def matrix_producten():
# pylint: disable=C0103
RNG().set(4)
u = random_tensor(r"\vec u", 3, ret=True, details=False)
RNG().consume_entropy(0x02, -0x14, 0x14)
M = random_tensor(r"\mathbf{M}", (3,2), ret=True, details=False)
N = random_tensor(r"\mathbf{N}", (2,3), ret=True, details=False)
O = random_tensor(r"\mathbf{O}", (2,2), ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(O.dot(N.dot(u)), r"\mathbf{O} (\mathbf{N} \vec u)", details=False)
latex_bmatrix(O.dot(N).dot(u), r"(\mathbf{O} \mathbf{N}) \vec u", details=False)
latex_bmatrix(O.dot(N), r"\mathbf{O} \mathbf{N}", details=False)
display(Markdown("<hr>"))
display(Markdown(r"$\mathbf{O}\mathbf{M} = \bot$"))
latex_bmatrix(O.dot(O), r"\mathbf{O} \mathbf{O}", details=False)
display(Markdown(r"$\mathbf{N}\mathbf{N} = \bot$"))
latex_bmatrix(N.dot(M), r"\mathbf{N} \mathbf{M}", details=False)
display(Markdown(r"$\mathbf{N}\mathbf{O} = \bot$"))
latex_bmatrix(M.dot(N), r"\mathbf{M} \mathbf{N}", details=False)
display(Markdown(r"$\mathbf{M}\mathbf{M} = \bot$"))
latex_bmatrix(M.dot(O), r"\mathbf{M} \mathbf{O}", details=False)
def matrix_producten():
RNG().set(4)
random_tensor(r"\vec u", 3)
RNG().consume_entropy(0x02, -0x14, 0x14)
random_tensor(r"\mathbf{M}", (3, 2))
random_tensor(r"\mathbf{N}", (2, 3))
random_tensor(r"\mathbf{O}", (2, 2))
def lineaire_combinaties():
RNG().set(1)
random_tensor(r"\vec u")
random_tensor(r"\vec v")
RNG().set(2)
random_tensor(r"\vec w", 2)
random_tensor(r"\vec x", 2)
random_tensor(r"\vec y", 4)
random_tensor(r"\vec z", 4)
def matrix_vector():
RNG().set(4)
random_tensor(r"\vec u", 3)
random_tensor(r"\vec v", 2)
random_tensor(r"\mathbf{M}", (3, 2))
random_tensor(r"\mathbf{N}", (2, 3))
random_tensor(r"\mathbf{O}", (2, 2))
RNG().set(2).consume_entropy(0x06, -0x14, 0x14)
random_tensor(r"\vec {p_a}", 2)
random_tensor(r"\vec {p_b}", 2)
random_tensor(r"\vec {q_a}", 4)
random_tensor(r"\vec {q_b}", 4)
def integrals():
# pylint: disable=C0103
RNG().set(9)
a, b = np.random.randint(2, 7, 2)
display(Markdown("**(a)**"))
display(Markdown(f"""\\begin{{align}}
\\int \\sqrt[{a}]x^{b}\\ dx
&= \\int x^{{\\frac{{{b}}}{{{a}}}}} \\ dx \\\\[2mm]
&= \\frac{{{a}}}{{{a+b}}} x^{{\\frac{{{a+b}}}{{{a}}}}} + C\\\\[2mm]
&= \\frac{{{a}}}{{{a+b}}} \\sqrt[{a}]x^{{{a+b}}} + C
\\end{{align}}"""))
display(Markdown("<hr>"))
a, b, c, d, e = np.random.randint(3, 9, 5)
a = (a % 3) + 2
b = (b % 3) + 1
a = a if a != b else a+1
a, b = max(a, b), min(a, b)
e = e if e != d else d+e
d, e = min(d, e), max(d, e)
display(Markdown("**(b)**"))
display(Markdown(f"""\\begin{{align}}
\\int_{{{d}}}^{{{e}}} {a*b}x^{{{a-1}}} - {b*c}x^{{{b-1}}}\\ dx
&= {b}x^{{{a}}} - {c}x^{{{b}}} \\ \\biggr\\rvert_{{{d}}}^{{{e}}} \\\\[2mm]
&= \\big({b}\\cdot{{{e}}}^{{{a}}} - {c}\\cdot{{{e}}}^{{{b}}}\\big) - \\big({b}\\cdot{{{d}}}^{{{a}}} - {c}\\cdot{{{d}}}^{{{b}}}\\big) \\\\[2mm]
