1245.sagews

f(x) = x^2 + 2*x + 1; f

x |--> x^2 + 2*x + 1

var('x,k,n')
(x, k, n)

(x, k, n)

taylor (f, x, 0, 1)

x |--> 2*x + 1

taylor (f, x, 0, 2)

x |--> x^2 + 2*x + 1

taylor (f, x, 0, 5)

x |--> x^2 + 2*x + 1

taylor (f, x, 1, 1)

x |--> 4*x

taylor (f, x, 1, 2)

x |--> (x - 1)^2 + 4*x

((x - 1)^2 + 4*x) - (x^2 + 2*x + 1)

(x - 1)^2 - x^2 + 2*x - 1

taylor (f, x, 1, 1)

x |--> 4*x

a(x) = 4*x; a

x |--> 4*x

abs(a(1) - f(1))

0

abs(a(5) - f(5))

16

abs(a(10) - f(10))

81

abs(a(100) - f(100))

9801

g(x) = 1/(1 + x); g

x |--> 1/(x + 1)

taylor (g, x, 1, 1)

x |--> -1/4*x + 3/4

[abs((taylor (g, x, 1, 1) - g) (1 + (1/2^k))) for k in range (6)]

[1/12, 1/40, 1/144, 1/544, 1/2112, 1/8320]

taylor (g, x, 1, 2)

x |--> 1/8*(x - 1)^2 - 1/4*x + 3/4

[abs((taylor (g, x, 1, 2) - g) (1 + (1/2^k))) for k in range (6)]

[1/24, 1/160, 1/1152, 1/8704, 1/67584, 1/532480]

taylor (g, x, (-9/10), 1)

x |--> -100*x - 80

[abs((taylor (g, x, (-9/10), 2) - g) ((-9/10) + (1/2^k))) for k in range (6)]

[10000/11, 625/3, 625/14, 625/72, 625/416, 625/2688]

plot((taylor (g, x, (-9/10), 2) - g), (-1,3))

h(x) = arctan(x); h

x |--> arctan(x)

t(x) = diff(h, x, 2); t

x |--> -2*x/(x^2 + 1)^2

u(x) = diff(h, x, 6); u

x |--> -3840*x^5/(x^2 + 1)^6 + 3840*x^3/(x^2 + 1)^5 - 720*x/(x^2 + 1)^4

v(x) = diff(h, x, 11); v

x |--> 3715891200*x^10/(x^2 + 1)^11 - 8360755200*x^8/(x^2 + 1)^10 + 6502809600*x^6/(x^2 + 1)^9 - 2032128000*x^4/(x^2 + 1)^8 + 217728000*x^2/(x^2 + 1)^7 - 3628800/(x^2 + 1)^6

plot(diff(h, x, 2), (0,1))

plot(diff(h, x, 6), (0,1))

plot(diff(h, x, 11), (0,1))

taylor (h, x, 0, 1);

Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_77.py", line 5, in <module> taylor (h, x, _sage_const_0 , _sage_const_1 ); File "/usr/local/sage/local/lib/python2.6/site-packages/sage/calculus/functional.py", line 359, in taylor return f.taylor(v=v,a=a,n=n) File "expression.pyx", line 2342, in sage.symbolic.expression.Expression.taylor (sage/symbolic/expression.cpp:12460) File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1404, in __call__ return self._obj.parent().function_call(self._name, [self._obj] + list(args), kwds) File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1312, in function_call return self.new(s) File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1096, in new return self(code) File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1031, in __call__ return cls(self, x, name=name) File "/usr/local/sage/local/lib/python2.6/site-packages/sage/interfaces/expect.py", line 1447, in __init__ raise TypeError, x TypeError: Error executing code in Maxima CODE: sage436 : taylor(sage432,sage433,sage434,sage435)$ Maxima ERROR: taylor: 1/2 cannot be a variable.

taylor (h, x, 0, 5)

x |--> 1/5*x^5 - 1/3*x^3 + x

taylor (h, x, 0, 10)

x |--> 1/9*x^9 - 1/7*x^7 + 1/5*x^5 - 1/3*x^3 + x

n(x) = x; n
n(1/2) - arctan(0.5)

0.0363523909991939

0.65*((1/2)^(2)/factorial(2))

0.0812500000000000

100*((1/2)^(2)/factorial(2))

25/2

3e6*((1/2)^(2)/factorial(2))

375000.000000000

n(x) = x; n
n(1) - arctan(1)

-1/4*pi + 1

0.65*((1)^(2)/factorial(2))

0.325000000000000

100*((1)^(2)/factorial(2))

50

3e6*((1)^(2)/factorial(2))

1.50000000000000e6

s(x) = 1/5*x^5 - 1/3*x^3 + x; s
s(1/2) - arctan(0.5)

0.000935724332527255

0.65*((1/2)^(6)/factorial(6))

0.0000141059027777778

100*((1/2)^(6)/factorial(6))

5/2304

3e6*((1/2)^(6)/factorial(6))

65.1041666666667

s(x) = 1/5*x^5 - 1/3*x^3 + x; s
s(1) - arctan(1)

-1/4*pi + 13/15

0.65*((1)^(6)/factorial(6))

0.000902777777777778

100*((1)^(6)/factorial(6))

5/36

3e6*((1)^(6)/factorial(6))

4166.66666666667

i(x)= 1/9*x^9 - 1/7*x^7 + 1/5*x^5 - 1/3*x^3 + x; i
i(0.5) - arctan(0.5)

0.0000366667928446973

0.65*((1/2)^(11)/factorial(11))

7.95110861842633e-12

100*((1/2)^(11)/factorial(11))

1/817496064

3e6*((1/2)^(11)/factorial(11))

0.0000366974243927369

i(x)= 1/9*x^9 - 1/7*x^7 + 1/5*x^5 - 1/3*x^3 + x; i
i(1) - arctan(1)

-1/4*pi + 263/315

0.65*((1)^(11)/factorial(11))

1.62838704505371e-8

100*((1)^(11)/factorial(11))

1/399168

3e6*((1)^(11)/factorial(11))

0.0751563251563252