Find the polynomial of lowest degree that will approximate $F ( x )$ throughout the given interval with an error of magnitude less than than $10 ^ { - 3 }$.
-$$\mathrm { F } ( \mathrm { x } ) = \int _ { 0 } ^ { \mathrm { x } } \frac { \tan ^ { - 1 } \mathrm { t } } { \mathrm { t } } \mathrm { dt } , [ 0,0.5 ]$$
A) $x - \frac { x ^ { 3 } } { 9 } + \frac { x ^ { 5 } } { 25 }$
B) $x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }$
C) $x - \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 5 } } { 60 }$
D) $x - \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 5 } } { 5 }$

Question

Quizplus · Accepted Answer

The given function $F(x) = \int_{0}^{x} \frac{\tan^{-1}t}{t} dt$ can be approximated by a Taylor series expansion of $\tan^{-1}t$ about $t=0$, which is $\tan^{-1}t = t - \frac{t^3}{3} + \frac{t^5}{5} - \cdots$. Integrating term by term from 0 to $x$, we get $F(x) = x - \frac{x^3}{9} + \frac{x^5}{25} - \cdots$. The polynomial $x - \frac{x^3}{9} + \frac{x^5}{25}$ is the lowest degree polynomial that approximates $F(x)$ within the given error margin for the interval $[0, 0.5]$, thus making choice A correct.

Find the Polynomial of Lowest Degree That Will Approximate F(x)F ( x )F(x)

Find the Polynomial of Lowest Degree That Will Approximate $F ( x )$