CoCalc -- zkouška II.sagews

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ubuntu2004

%typeset_mode True
#ZK II
#1
CC = ComplexField()
v = x^2+1
e.expand()
assume(v, "complex")
v.rectform()
expand(x^2+1, "complex")

$\displaystyle e$

Error in lines 5-5
Traceback (most recent call last):
  File "/cocalc/lib/python3.10/site-packages/smc_sagews/sage_server.py", line 1244, in execute
    exec(
  File "", line 1, in <module>
  File "/ext/sage/9.8/src/sage/symbolic/assumptions.py", line 676, in assume
    x.assume()
  File "sage/symbolic/expression.pyx", line 2380, in sage.symbolic.expression.Expression.assume
    raise TypeError("self (=%s) must be a relational expression" % self)
TypeError: self (=x^2 + 1) must be a relational expression

#2
reset()
var('y')
def f(x,y):
    if x<y: return 6
    else: return 5
f(9,2)
f(0,1)

$\displaystyle y$

$\displaystyle 5$

$\displaystyle 6$

#3
reset()
var('y')
vyr = y^3 + 2*y + 1 + y/(y+1)^3;vyr
vyr.subs(vyr.operands()[2] == vyr.operands()[2].simplify_full())

$\displaystyle y$

$\displaystyle y^{3} + 2 \, y + \frac{y}{{\left(y + 1\right)}^{3}} + 1$

$\displaystyle y^{3} + 2 \, y + \frac{y}{y^{3} + 3 \, y^{2} + 3 \, y + 1} + 1$

#4
reset()
eqn = cos(x)==0;eqn
res = solve(eqn, x);res
check = eqn.subs(res);check
#chybí další řešení

$\displaystyle \cos\left(x\right) = 0$

[$\displaystyle x = \frac{1}{2} \, \pi$]

$\displaystyle 0 = 0$

#5
reset()
var('x,y')
implicit_plot((x - 1)^2+y^2==9, (x,-2,4), (y,-3,4))

($\displaystyle x$, $\displaystyle y$)

#6
n(pi^e, digits=13)
x=n(pi^e, digits=13);"x=";x

$\displaystyle 22.45915771836$

#7
U={i^2 for i in range(1,6)};print("U =");U;
V = {n^n for n in range(1,10)};print("V ="); V
print("U průnik V = ");U.intersection(V)

U =
{1, 4, 9, 16, 25}
V =
{256, 1, 46656, 16777216, 4, 387420489, 3125, 823543, 27}
U průnik V = 
{1, 4}

#8
A=matrix( 5, 5, lambda i, j: gcd(i+1,j+1));A
"|A| ="; det(A);

$\displaystyle \left(\begin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 1 & 2 & 1 \\ 1 & 1 & 3 & 1 & 1 \\ 1 & 2 & 1 & 4 & 1 \\ 1 & 1 & 1 & 1 & 5 \end{array}\right)$

|A| =

$\displaystyle 16$

#9
reset()
f3 = x^3*cos(x);f3
g(x) = integrate(f3,x);g(x)
g(0)

$\displaystyle x^{3} \cos\left(x\right)$

$\displaystyle 3 \, {\left(x^{2} - 2\right)} \cos\left(x\right) + {\left(x^{3} - 6 \, x\right)} \sin\left(x\right)$

$\displaystyle -6$

#10
reset()
def R(n,x):
    if(n==0):
        result = 1
    elif (n==1):
        result = x
    else:
        result = 2*n*R(n-1,x)+2*R(n-2,x)
    return result

var('y')
R(4,y)
g = R(4,y);
h(x) = diff(g, x,x);h(x)
h(1)

$\displaystyle y$

$\displaystyle 216 \, y + 100$

$\displaystyle 0$

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