Respuestas

2013-07-15T06:00:25+02:00
F(x)~ f(a) + f'(a)/1! · (x-a) + .... + fn) (a)/n! · (x-a)^n +...

1) f(x)= 1+x+x² --> f(2)=7
f'(x)= 1+ 2x --> f'(2)=5
f''(x)= 2 --> f''(2)=2
fn) (x)=0 para todo n>2

==> 1+x+x² = 7 + 5·(x-2) + 2/2! ·(x-2)² = 7 + 5·(x-2) + (x-2)²

2) f(x)= senx --> f(pi/4)=√2/2
f'(x)= cosx --> f'(pi/4)= √2/2
f''(x)= -senx --> f''(pi/4)= -√2/2
f'''(x)= -cosx --> f'''(pi/4)= -√2/2
Y a partir de aquí empiezan a repetirse

==> senx ~ √2/2 ·[1+ 1/1! ·(x-pi/4) - 1/2! (x-pi/4)² - 1/3! ·(x-pi/4)³ + 1/4!· (x-pi/4)^4+......] ~ ∑ _{n=0, inf} (-1)^E(n/2) ·√2/2 · 1/n! · (x-pi/4)^n

3) g(x)= e^x --> g(0)=1
gn) (x)= e^x --> gn) (0)=1

==> e^x ~ ∑ _{n=0, inf} 1/n! ·x^n
e^(-x) ~ ∑ _{n=0, inf} (-1)^n/n! ·x^n

f(x)= x²·e^(-x) ~ ∑ _{n=0, inf} (-1)^n/n! ·x^(n+2)