Estimate of √1,1 with Taylor

Problem

Estimate $\sqrt{1{,}1}$ with the second-order Taylor polynomial of $(1+x)^{1/2}$ at $x_0 = 0$ and bound the error with the Lagrange remainder.

Solution

Expansion of $(1+x)^{1/2}$ at $x_0=0$: $f(0)=1$, $f'(x)=\frac{1}{2}(1+x)^{-1/2}$, $f'(0)=\frac12$; $f''(x)=-\frac{1}{4}(1+x)^{-3/2}$, $f''(0)=-\frac14$. Hence $T_2(x) = 1 + \frac{x}{2} - \frac{x^2}{8}.$ With $x=0{,}1$: $T_2(0{,}1) = 1 + 0{,}05 - 0{,}00125 = \boxed{1{,}04875}$ (true value $\approx 1{,}048809$).

Lagrange remainder: $R_2(x) = \dfrac{f'''(c)}{6}x^3$ with $f'''(x)=\frac{3}{8}(1+x)^{-5/2}$ and $c\in(0;0{,}1)$. Bounding $|f'''(c)|\le\frac{3}{8}$: $|R_2(0{,}1)| \le \frac{3/8}{6}\cdot (0{,}1)^3 = \frac{1}{16}\cdot 10^{-3} \approx 6{,}3\cdot 10^{-5}.$ The actual error ($\approx 5{,}9\cdot 10^{-5}$) is consistent with the estimate.

Links

Topics: Derivatives
Concepts: Taylor polynomial · Taylor remainder
Methods: Taylor Lagrange remainder
Skills: Estimate
Exercise type: Derivative computation