According to Taylor's remainder estimate, the maximum possible error in the use of
4
Σ (x^n) / (n!)
n=0
to approximate e^x on the interval [-1,1] is which value?
4
Σ (x^n) / (n!)
n=0
to approximate e^x on the interval [-1,1] is which value?
-
The error is given by
|f '''''(c) x^5/5!| for some c between 0 and x.
Since f(x) = e^x, f '''''(x) = e^x.
So, |f '''''(c) x^5/5!|
= e^c |x|^5 / 5!
≤ e^c * 1^5 / 5!, since |x| ≤ 1.
< e^1 / 5!, since c < x ≤ 1
= e/120.
Note: If you want your answer without e in it, use a crude bound like e < 3
==> |Error| < 3/120 = 1/40.
I hope this helps!
|f '''''(c) x^5/5!| for some c between 0 and x.
Since f(x) = e^x, f '''''(x) = e^x.
So, |f '''''(c) x^5/5!|
= e^c |x|^5 / 5!
≤ e^c * 1^5 / 5!, since |x| ≤ 1.
< e^1 / 5!, since c < x ≤ 1
= e/120.
Note: If you want your answer without e in it, use a crude bound like e < 3
==> |Error| < 3/120 = 1/40.
I hope this helps!