1)), Integers), Interval.open(-pi/2, pi/2)))
f(-1) = 0
f(1) = 0

from sympy import symbols, cos, sin, diff, pi, simplify
x = symbols('x', real=True)
f = 3 - 3*cos(1-x**2)
fp = simplify(diff(f, x))
critical = [-1, 0, 1]
print("f'(x) =", fp)
for c in critical:
    print("x =", c, ": f'(x) =", simplify(fp.subs(x,c)), ", f(x) =", f.subs(x,c))
print("Numerisk f(0) =", f.subs(x,0).evalf())
assert all(simplify(fp.subs(x,c)) == 0 for c in critical)

Utskrift
f'(x) = 6*x*sin(x**2 - 1)
x = -1 : f'(x) = 0 , f(x) = 0
x = 0 : f'(x) = 0 , f(x) = 3 - 3*cos(1)
x = 1 : f'(x) = 0 , f(x) = 0
Numerisk f(0) = 1.37909308239558

from sympy import symbols, cos, simplify
x = symbols('x', real=True)
f = 3 - 3*cos(1-x**2)
print("f(-x)-f(x) =", simplify(f.subs(x,-x)-f))
print("f(-1) =", f.subs(x,-1))
print("f(0) =", f.subs(x,0), "≈", f.subs(x,0).evalf())
print("f(1) =", f.subs(x,1))
for value in [-1.4,-.5,.5,1.4]:
    print("f(", value, ") ≈", float(f.subs(x,value)))
assert simplify(f.subs(x,-x)-f) == 0

Utskrift
f(-x)-f(x) = 0
f(-1) = 0
f(0) = 3 - 3*cos(1) ≈ 1.37909308239558
f(1) = 0
f( -1.4 ) ≈ 1.2794400417826293
f( -0.5 ) ≈ 0.8049333933785374
f( 0.5 ) ≈ 0.8049333933785374
f( 1.4 ) ≈ 1.2794400417826293

from sympy import symbols, sin, cos, expand_trig, simplify
x = symbols('x', real=True)
expanded = expand_trig(cos(2*x))
expected = cos(x)**2 - sin(x)**2
print("Utvidet cos(2x) =", expanded)
print("Forventet uttrykk =", expected)
print("Kontroll =", simplify(expanded - expected))
assert simplify(expanded - expected) == 0

Utskrift
Utvidet cos(2x) = 2*cos(x)**2 - 1
Forventet uttrykk = -sin(x)**2 + cos(x)**2
Kontroll = 0

from sympy import symbols, sin, cos, trigsimp, simplify
x = symbols('x', real=True)
expr = cos(x)**4 - sin(x)**4
result = trigsimp(expr)
print("Opprinnelig uttrykk =", expr)
print("Trigonometrisk forenkling =", result)
print("Kontroll mot cos(2x) =", simplify(expr - cos(2*x)))
assert simplify(expr - cos(2*x)) == 0

Utskrift
Opprinnelig uttrykk = -sin(x)**4 + cos(x)**4
Trigonometrisk forenkling = cos(2*x)
Kontroll mot cos(2x) = 0

from sympy import symbols, sin, cos, pi, solveset, Interval, simplify
x = symbols('x', real=True)
expr = sin(x) + cos(x) - 1
solutions = solveset(expr, x, domain=Interval(0, 2*pi))
print("Likning: sin(x) + cos(x) = 1")
print("Losninger i [0, 2*pi] =", solutions)
for value in [0, pi/2, 2*pi]:
    print("x =", value, "gir venstreside =", simplify(sin(value)+cos(value)))
assert all(simplify(expr.subs(x,v)) == 0 for v in [0,pi/2,2*pi])

Utskrift
Likning: sin(x) + cos(x) = 1
Losninger i [0, 2*pi] = {0, pi/2, 2*pi}
x = 0 gir venstreside = 1
x = pi/2 gir venstreside = 1
x = 2*pi gir venstreside = 1

from sympy import Rational
s = 125
v = 26 - Rational(8,100)*s
print("Tilbakelagt strekning =", s, "km")
print("Fart v = 26 - 0.08*s =", v, "km/h")
print("Fartsreduksjon =", 26-v, "km/h")
assert v == 16