&= \\big({b}\\cdot{{{e**a}}} - {c}\\cdot{{{e**b}}}\\big) - \\big({b}\\cdot{{{d**a}}} - {c}\\cdot{{{d**b}}}\\big) \\\\[2mm]
&= \\big({b*e**a} - {c*e**b}\\big) - \\big({b*d**a} - {c*d**b}\\big) \\\\[2mm]
&= {b*e**a-c*e**b} - {b*d**a-c*d**b} \\\\[2mm]
&= {(b*e**a-c*e**b)-(b*d**a-c*d**b)} \\\\[2mm]
\\end{{align}}"""))
def rank():
# pylint: disable=C0103
RNG().set(7)
def stats(M, label):
def basis(vs):
res = r"\left\{"
res += ", ".join([r"\begin{bmatrix}" + r"\\".join([str(c) for c in v]) + r"\end{bmatrix}" for v in vs])
res += r"\right\}"
return res
U = lu(M)[2]
cols = []
for i in range(U.shape[0]):
r = np.flatnonzero(U[i, :])
if r.size > 0:
cols.append(r[0])
cols = dict.fromkeys(cols)
display(Markdown(f"$\\text{{I}}(\\textbf{{{label}}}) = {basis([M[:,c] for c in cols])}$"))
display(Markdown(f"$\\text{{Rank}}(\\textbf{{{label}}}) = {np.linalg.matrix_rank(M)}$"))
display(Markdown(f"$\\text{{Ker}}(\\textbf{{{label}}}) = {basis(Matrix(M).nullspace())}$"))
display(Markdown(f"$\\text{{Nullity}}(\\textbf{{{label}}}) = {M.shape[1] - np.linalg.matrix_rank(M)}$"))
M = random_tensor(r"\textbf{M}", (2,2), singular=NONDEGENERATE, interval=(0,5), ret=True, details=False)
stats(M, 'M')
display(Markdown("<hr>"))
N = random_tensor(r"\textbf{N}", (2,2), singular=DEGENERATE, interval=(0,5), ret=True, details=False)
stats(N, 'N')
display(Markdown("<hr>"))
O = random_tensor(r"\textbf{O}", (3,3), singular=NONDEGENERATE, interval=(0,5), ret=True, details=False)
stats(O, 'O')
display(Markdown("<hr>"))
P = random_tensor(r"\textbf{P}", (3,3), singular=DEGENERATE, interval=(0,5), ret=True, details=False)
stats(P, 'P')
display(Markdown("<hr>"))
Q = random_tensor(r"\textbf{Q}", (2,3), singular=DONT_CARE, interval=(0,5), ret=True, details=False)
stats(Q, 'Q')
display(Markdown("<hr>"))
R = random_tensor(r"\textbf{R}", (2,3), singular=DONT_CARE, interval=(0,5), ret=True, details=False)
stats(R, 'R')
RNG().set(1)
def inwendige_producten():
RNG().set(3)
random_tensor(r"\vec u", 2)
random_tensor(r"\vec v", 3)
random_tensor(r"\vec w", 2)
random_tensor(r"\vec x", 4)
random_tensor(r"\vec y", 4)
def negatieven_en_sommen():
RNG().set(0)
random_tensor(r"\vec u")
random_tensor(r"\vec v")
random_tensor(r"\vec w", 3)
random_tensor(r"\vec x", 3)
random_tensor(r"\vec y", 5)
random_tensor(r"\vec z", 5)
def determinanten():
RNG().set(6)
random_tensor(r"\textbf{M}", (2, 2), singular=NONDEGENERATE)
random_tensor(r"\textbf{N}", (2, 2), singular=DEGENERATE)
random_tensor(r"\textbf{O}", (2, 2), singular=NONDEGENERATE)
random_tensor(r"\textbf{P}", (3, 3),
singular=NONDEGENERATE,
interval=(0, 5))
random_tensor(r"\textbf{Q}", (3, 3), singular=DEGENERATE, interval=(0, 5))
def rank():
RNG().set(7)
random_tensor(r"\textbf{M}", (2, 2),
singular=NONDEGENERATE,
interval=(0, 5))
random_tensor(r"\textbf{N}", (2, 2), singular=DEGENERATE, interval=(0, 5))
random_tensor(r"\textbf{O}", (3, 3),
singular=NONDEGENERATE,
interval=(0, 5))
