Answer
$a)$ = $366$
$b)$ = $26$
Work Step by Step
$a)$
$f'(x)$ = -$x^{-2}$
$f''(x)$ = $2x^{-3}$
$f''(1)$ = $2(1)^{-3}$ = $2$
$f''(3)$ = $2(3)^{-3}$ = $\frac{2}{27}$
$|E|$ $\leq$ $\frac{(b-a)^{3}}{12n^{2}}$$max|f''(x)|$
$E$ $less$ $than$ $0.00001$ $then$
$0.00001$ $\geq$ $\frac{(3-1)^{3}}{12n^{2}}$$(2)$
$n^{2}$ $\geq$ $\frac{(2)^{3}(2)}{12(0.00001)}$
$n$ $\geq$ $365.15$
$n$ = $366$
$b)$
$f^{3}(x)$ = -$6x^{-4}$
$f^{4}(x)$ = $24x^{-5}$
$f^{4}(1)$ = $24(1)^{-5}$ = $24$
$f^{4}(3)$ = $24(3)^{-5}$ = $\frac{24}{3^{5}}$ = $\frac{24}{243}$
$|E|$ $\leq$ $\frac{(b-a)^{5}}{180n^{4}}$$max|f^{4}(x)|$
$E$ $less$ $than$ $0.00001$ $then$
$0.00001$ $\geq$ $\frac{(3-1)^{5}}{180n^{4}}$$(24)$
$n^{4}$ $\geq$ $\frac{(2)^{5}(24)}{180(0.00001)}$
$n$ $\geq$ $25.558$
$n$ = $26$