Utskrift
Tilbakelagt strekning = 125 km
Fart v = 26 - 0.08*s = 16 km/h
Fartsreduksjon = 10 km/h

from sympy import symbols, Function, Eq, dsolve, diff, exp, Rational, simplify
t = symbols('t', real=True)
s = Function('s')
ode = Eq(diff(s(t), t), 26 - Rational(8,100)*s(t))
solution = dsolve(ode, ics={s(0):0})
print("Differensiallikning:", ode)
print("Losning:", solution)
rhs = solution.rhs
print("s(0) =", rhs.subs(t,0))
print("Restledd =", simplify(diff(rhs,t) - (26-Rational(8,100)*rhs)))
assert rhs.subs(t,0) == 0
assert simplify(diff(rhs,t) - (26-Rational(8,100)*rhs)) == 0

Utskrift
Differensiallikning: Eq(Derivative(s(t), t), 26 - 2*s(t)/25)
Losning: Eq(s(t), 325 - 325*exp(-2*t/25))
s(0) = 0
Restledd = 0

from sympy import symbols, exp, log, Rational, N, solve, Eq, simplify
t = symbols('t', positive=True)
s = 325*(1-exp(-Rational(8,100)*t))
first_hour = N(s.subs(t,1), 10)
time_exact = log(Rational(13,8))/Rational(8,100)
time_num = N(time_exact, 10)
print("s(1) =", first_hour, "km")
print("Eksakt tid til 125 km =", time_exact)
print("Numerisk tid =", time_num, "timer")
print("Kontroll s(tid) =", N(s.subs(t,time_exact), 12))
assert simplify(s.subs(t,time_exact)-125) == 0

Utskrift
s(1) = 24.98718742 km
Eksakt tid til 125 km = 25*log(13/8)/2
Numerisk tid = 6.068847697 timer
Kontroll s(tid) = 125.000000000

from sympy import symbols, Matrix, Rational, simplify
t = symbols('t', real=True)
AB = Matrix([1,0,-1])
AC = Matrix([1,1,0])
AP = Matrix([t,2*t+1,t**2+2])
determinant = Matrix.hstack(AB,AC,AP).det()
V = simplify(determinant/6)
print("Determinant =", determinant)
print("Kvadratkomplettert uttrykk =", (t-Rational(1,2))**2 + Rational(3,4))
print("Volumfunksjon V(t) =", V)
assert simplify(determinant-(t**2-t+1)) == 0
assert simplify(V-(t**2-t+1)/6) == 0

Utskrift
Determinant = t**2 - t + 1
Kvadratkomplettert uttrykk = (t - 1/2)**2 + 3/4
Volumfunksjon V(t) = t**2/6 - t/6 + 1/6

from sympy import symbols, Eq, solve, Rational, simplify
t = symbols('t', real=True)
V = (t**2-t+1)/6
roots = solve(Eq(V,Rational(7,2)),t)
print("Parameterverdier =", roots)
for r in roots:
    P = (r, 2*r+1, r**2+2)
    print("t =", r, "gir P =", P, "og V =", simplify(V.subs(t,r)))
assert set(roots) == {-4,5}

Utskrift
Parameterverdier = [-4, 5]
t = -4 gir P = (-4, -7, 18) og V = 7/2
t = 5 gir P = (5, 11, 27) og V = 7/2

from sympy import symbols, diff, solve, Rational, simplify
t = symbols('t', real=True)
V = (t**2-t+1)/6
critical = solve(diff(V,t),t)[0]
P = (critical, 2*critical+1, critical**2+2)
print("V'(t) =", diff(V,t))
print("Kritisk t =", critical)
print("V''(t) =", diff(V,t,2))
print("Punkt P =", P)
print("Minste volum =", simplify(V.subs(t,critical)))
assert critical == Rational(1,2)
assert P == (Rational(1,2),2,Rational(9,4))
assert simplify(V.subs(t,critical)) == Rational(1,8)

Utskrift
V'(t) = t/3 - 1/6
Kritisk t = 1/2
V''(t) = 1/3
Punkt P = (1/2, 2, 9/4)
Minste volum = 1/8

from sympy import symbols, cos, pi, Eq, solveset, Interval, N, acos, Rational
t = symbols('t', real=True)
h = -Rational(135,2)*cos(pi*t/15) + Rational(139,2)
t1 = 15*acos(Rational(13,45))/pi
t2 = 30-t1
print("h(t) =", h)
print("Eksakt t1 =", t1)
print("Eksakt t2 =", t2)
print("Numeriske tider =", N(t1,10), "og", N(t2,10))
print("Kontroll h(t1) =", N(h.subs(t,t1),12))
print("Kontroll h(t2) =", N(h.subs(t,t2),12))
assert (h.subs(t,t1)-50).simplify() == 0
assert (h.subs(t,t2)-50).simplify() == 0