random_tensor(r"\textbf{P}", (3, 3), singular=DEGENERATE, interval=(0, 5))
random_tensor(r"\textbf{Q}", (2, 3), singular=DONT_CARE, interval=(0, 5))
random_tensor(r"\textbf{R}", (2, 3), singular=DONT_CARE, interval=(0, 5))
def inwendige_producten():
# pylint: disable=C0103
RNG().set(3)
u = random_tensor(r"\vec u", 2, ret=True, details=False)
v = random_tensor(r"\vec v", 3, ret=True, details=False)
w = random_tensor(r"\vec w", 2, ret=True, details=False)
x = random_tensor(r"\vec x", 4, ret=True, details=False)
y = random_tensor(r"\vec y", 4, ret=True, details=False)
display(Markdown("<hr>"))
dot = lambda u, v: (u.T@v)[0][0]
table = f"""$\\begin{{array}}{{|r|c|c|c|c|c|}} \\hline
& \\vec u & \\vec v & \\vec w & \\vec x & \\vec y \\\\ \\hline
\\vec u & {dot(u,u)}
& \\bot
& {dot(u,w)}
& \\bot
& \\bot
\\\\ \\hline
\\vec v & \\bot
& {dot(v,v)}
& \\bot
& \\bot
& \\bot
\\\\ \\hline
\\vec w & {dot(w,u)}
& \\bot
& {dot(w,w)}
& \\bot
& \\bot
\\\\ \\hline
\\vec x & \\bot
& \\bot
& \\bot
& {dot(x,x)}
& {dot(x,y)}
\\\\ \\hline
\\vec y & \\bot
& \\bot
& \\bot
& {dot(y,x)}
& {dot(y,y)}
\\\\ \\hline
\\end{{array}}$"""
display(Markdown(table))
display(Markdown("<hr>"))
display(Markdown(r"**Wat valt je op qua symmetrie aan de tabel? Is $\langle \vec u | \vec v\rangle$ hetzelfde als $\langle \vec v | \vec u \rangle$?**"))
display(Markdown(r"*Ja, dus de volgorde maakt niet uit / het inwendig product is hier commutatief.*"))
display(Markdown(r"**Heeft $\langle \vec u | \vec u \rangle$ een speciale betekenis? Of de wortel daarvan?**"))
display(Markdown(r"*De wortel van het inwendig product van een vector met zichzelf is de lengte van de vector (stelling van Pythagoras)*"))
def inverses():
# pylint: disable=C0103
RNG().set(6)
def fr_matrix(M, divisor=1, label=None):
def fraction(n):
n = int(n)
gcd = np.gcd(n,divisor)
s = '' if divisor*n > 0 else '-'
n = abs((n/gcd).round(0).astype(int))
d = abs((divisor/gcd).round(0).astype(int))
if d == 1:
return s+str(n)
else:
return f"{s}\\frac{{{n}}}{{{d}}}"
if len(M.shape) > 2:
raise ValueError('bmatrix can at most display two dimensions')
lines = str(M).replace("[", "").replace("]", "").splitlines()
if label:
result = [label + " = "]
else:
result = [""]
result += [r"\begin{bmatrix}"]
result += [" " + " & ".join(map(fraction, l.split())) + r"\\" for l in lines]
result += [r"\end{bmatrix}"]
display(Math("\n".join(result)))
adj = lambda M: np.array(((M[1][1], -M[0][1]),(-M[1][0], M[0][0])))
M = random_tensor(r"\textbf{M}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
det = np.linalg.det(M).round(0).astype(int)
latex_bmatrix(adj(M), r"$\text{adj}(\mathbf{M})", details=False)
display(Markdown(f"$\\text{{det}}(\\mathbf{{M}}) = {det}$"))
fr_matrix(adj(M), det, r"$\mathbf{M}^{-1}")
display(Markdown("<hr>"))
N = random_tensor(r"\textbf{N}", (2,2), singular=DEGENERATE, ret=True, details=False)
latex_bmatrix(adj(N), r"$\text{adj}(\mathbf{N})", details=False)
