Question-14

Find the best linear approximation of \(\displaystyle f( x) =\sqrt{1+x}\) at \(\displaystyle x=0\).

• \(\sqrt{1 + x}\)

• \(1 + x\)

• \(x\)

• \(1 + \cfrac{x}{2}\)

Answer

• \(\sqrt{1 + x}\)

• \(1 + x\)

• \(x\)

• \(1 + \cfrac{x}{2}\)

Solution

Using Taylor series, the best linear approximation of \(\displaystyle f\) at \(\displaystyle x=a\) is given by:

\[ \begin{equation*} L( x)\Bigl|_{x=a} =f( a) +f^{\prime }( a)( x-a) \end{equation*} \]

For \(\displaystyle x=0\) and \(\displaystyle f( x) =\sqrt{1+x}\), we have:

\[ \begin{equation*} L( x)\Bigl|_{x=0} =1+\frac{x}{2} \end{equation*} \]

We have used the fact that \(\displaystyle f^{\prime }( x) =\frac{1}{2\sqrt{1+x}}\) and \(\displaystyle f^{\prime }( 0) =\frac{1}{2}\).