CoCalc -- SageWeek06.sagews

Views: ⁵³

2 + 2

4

Q2.

cos?

   File: /usr/local/sage/sage-6.3.beta6/local/lib/python2.7/site-packages/sage/functions/trig.py
   Docstring:
      The cosine function.

   EXAMPLES:

      sage: cos(pi)
      -1
      sage: cos(x).subs(x==pi)
      -1
      sage: cos(2).n(100)
      -0.41614683654714238699756822950
      sage: loads(dumps(cos))
      cos

   We can prevent evaluation using the "hold" parameter:

      sage: cos(0,hold=True)
      cos(0)

   To then evaluate again, we currently must use Maxima via
   "sage.symbolic.expression.Expression.simplify()":

      sage: a = cos(0,hold=True); a.simplify()
      1

   TESTS:

      sage: conjugate(cos(x))
      cos(conjugate(x))
      sage: cos(complex(1,1))     # rel tol 1e-15
      (0.8337300251311491-0.9888977057628651j)

sum?

   File: /usr/local/sage/sage-6.3.beta6/local/lib/python2.7/site-packages/sage/misc/functional.py
   Docstring:
      Returns the symbolic sum sum_{v = a}^b expression with respect to
   the variable v with endpoints a and b.

   INPUT:

   * "expression" - a symbolic expression

   * "v" - a variable or variable name

   * "a" - lower endpoint of the sum

   * "b" - upper endpoint of the sum

   * "algorithm" - (default: 'maxima')  one of - 'maxima' - use Maxima
     (the default) - 'maple' - (optional) use Maple - 'mathematica' -
     (optional) use Mathematica

   EXAMPLES:

      sage: k, n = var('k,n')
      sage: sum(k, k, 1, n).factor()
      1/2*(n + 1)*n

      sage: sum(1/k^4, k, 1, oo)
      1/90*pi^4

      sage: sum(1/k^5, k, 1, oo)
      zeta(5)

   Warning: This function only works with symbolic expressions. To sum any
     other objects like list elements or function return values,
     please use python summation, see
     http://docs.python.org/library/functions.html#sumIn particular,
     this does not work:

        sage: n = var('n')
        sage: list=[1,2,3,4,5]
        sage: sum(list[n],n,0,3)
        Traceback (most recent call last):
        ...
        TypeError: unable to convert x (=n) to an integer

     Use python "sum()" instead:

        sage: sum(list[n] for n in range(4))
        10

     Also, only a limited number of functions are recognized in
     symbolic sums:

        sage: sum(valuation(n,2),n,1,5)
        Traceback (most recent call last):
        ...
        AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'valuation'

     Again, use python "sum()":

        sage: sum(valuation(n+1,2) for n in range(5))
        3

     (now back to the Sage "sum" examples)

   A well known binomial identity:

      sage: sum(binomial(n,k), k, 0, n)
      2^n

   The binomial theorem:

      sage: x, y = var('x, y')
      sage: sum(binomial(n,k) * x^k * y^(n-k), k, 0, n)
      (x + y)^n

      sage: sum(k * binomial(n, k), k, 1, n)
      2^(n - 1)*n

      sage: sum((-1)^k*binomial(n,k), k, 0, n)
      0

      sage: sum(2^(-k)/(k*(k+1)), k, 1, oo)
      -log(2) + 1

   Another binomial identity (trac #7952):

      sage: t,k,i = var('t,k,i')
      sage: sum(binomial(i+t,t),i,0,k)
      binomial(k + t + 1, t + 1)

   Summing a hypergeometric term:

      sage: sum(binomial(n, k) * factorial(k) / factorial(n+1+k), k, 0, n)
      1/2*sqrt(pi)/factorial(n + 1/2)

   We check a well known identity:

      sage: bool(sum(k^3, k, 1, n) == sum(k, k, 1, n)^2)
      True

   A geometric sum:

      sage: a, q = var('a, q')
      sage: sum(a*q^k, k, 0, n)
      (a*q^(n + 1) - a)/(q - 1)

   The geometric series:

      sage: assume(abs(q) < 1)
      sage: sum(a*q^k, k, 0, oo)
      -a/(q - 1)

   A divergent geometric series.  Don't forget to forget your
   assumptions:

      sage: forget()
      sage: assume(q > 1)
      sage: sum(a*q^k, k, 0, oo)
      Traceback (most recent call last):
      ...
      ValueError: Sum is divergent.