display(Markdown(f"$\\text{{det}}(\\mathbf{{N}}) = {np.linalg.det(N).round(0).astype(int)}$"))
display(Markdown(r"$\mathbf{N}^{-1} = \bot$"))
display(Markdown("<hr>"))
O = random_tensor(r"\textbf{O}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
det = np.linalg.det(O).round(0).astype(int)
latex_bmatrix(adj(O), r"$\text{adj}(\mathbf{O})", details=False)
display(Markdown(f"$\\text{{det}}(\\mathbf{{O}}) = {det}$"))
fr_matrix(adj(O), det, r"$\mathbf{O}^{-1}")
def matrix_vector():
# pylint: disable=C0103
RNG().set(4)
u = random_tensor(r"\vec u", 3, ret=True, details=False)
v = random_tensor(r"\vec v", 2, ret=True, details=False)
M = random_tensor(r"\mathbf{M}", (3,2), ret=True, details=False)
N = random_tensor(r"\mathbf{N}", (2,3), ret=True, details=False)
O = random_tensor(r"\mathbf{O}", (2,2), ret=True, details=False)
RNG().set(2).consume_entropy(0x06, -0x14, 0x14)
pa = random_tensor(r"\vec {p_a}", 2, ret=True, details=False)
pb = random_tensor(r"\vec {p_b}", 2, ret=True, details=False)
qa = random_tensor(r"\vec {q_a}", 4, ret=True, details=False)
qb = random_tensor(r"\vec {q_b}", 4, ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(M.dot(v), r"\mathbf{M}\vec{v}", details=False)
display(Markdown(r"$\mathbf{M}\vec{u} = \bot$"))
display(Markdown(r"$\mathbf{N}\vec{v} = \bot$"))
latex_bmatrix(N.dot(u), r"\mathbf{N}\vec{u}", details=False)
latex_bmatrix(O.dot(N.dot(u)), r"\mathbf{O} (\mathbf{N} \vec u)", details=False)
display(Markdown("<hr>"))
P = np.hstack((pa,pb))
Q = np.hstack((qa,qb))
latex_bmatrix(P, r"\mathbf{P}", details=False)
latex_bmatrix(Q, r"\mathbf{Q}", details=False)
display(Markdown("<hr>"))
latex_bmatrix(P.dot(np.array((3,4))).reshape(2,1), r"\mathbf{P} \begin{bmatrix}3 \\ 4\end{bmatrix}", details=False)
latex_bmatrix(Q.dot(np.array((8,-0.5))).reshape(4,1), r"\mathbf{Q} \begin{bmatrix}8 \\ -\frac{1}{2}\end{bmatrix}", details=False)
def negatieven_en_sommen():
# pylint: disable=C0103
RNG().set(0)
u = random_tensor(r"\vec u", ret=True, details=False)
v = random_tensor(r"\vec v", ret=True, details=False)
w = random_tensor(r"\vec w", 3, ret=True, details=False)
x = random_tensor(r"\vec x", 3, ret=True, details=False)
y = random_tensor(r"\vec y", 5, ret=True, details=False)
z = random_tensor(r"\vec z", 5, ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(-u, r"-\vec u", details=False)
latex_bmatrix(-v, r"-\vec v", details=False)
latex_bmatrix(-w + x, r"-\vec w + \vec x", details=False)
latex_bmatrix(-y-z, r"- \vec y - \vec z", details=False)
def implicit_diff():
# pylint: disable=C0103
RNG().set(13)
a, b, c, d = np.random.randint(2, 7, 4)
text = f"Gegeven ${hide_one(a-1)}x^3y - {hide_one(b-1)}x^2 + {hide_one(c-1)}y^4 = {2*d}$, geef $\\frac{{dy}}{{dx}}$"
display(Markdown("**(a)** " + text + r"$\\\\[3mm]$"))
display(Markdown(f"""\\begin{{align}}
\\frac{{dy}}{{dx}}({hide_one(a-1)}x^3y - {hide_one(b-1)}x^2 + {hide_one(c-1)}y^4) &= \\frac{{dy}}{{dx}} {2*d} \\\\[2mm]
