We can apply the Taylor expansion for the function:
[tex]f(x)=\sqrt[]{x}[/tex]around x0=100.
The Taylor expansion with 2 terms (one differential) can be written as:
[tex]f(x)\approx f(x_0)+f^{\prime}(x_0)(x-x_0)[/tex]The value of f(x0) is 10.
We have to calculate the first derivative of f(x):
[tex]f^{\prime}(x)=\frac{d}{dx}\lbrack x^{\frac{1}{2}}\rbrack=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt[]{x}}[/tex]Now, we can calculate f'(x0):
[tex]f^{\prime}(x_0)=f^{\prime}(100)=\frac{1}{2\sqrt[]{100}}=\frac{1}{2\cdot10}=\frac{1}{20}[/tex]We then can calculate:
[tex]\begin{gathered} f(x)\approx10+\frac{(x-100)}{20} \\ f(101)\approx10+\frac{(101-100)}{20}=10+\frac{1}{20}=10.0500 \end{gathered}[/tex]Answer: 10.0500.