   This summation only Mathematica can perform:

      sage: sum(1/(1+k^2), k, -oo, oo, algorithm = 'mathematica')     # optional - mathematica
      pi*coth(pi)

   Use Maple as a backend for summation:

      sage: sum(binomial(n,k)*x^k, k, 0, n, algorithm = 'maple')      # optional - maple
      (x + 1)^n

   Python ints should work as limits of summation (trac #9393):

      sage: sum(x, x, 1r, 5r)
      15

   Note: 1. Sage can currently only understand a subset of the output of
        Maxima, Maple and Mathematica, so even if the chosen backend
        can perform the summation the result might not be convertable
        into a Sage expression.

Q3.

cos(3)

cos(3)

cos(3.)

-0.989992496600445

sqrt(4)

2

sqrt(8)

2*sqrt(2)

sqrt(8.)

2.82842712474619

I ** 2

-1

sqrt(-53.)

7.28010988928052*I

Q4.

i = var('i')  # Students won't know that they need to do this.
expression = sum(1 / sqrt(i), i, 1, 5)
print expression

1/60*(((3*sqrt(2) + 2)*sqrt(3) + 2*sqrt(2))*sqrt(5) + 2*sqrt(2)*sqrt(3))*sqrt(2)*sqrt(3)*sqrt(5)

factor(expression)

1/60*(3*sqrt(2)*sqrt(3)*sqrt(5) + 2*sqrt(2)*sqrt(3) + 2*sqrt(2)*sqrt(5) + 2*sqrt(3)*sqrt(5))*sqrt(2)*sqrt(3)*sqrt(5)

simplify(expression)

1/60*(((3*sqrt(2) + 2)*sqrt(3) + 2*sqrt(2))*sqrt(5) + 2*sqrt(2)*sqrt(3))*sqrt(2)*sqrt(3)*sqrt(5)

expand(expression)

1/2*sqrt(2) + 1/3*sqrt(3) + 1/5*sqrt(5) + 3/2

Q5.

factor(234398)

2 * 233 * 503

f = 234398.factor()
len(f)

3

nbroffactors = []
n = 1
while n < 1000:  # Student might try a `for n in range` approach but this gives python int type objects which do not have a factor method.
    nbroffactors.append(len(n.factor()))
    n += 1
print mean(nbroffactors)

236/111

Q6.

k = var('k')
k = 3
while k < 100:  # This is one approach of doing this that makes nice use of the string substitution below but there are many other ways.
    print "For k =%s, k ^ 2 - 79 * k + 1601 = %s is prime: %s" % (k, k^2 - 79 * k +1601, (k^2 - 79 * k +1601).is_prime())
    k += 1