{hide_one(a-1)}\\frac{{dy}}{{dx}}x^3y - {hide_one(b-1)} \\frac{{dy}}{{dx}} x^2 + {hide_one(c-1)}\\frac{{dy}}{{dx}}y^4 &= 0 \\\\[2mm]
{hide_one(a-1)}(3x^2y + x^3 \\frac{{dy}}{{dx}}) - {hide_one(b-1)}(2x) + {hide_one(c-1)}(4y^3) \\frac{{dy}}{{dx}} &= 0 \\\\[2mm]
{hide_one(3*(a-1))}x^2y + {hide_one(a-1)}x^3 \\frac{{dy}}{{dx}} - {hide_one(2*(b-1))}x + {hide_one(4*(c-1))}y^3 \\frac{{dy}}{{dx}} &= 0 \\\\[2mm]
{hide_one(a-1)}x^3 \\frac{{dy}}{{dx}} + {hide_one(4*(c-1))}y^3 \\frac{{dy}}{{dx}} &= {hide_one(2*(b-1))}x - {hide_one(3*(a-1))}x^2y \\\\[2mm]
({hide_one(a-1)}x^3 + {hide_one(4*(c-1))}y^3) \\frac{{dy}}{{dx}} &= {hide_one(2*(b-1))}x - {hide_one(3*(a-1))}x^2y \\\\[2mm]
\\frac{{dy}}{{dx}} &= \\frac{{{hide_one(2*(b-1))}x - {hide_one(3*(a-1))}x^2y}}{{{hide_one(a-1)}x^3 + {hide_one(4*(c-1))}y^3}} \\\\[2mm]
\\end{{align}}"""))
def lineaire_combinaties():
# pylint: disable=C0103
RNG().set(1)
u = random_tensor(r"\vec u", ret=True, details=False)
v = random_tensor(r"\vec v", ret=True, details=False)
w = random_tensor(r"\vec w", 2, ret=True, details=False)
x = random_tensor(r"\vec x", 2, ret=True, details=False)
y = random_tensor(r"\vec y", 4, ret=True, details=False)
z = random_tensor(r"\vec z", 4, ret=True, details=False)
display(Markdown("<hr>"))
latex_bmatrix(3*u, r"3\vec{u}", details=False)
latex_bmatrix(-5*v, r"-5\vec{v}", details=False)
latex_bmatrix(v/2, r"\frac{1}{2} \vec{v}", details=False)
latex_bmatrix(3*w + 4*x, r"3\vec{w} + 4\vec{x}", details=False)
latex_bmatrix(8*y-z/2, r"8\vec{y} - \frac{1}{2}\vec{z}", details=False)
def dif_eq():
# pylint: disable=C0103
RNG().set(11)
a, b, c, d, e, f, g, h = np.random.randint(2, 9, 8)
b = int(b/2)
c = c*d
d = 3*e*f
e = 2*(g-4)
f = h*2 - 1
deriv = f"f^\\prime(x) = {a*b}x^{b-1}+{c}e^x"
val = f"f({e}) = {a*(e**b)-d}+{c}e^{{{e}}}"
form = f"f(x) = {a}x^{b} + {c}e^x-{d}"
ant = f"{a*(f**b)-d}+{c}e^{{{f}}}"
display(Markdown(f"Vind $f({f})$ gegeven de volgende afgeleidde en waarde:\n\n$${deriv},\\quad {val}$$"))
display(Markdown("<hr>"))
display(Markdown(f"""\\begin{{align}} f(x)
&= \\int f ^\\prime(x)\\ dx \\\\[1mm]
&= \\int {a*b}x^{b-1}+{c}e^x \\ dx \\\\[1mm]
&= {a}x^{b}+{c}e^x + C
\\end{{align}}"""))
display(Markdown("<hr>"))
display(Markdown(f"""\\begin{{align}}
f({e}) &= {a}({e})^{b}+{c}e^{{{e}}} + C \\\\[1mm]
{a*(e**b)-d}+{c}e^{{{e}}} &= {a}({e})^{b}+{c}e^{{{e}}} + C \\\\[1mm]
{a*(e**b)-d} &= {a}({e**b})+ C \\\\[1mm]
{a*(e**b)-d} &= {a*e**b}+ C \\\\[1mm]
C &= {-d} \\\\[1mm]
\\end{{align}}"""))
display(Markdown("<hr>"))
display(Markdown(f"""\\begin{{align}}
f(x) &= {a}x^{b}+{c}e^x - {d} \\\\[1mm]
f({f}) &= {a}({f})^{b}+{c}e^{{{f}}} - {d} \\\\[1mm]
&= {a}({f**b})+{c}e^{{{f}}} - {d} \\\\[1mm]
&= {a*f**b}+{c}e^{{{f}}} - {d} \\\\[1mm]
&= {a*f**b-d}+{c}e^{{{f}}} \\\\[1mm]
\\end{{align}}"""))
display(Markdown("<hr>"))
display(Markdown(f"**Antwoord:** ${ant}$, met ${form}$."))