Utskrift
h(t) = 139/2 - 135*cos(pi*t/15)/2
Eksakt t1 = 15*acos(13/45)/pi
Eksakt t2 = -15*acos(13/45)/pi + 30
Numeriske tider = 6.100712740 og 23.89928726
Kontroll h(t1) = 50.0000000000
Kontroll h(t2) = 50.0000000000

from sympy import symbols, cos, sin, pi, Rational, diff, N, simplify
t = symbols('t', real=True)
h = -Rational(135,2)*cos(pi*t/15) + Rational(139,2)
hp = diff(h,t)
hpp = diff(hp,t)
for value in [Rational(15,2), Rational(45,2)]:
    print("t =", value)
    print("  h(t) =", simplify(h.subs(t,value)))
    print("  h'(t) =", simplify(hp.subs(t,value)), "≈", N(hp.subs(t,value),8))
    print("  h''(t) =", simplify(hpp.subs(t,value)))
assert simplify(hpp.subs(t,Rational(15,2))) == 0
assert simplify(hpp.subs(t,Rational(45,2))) == 0

Utskrift
t = 15/2
  h(t) = 139/2
  h'(t) = 9*pi/2 ≈ 14.137167
  h''(t) = 0
t = 45/2
  h(t) = 139/2
  h'(t) = -9*pi/2 ≈ -14.137167
  h''(t) = 0

from sympy import symbols, expand, simplify
x, a, b, s, t = symbols('x a b s t', real=True)
f = x**2 + a*x + b
fs = s**2+a*s+b
ft = t**2+a*t+b
g = expand(fs + (2*s+a)*(x-s))
h = expand(ft + (2*t+a)*(x-t))
print("Tangent g(x) =", g)
print("Tangent h(x) =", h)
print("Kontroll g =", simplify(g-((a+2*s)*x+b-s**2)))
print("Kontroll h =", simplify(h-((a+2*t)*x+b-t**2)))
assert simplify(g-((a+2*s)*x+b-s**2)) == 0
assert simplify(h-((a+2*t)*x+b-t**2)) == 0

Utskrift
Tangent g(x) = a*x + b - s**2 + 2*s*x
Tangent h(x) = a*x + b - t**2 + 2*t*x
Kontroll g = 0
Kontroll h = 0

from sympy import symbols, Eq, solve, simplify
x, a, b, s, t = symbols('x a b s t', real=True)
g = (a+2*s)*x+b-s**2
h = (a+2*t)*x+b-t**2
solution = solve(Eq(g,h),x)[0]
print("g(x) =", g)
print("h(x) =", h)
print("Losning for x =", solution)
print("Kontroll =", simplify(solution-(s+t)/2))
assert simplify(solution-(s+t)/2) == 0

Utskrift
g(x) = b - s**2 + x*(a + 2*s)
h(x) = b - t**2 + x*(a + 2*t)
Losning for x = s/2 + t/2
Kontroll = 0

from sympy import symbols, integrate, simplify, factor
x, a, b, s, t = symbols('x a b s t', real=True)
f = x**2+a*x+b
g = (a+2*s)*x+b-s**2
h = (a+2*t)*x+b-t**2
xp = (s+t)/2
left_gap = simplify(f-g)
right_gap = simplify(f-h)
A1 = factor(integrate(left_gap,(x,s,xp)))
A2 = factor(integrate(right_gap,(x,xp,t)))
print("f(x)-g(x) =", left_gap)
print("f(x)-h(x) =", right_gap)
print("Venstre areal A1 =", A1)
print("Hoyre areal A2 =", A2)
print("Differanse A1-A2 =", simplify(A1-A2))
assert simplify(left_gap-(x-s)**2) == 0
assert simplify(right_gap-(x-t)**2) == 0
assert simplify(A1-A2) == 0

Utskrift
f(x)-g(x) = s**2 - 2*s*x + x**2
f(x)-h(x) = t**2 - 2*t*x + x**2
Venstre areal A1 = -(s - t)**3/24
Hoyre areal A2 = -(s - t)**3/24
Differanse A1-A2 = 0