For k =3, k ^ 2 - 79 * k + 1601 = 1373 is prime: True
For k =4, k ^ 2 - 79 * k + 1601 = 1301 is prime: True
For k =5, k ^ 2 - 79 * k + 1601 = 1231 is prime: True
For k =6, k ^ 2 - 79 * k + 1601 = 1163 is prime: True
For k =7, k ^ 2 - 79 * k + 1601 = 1097 is prime: True
For k =8, k ^ 2 - 79 * k + 1601 = 1033 is prime: True
For k =9, k ^ 2 - 79 * k + 1601 = 971 is prime: True
For k =10, k ^ 2 - 79 * k + 1601 = 911 is prime: True
For k =11, k ^ 2 - 79 * k + 1601 = 853 is prime: True
For k =12, k ^ 2 - 79 * k + 1601 = 797 is prime: True
For k =13, k ^ 2 - 79 * k + 1601 = 743 is prime: True
For k =14, k ^ 2 - 79 * k + 1601 = 691 is prime: True
For k =15, k ^ 2 - 79 * k + 1601 = 641 is prime: True
For k =16, k ^ 2 - 79 * k + 1601 = 593 is prime: True
For k =17, k ^ 2 - 79 * k + 1601 = 547 is prime: True
For k =18, k ^ 2 - 79 * k + 1601 = 503 is prime: True
For k =19, k ^ 2 - 79 * k + 1601 = 461 is prime: True
For k =20, k ^ 2 - 79 * k + 1601 = 421 is prime: True
For k =21, k ^ 2 - 79 * k + 1601 = 383 is prime: True
For k =22, k ^ 2 - 79 * k + 1601 = 347 is prime: True
For k =23, k ^ 2 - 79 * k + 1601 = 313 is prime: True
For k =24, k ^ 2 - 79 * k + 1601 = 281 is prime: True
For k =25, k ^ 2 - 79 * k + 1601 = 251 is prime: True
For k =26, k ^ 2 - 79 * k + 1601 = 223 is prime: True
For k =27, k ^ 2 - 79 * k + 1601 = 197 is prime: True
For k =28, k ^ 2 - 79 * k + 1601 = 173 is prime: True
For k =29, k ^ 2 - 79 * k + 1601 = 151 is prime: True
For k =30, k ^ 2 - 79 * k + 1601 = 131 is prime: True
For k =31, k ^ 2 - 79 * k + 1601 = 113 is prime: True
For k =32, k ^ 2 - 79 * k + 1601 = 97 is prime: True
For k =33, k ^ 2 - 79 * k + 1601 = 83 is prime: True
For k =34, k ^ 2 - 79 * k + 1601 = 71 is prime: True
For k =35, k ^ 2 - 79 * k + 1601 = 61 is prime: True
For k =36, k ^ 2 - 79 * k + 1601 = 53 is prime: True
For k =37, k ^ 2 - 79 * k + 1601 = 47 is prime: True
For k =38, k ^ 2 - 79 * k + 1601 = 43 is prime: True
For k =39, k ^ 2 - 79 * k + 1601 = 41 is prime: True
For k =40, k ^ 2 - 79 * k + 1601 = 41 is prime: True
For k =41, k ^ 2 - 79 * k + 1601 = 43 is prime: True
For k =42, k ^ 2 - 79 * k + 1601 = 47 is prime: True
For k =43, k ^ 2 - 79 * k + 1601 = 53 is prime: True
For k =44, k ^ 2 - 79 * k + 1601 = 61 is prime: True
For k =45, k ^ 2 - 79 * k + 1601 = 71 is prime: True
For k =46, k ^ 2 - 79 * k + 1601 = 83 is prime: True
For k =47, k ^ 2 - 79 * k + 1601 = 97 is prime: True
For k =48, k ^ 2 - 79 * k + 1601 = 113 is prime: True
For k =49, k ^ 2 - 79 * k + 1601 = 131 is prime: True
For k =50, k ^ 2 - 79 * k + 1601 = 151 is prime: True
For k =51, k ^ 2 - 79 * k + 1601 = 173 is prime: True
For k =52, k ^ 2 - 79 * k + 1601 = 197 is prime: True
For k =53, k ^ 2 - 79 * k + 1601 = 223 is prime: True
For k =54, k ^ 2 - 79 * k + 1601 = 251 is prime: True
For k =55, k ^ 2 - 79 * k + 1601 = 281 is prime: True
For k =56, k ^ 2 - 79 * k + 1601 = 313 is prime: True
For k =57, k ^ 2 - 79 * k + 1601 = 347 is prime: True
For k =58, k ^ 2 - 79 * k + 1601 = 383 is prime: True
For k =59, k ^ 2 - 79 * k + 1601 = 421 is prime: True
For k =60, k ^ 2 - 79 * k + 1601 = 461 is prime: True
For k =61, k ^ 2 - 79 * k + 1601 = 503 is prime: True
For k =62, k ^ 2 - 79 * k + 1601 = 547 is prime: True
For k =63, k ^ 2 - 79 * k + 1601 = 593 is prime: True
For k =64, k ^ 2 - 79 * k + 1601 = 641 is prime: True
For k =65, k ^ 2 - 79 * k + 1601 = 691 is prime: True
For k =66, k ^ 2 - 79 * k + 1601 = 743 is prime: True
For k =67, k ^ 2 - 79 * k + 1601 = 797 is prime: True
For k =68, k ^ 2 - 79 * k + 1601 = 853 is prime: True
For k =69, k ^ 2 - 79 * k + 1601 = 911 is prime: True
For k =70, k ^ 2 - 79 * k + 1601 = 971 is prime: True
For k =71, k ^ 2 - 79 * k + 1601 = 1033 is prime: True
For k =72, k ^ 2 - 79 * k + 1601 = 1097 is prime: True
For k =73, k ^ 2 - 79 * k + 1601 = 1163 is prime: True
For k =74, k ^ 2 - 79 * k + 1601 = 1231 is prime: True
For k =75, k ^ 2 - 79 * k + 1601 = 1301 is prime: True
For k =76, k ^ 2 - 79 * k + 1601 = 1373 is prime: True
For k =77, k ^ 2 - 79 * k + 1601 = 1447 is prime: True
For k =78, k ^ 2 - 79 * k + 1601 = 1523 is prime: True
For k =79, k ^ 2 - 79 * k + 1601 = 1601 is prime: True
For k =80, k ^ 2 - 79 * k + 1601 = 1681 is prime: False
For k =81, k ^ 2 - 79 * k + 1601 = 1763 is prime: False
For k =82, k ^ 2 - 79 * k + 1601 = 1847 is prime: True
For k =83, k ^ 2 - 79 * k + 1601 = 1933 is prime: True
For k =84, k ^ 2 - 79 * k + 1601 = 2021 is prime: False
For k =85, k ^ 2 - 79 * k + 1601 = 2111 is prime: True
For k =86, k ^ 2 - 79 * k + 1601 = 2203 is prime: True
For k =87, k ^ 2 - 79 * k + 1601 = 2297 is prime: True
For k =88, k ^ 2 - 79 * k + 1601 = 2393 is prime: True
For k =89, k ^ 2 - 79 * k + 1601 = 2491 is prime: False
For k =90, k ^ 2 - 79 * k + 1601 = 2591 is prime: True
For k =91, k ^ 2 - 79 * k + 1601 = 2693 is prime: True
For k =92, k ^ 2 - 79 * k + 1601 = 2797 is prime: True
For k =93, k ^ 2 - 79 * k + 1601 = 2903 is prime: True
For k =94, k ^ 2 - 79 * k + 1601 = 3011 is prime: True
For k =95, k ^ 2 - 79 * k + 1601 = 3121 is prime: True
For k =96, k ^ 2 - 79 * k + 1601 = 3233 is prime: False
For k =97, k ^ 2 - 79 * k + 1601 = 3347 is prime: True
For k =98, k ^ 2 - 79 * k + 1601 = 3463 is prime: True
For k =99, k ^ 2 - 79 * k + 1601 = 3581 is prime: True