def determinanten():
# pylint: disable=C0103
RNG().set(6)
M = random_tensor(r"\textbf{M}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
N = random_tensor(r"\textbf{N}", (2,2), singular=DEGENERATE, ret=True, details=False)
O = random_tensor(r"\textbf{O}", (2,2), singular=NONDEGENERATE, ret=True, details=False)
P = random_tensor(r"\textbf{P}", (3,3), singular=NONDEGENERATE, interval=(0,5), ret=True, details=False)
Q = random_tensor(r"\textbf{Q}", (3,3), singular=DEGENERATE, interval=(0,5), ret=True, details=False)
display(Markdown("<hr>"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{M}}) = {np.linalg.det(M).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{N}}) = {np.linalg.det(N).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{O}}) = {np.linalg.det(O).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{P}}) = {np.linalg.det(P).round(0).astype(int)}$"))
display(Markdown(f"$\\text{{det}}(\\mathbf{{Q}}) = {np.linalg.det(Q).round(0).astype(int)}$"))
def derivatives():
# pylint: disable=C0103
RNG().set(8)
a, b, c = np.random.randint(2, 7, 3)
text = f"Gegeven $f(x) = ({a-1}- {hide_one(b)}x)^{hide_one(c)}$, bepaal $f^\\prime(x)$"
display(Markdown("**(a)** " + text + r"$\\\\[3mm]$"))
if c-1 == 1:
xterm = "x"
else:
xterm = f"x^{c-1}"
display(Markdown(f"""\\begin{{align}}f^\\prime(x)
&= {-b}\\cdot{c}({a-1}- {hide_one(b)}{xterm}) \\\\[1mm]
&= {hide_one(b*b*c)}{xterm} {-b*c*(a-1)}
\\end{{align}}"""))
display(Markdown("*Je kan er voor kiezen de polynoom eerst helemaal uit te schrijven en hier de afgeleidde van te nemen; deze kan daarna wel of niet versimpeld worden.*"))
display(Markdown("<hr>"))
a, b, c, d = np.random.randint(2, 7, 4)
text = f"Gegeven $g(x) = {hide_one(a-1)}x^{hide_one(b)}\\ \\text{{tan}}({hide_one(c)}x^{hide_one(d)})$, geef $g^\\prime(x)$"
display(Markdown("**(b)** " + text + r"$\\\\[3mm]$"))
inner = f"{hide_one(c)}x^{hide_one(d)}"
tan = r"\text{tan}(" + inner + ")"
sec = r"\text{sec}^2(" + inner + ")"
g = (a-1) * np.gcd(b, c*d)
display(Markdown(f"""\\begin{{align}}g^\\prime(x)
&= {(a-1)*b}x^{hide_one(b-1)}\\ {tan} + {a-1}x^{b}{c*d}x^{hide_one(d-1)}\\ {sec} \\\\[1mm]
&= {(a-1)*b}x^{hide_one(b-1)}\\ {tan} + {(a-1)*c*d}x^{{{hide_one(b+d-1)}}}\\ {sec} \\\\[1mm]
&= {hide_one(g)}x^{{{hide_one(b-1)}}}\\ \\big({hide_one((a-1)*b/g)}{tan} + {hide_one((a-1)*c*d/g)}x^{{{d}}}\\ {sec}\\big)
\\end{{align}}"""))
display(Markdown("<hr>"))
a, b, c, d = np.random.randint(2, 7, 4)
text = f"Gegeven $h(x) = \\text{{log}}_{a}({b-1}x-{c}x^{d})$, geef $h^\\prime(x)$"
display(Markdown("**(c)** " + text + r"$\\\\[3mm]$"))
g = np.gcd(c,b-1)
if g != 1:
simplification = f"\\\\[2mm] &= \\frac{{{int((b-1)/g)}-{hide_one(c*d/g)}x^{{{d-1}}}}}{{({hide_one((b-1)/g)}x-{hide_one(c/g)}x^{d})\\ \\text{{ln}}({a})}}"
else:
simplification = ""
display(Markdown(f"""\\begin{{align}}h^\\prime(x)
&= \\frac{{{b-1}-{hide_one(c*d)}x^{{{d-1}}}}}{{({b-1}x-{c}x^{d})\\ \\text{{ln}}({a})}} {simplification} \\end{{align}}"""))
