1247.sagews

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

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

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

abs(taylor (f,x,1,1) - f(x))(1);

0

abs(taylor (f,x,1,1) - f(x))(5);

16

abs(taylor (f,x,1,1) - f(x))(10);

81

abs(taylor (f,x,1,1) - f(x))(100);

9801

g(x) = 1/

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

taylor(g,x,10,1);

x |--> -1/121*x + 21/121

R1 = g(x) - taylor(g,x,10,1);

[abs(R1(1 + (1/(2^k)))) for k in range (6)];

[64/363, 289/1210, 1225/4356, 5041/16456, 169/528, 82369/251680]

R2 =  g(x) - taylor(g,x,10,2);

[abs(R2(1 + (1/(2^k)))) for k in range (6)];

[512/3993, 4913/26620, 42875/191664, 357911/1448128, 2197/8448, 23639903/88591360]

R1a = g(x) - taylor(g,x,(-9/10),1);

[abs(R1a(1 + (1/(2^k)))) for k in range (6)];

[841/3, 1152/5, 1849/9, 6561/34, 24649/132, 95481/520]

R2a = g(x) - taylor(g,x,(-9/10),2);

[abs(R2a(1 + (1/(2^k)))) for k in range (6)];

[24389/3, 27648/5, 79507/18, 531441/136, 3869893/1056, 29503629/8320]

R1.plot(x,-1,3);

R2a.plot(x,-1,3);

h(x) = arctan(x);h

x |--> arctan(x)

d2(x) = derivative(h,x,2);d2

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

d6(x) = derivative(h,x,6);d6

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

d11(x) = derivative(h,x,11);d11

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

d2.plot(x, 0, 1);

d6.plot(x,0,1);

d11.plot(x,0,1);

T1 = taylor(h,x,0,1);T1

x |--> x

T5 = taylor(h,x,0,5);T5

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

T10 = taylor(h,x,0,10);T10

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

RR(abs(T1(1/2)-h(1/2)))

0.0363523909991939

RR((.5^2)/2)

0.125000000000000

RR(abs(T1(1)-h(1)))

0.214601836602552

RR(1/2)

0.500000000000000

RR(abs(T5(1/2)-h(1/2)))

0.000935724332527255

RR((120)*((abs(1/2)^6)/factorial(6)))

0.00260416666666667

RR(abs(T5(1)-h(1)))

0.0812685032692184

RR((120)*((abs(1)^6)/factorial(6)))

0.166666666666667

RR(abs(T10(1/2)-h(1/2)))

0.0000366667928446973

RR((4e6)*((abs(1/2)^11)/factorial(11)))

0.0000489298991903159

RR(abs(T10(1)-h(1)))

0.0495224715231867

RR((4e6)*((abs(1)^11)/factorial(11)))

0.100208433541767