What is the smallest value of n that will guarantee (according to Taylor's Formula) that the Taylor polynomial $T _ { n }$ at the number 0 will be within 0.0001 of $e ^ { x }$ for $0 \leq x \leq 1$ ?
A)4
B)5
C)8
D)6
E)7
F)2
G)3
H)9

Question

Quizplus · Accepted Answer

The error in the Taylor polynomial approximation of $e^x$ is given by the remainder term $R_n(x) = \frac{e^c x^{n+1}}{(n+1)!}$ for some $c$ between 0 and $x$. To ensure the error is less than 0.0001 for $0 \leq x \leq 1$, we need $\frac{e x^{n+1}}{(n+1)!} < 0.0001$. Testing values, $n = 7$ satisfies this condition.

What Is the Smallest Value of N That Will Guarantee TnT _ { n }Tn​

What Is the Smallest Value of N That Will Guarantee $T _ { n }$