def gauss_jordan():
# pylint: disable=C0103
RNG().set(5)
(A,b) = random_sys_of_eq(details=False, ret=True)
b = (np.linalg.det(A)*b).astype(int)
display(Markdown("<hr>"))
display(Markdown(f"${A[0][0]} x_0 + {A[0][1]} x_1 + {A[0][2]} x_2 = {b[0]}$"))
display(Markdown(f"${A[1][0]} x_0 + {A[1][1]} x_1 + {A[1][2]} x_2 = {b[1]}$"))
display(Markdown(f"${A[2][0]} x_0 + {A[2][1]} x_1 + {A[2][2]} x_2 = {b[2]}$"))
x = np.linalg.solve(A, b)
display(Markdown("<hr>"))
display(Markdown("**Hier moet de je de Gauss-Jordan eliminatie uitvoeren; hier zijn meerdere wegen mogelijk, dus check vooral of je logische stappen uitvoert en of het eindresultaat klopt met wat hieronder staat gegeven.**"))
display(Markdown(f"$x_0 = {x[0].round(0).astype(int)}$"))
display(Markdown(f"$x_1 = {x[1].round(0).astype(int)}$"))
display(Markdown(f"$x_2 = {x[2].round(0).astype(int)}$"))
def double_derivatives():
# pylint: disable=C0103
RNG().set(12)
a, b = np.random.randint(2, 7, 2)
text = f"Gegeven $k(x) = \\frac{{{a}}}{{x^{b}}}$, geef $k^{{\\prime\\prime}}(x)$"
display(Markdown("**(a)** " + text + r"$\\\\[3mm]$"))
display(Markdown(f"""\\begin{{align}}
k(x) &= {a}x^{{{-b}}} \\\\[1mm]
k^\\prime(x) &= {a*-b}x^{{{-b-1}}} \\\\[1mm]
k^{{\\prime\\prime}}(x) &= {a*b*(b+1)}x^{{{-b-2}}} \\\\[2mm]
&= \\frac{{{a*b*(b+1)}}}{{x^{{{b+2}}}}}
\\end{{align}}"""))
display(Markdown("<hr>"))
a, b = np.random.randint(2, 7, 2)
text = f"Gegeven $\\frac{{dy}}{{dx}} = x^{a} - {hide_one(b-1)}y$, geef $\\frac{{d^2y}}{{dx^2}}$"
display(Markdown("**(b)** " + text + r"$\\\\[3mm]$"))
display(Markdown(f"""\\begin{{align}}\\frac{{d^2y}}{{dx^2}}
&= \\frac{{dy}}{{dx}}x^{a} - {hide_one(b-1)}\\frac{{dy}}{{dx}} y \\\\[1mm]
&= {a}x^{hide_one(a-1, True)} - {hide_one(b-1)} (x^{a} - {hide_one(b-1)}y) \\\\[1mm]
&= {a}x^{hide_one(a-1, True)} - {hide_one(b-1)}x^{a} + {hide_one((b-1)**2)}y
\\end{{align}}"""))
def integrals_billy():
# pylint: disable=C0103
RNG().set(10)
a, b, c = np.random.randint(2, 7, 3)
display(Markdown("**(b)**"))
display(Markdown(f"""\\begin{{align}}
\\int_{min(a,b)}^{max(a,b)+2} {c}e^x\\ dx
&= {c} \\int_{min(a,b)}^{max(a,b)+2} e^x\\ dx \\\\[1mm]
&= {c}\\cdot e^x\\ \\biggr\\rvert_{min(a,b)}^{max(a,b)+2} \\\\[1mm]
&= {c} (e^{max(a,b)+2} - e^{min(a,b)})\\\\[1mm]
&= {c} e^{max(a,b)+2} - {c} e^{min(a,b)}
\\end{{align}}"""))
display(Markdown("<hr>"))
a, b, c = np.random.randint(2, 5, 3)
display(Markdown("**(a)**"))
display(Markdown(f"""\\begin{{align}}
\\int_{{{halve(min(a,b))}\\pi}}^{{{halve(max(a,b)+2)}\\pi}} -{c} \\text{{sin}}(x)\\ dx
&= -{c} \\int_{{{halve(min(a,b))}\\pi}}^{{{halve(max(a,b)+2)}\\pi}} \\text{{sin}}(x)\\ dx \\\\[1mm]
&= -{c}\\cdot -\\text{{cos}}(x)\\ \\biggr\\rvert_{{{halve(min(a,b))}\\pi}}^{{{halve(max(a,b)+2)}\\pi}} \\\\[1mm]
&= {c} \\text{{cos}}(x)\\ \\biggr\\rvert_{{{halve(min(a,b))}\\pi}}^{{{halve(max(a,b)+2)}\\pi}} \\\\[1mm]
&= {c} \\left(\\text{{cos}}\\left({halve(max(a,b)+2)}\\pi\\right) - \\text{{cos}}\\left({halve(min(a,b))}\\pi\\right)\\right) \\\\[1mm]
