Question

Use an appropriate Taylor polynomial about 0 and the Lagrange Remainder Formula to approximate sin(4/7) with an error less than 0.0001.

Answer 1

For a taylor polynomial 

f(x) = Sum a_n x^(n) 

= Pn + epsilon 

epsilon = Sum a_n+1 (x-x_0)^(n+1) 


Lagrange remainder 
a_n = f^(n) (x_0) / n! 

|epsilon| < ∫ [x_0 to x] f^(n+1)(t) / (t - x_0)^n / n! dt 
< max (|f^(n+1)|) (x-x_0)^n+1 / n+1! 

In the Taylor expansion of sin x 

sin x = x - x^3/3! ... 

find n such that epsilon < 0.0001 

epsilon < max (|f^(n+1)|) (x-x_0)^n+1 / n+1! < 0.0001 
for sin x, max (|f^(n+1)|) < 1 
(6/7) < 1 so (6/7)^n <1 

1/(n+1)! < 0.0001 
n+1! > 0.0001 

(6/7)^7/(n+1)! < 0.0001 
the coefficient of x^6 = 0 in the expansion of sin x 

sin x = x - x^3 + x^5 +- 0.0001