Q7.

y = var('y')
myexp = x ^ 2 - 5 * x ^ 2 + 12 * x * y - 9 * y ^ 2
myexp.factor()

-(2*x - 3*y)^2

Q8.

f(x) =  x^3 + pi*x^2 - 23/2*x^2 - 23/2*pi*x  + 15*x+ 15*pi
plot(f, -15, 15)

f.roots()  # If students don't understand the output invite them to try `f.roots?`

[(-pi, 1), (3/2, 1), (10, 1)]

f.factor()

x |--> 1/2*(pi + x)*(2*x - 3)*(x - 10)

solve(f(x) == 0, x)

[x == -pi, x == (3/2), x == 10]

Q9.

solve(x^2 == -1, x)

[x == -I, x == I]

a = var('a')
solve(x^2 - 53 * x + 2 * a == 0, x)

[x == -1/2*sqrt(-8*a + 2809) + 53/2, x == 1/2*sqrt(-8*a + 2809) + 53/2]

solve(sin(x) == x - 1, x)  # This won't work as `solve` finds analytical solutions to equations
plot(sin(x) - x + 1, -10, 10)  # Invite students to plot this to attempt to find a root, once they have done this encourage them to look at `find_root?`

[x == sin(x) + 1]

find_root(sin(x) - x + 1, 0, 5)

1.934563210752024

solve(x^5 + sin(x) - 2 * x == .5 ,x)  # Again no result can be obtained
plot(x^5 + sin(x) - 2 * x - .5, x, -5, 5)

[0 == 2*x^5 - 4*x + 2*sin(x) - 1]

find_root(x^5 + sin(x) - 2 * x - .5, -2, 2)

1.131990740991698

y, z = var('y', 'z')
solve([x + y == z, 3*x - y == 0, y + z ==1], [x, y, z])

[[x == (1/7), y == (3/7), z == (4/7)]]