&= {c} ( {int(np.round(np.cos((0.5*np.pi*(max(a,b)+2)))))} - {int(np.round(np.cos((0.5*np.pi*min(a,b)))))} ) \\\\[1mm]
&= {c} ( {int(np.round(np.cos((0.5*np.pi*(max(a,b)+2))))) - int(np.round(np.cos((0.5*np.pi*min(a,b)))))} ) \\\\[1mm]
&= {c* ( int(np.round(np.cos((0.5*np.pi*(max(a,b)+2))))) - int(np.round(np.cos((0.5*np.pi*min(a,b))))))}
\\end{{align}}"""))
display(Markdown("<hr>"))
a, b, c, d = np.random.randint(3, 9, 4)
display(Markdown("**(b)**"))
display(Markdown(f"""\\begin{{align}}
\\int ({a*b}x^{b-1})({a}x^{b}+{c})^{d}\\ dx
&= \\int u^{d}\\ du \\qquad \\text{{via $u$-substitutie met}}\\quad u ={a}x^{b}+{c};\\ du = {a*b}x^{b-1} \\\\[1mm]
&= \\frac{{1}}{{{d+1}}} u^{d+1} + C\\\\[1mm]
&= \\frac{{1}}{{{d+1}}} ({a}x^{b}+{c})^{d+1} + C
\\end{{align}}"""))
display(Markdown("<hr>"))
a, b, c = np.random.randint(2, 7, 3)
display(Markdown("**(c)**"))
display(Markdown(f"""\\begin{{align}}
\\int ({a}x^{b})\\text{{log}}_{c}(x)\\ dx
&= \\int \\frac{{({a}x^{b})\\text{{ln}}(x)}}{{\\text{{ln}}({c})}}\\ dx \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\int \\text{{ln}}(x)\\ x^{b} \\ dx \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\left(\\int u\\ dv\\right) \\
\\text{{via partiële integratie met}}\\quad u = \\text{{ln}}(x);\\ du = \\frac{{1}}{{x}}\\ dx;\\
dv = x^{{{b}}}\\ dx;\\ v = \\frac{{x^{b+1}}}{{{b+1}}} \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\left(uv - \\int v\\ du\\right) \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\left(\\text{{ln}}(x)\\frac{{x^{b+1}}}{{{b+1}}}
- \\int \\frac{{x^{b+1}}}{{{b+1}}}\\ \\frac{{1}}{{x}}\\ dx\\right) \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\left(\\frac{{x^{b+1}\\text{{ln}}(x)}}{{{b+1}}}
- \\frac{{1}}{{{b+1}}}\\int x^{b}\\ dx \\right) \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\left(\\frac{{x^{b+1}\\text{{ln}}(x)}}{{{b+1}}}
- \\frac{{1}}{{{b+1}}}\\cdot \\frac{{x^{b+1}}}{{{b+1}}} + C \\right) \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\left(\\frac{{{b+1}x^{b+1}\\text{{ln}}(x)}}{{{(b+1)**2}}}
- \\frac{{x^{b+1}-x^{b+1}}}{{{(b+1)**2}}} + C \\right) \\\\[1mm]
&= \\frac{{{a}}}{{\\text{{ln}}({c})}}\\ \\left(\\frac{{x^{b+1}\\ ({b+1}\\ \\text{{ln}}(x)-1)}}{{{(b+1)**2}}} + C \\right) \\\\[1mm]
&= \\frac{{{a}x^{b+1}\\ ({b+1}\\ \\text{{ln}}(x)-1)}}{{{(b+1)**2}\\ \\text{{ln}}({c})}} + C \\\\[1mm]
\\end{{align}}"""))
display(Markdown("<hr>"))
def integrals():
RNG().set(9)
random_integrals()
def double_derivatives():
RNG().set(12)
random_double_derivatives()
def implicit_diff():
RNG().set(13)
random_implicit_diff()
def dif_eq():
RNG().set(11)
random_de()
def derivatives():
RNG().set(8)
random_derivatives()
def integrals_billy():
RNG().set(10)
random_integrals_extra()
def gauss_jordan():
RNG().set(5)
random_sys_of_eq()
def inverses():
RNG().set(6)
random_tensor(r"\textbf{M}", (2, 2), singular=NONDEGENERATE)
random_tensor(r"\textbf{N}", (2, 2), singular=DEGENERATE)
random_tensor(r"\textbf{O}", (2, 2), singular=NONDEGENERATE)
