The Taylor series of $f$, centered at $a$, is $f(x) = f(a) + f'(a) (x-a) + \frac12 f''(a) (x-a)^2 + \cdots$

In this problem you will use this formula with $f(x) = \sqrt{x}$ and $a =$ to approximate $\sqrt{101}$.
This is the best value of $a$ to use since it is close to 101, and you can compute $f(a)$ without a calculator. In fact, you can easily compute $f'(a)$ and $f''(a)$ without a calculator, too. Try to do this whole problem without a calculator.

The first approximation is

$\sqrt{101} \approx T_1(101) = f(a) + f'(a) (x-a) =$

You might remember this as the linear approximation from Calculus I. The quadratic approximation to $f$ near $x = a$ gives

$\sqrt{101} \approx T_2(101) = f(a) + f'(a) (x-a) + \frac12 f''(a) (x-a)^2 =$

Warning! WeBWorK might have marked those approximations as correct even if they are wrong. This is because WeBWorK marks an answer correct if it is within 0.1 per cent of the correct answer. A good way to see if you did it right it to check the errors of the approximations. You don't need a calculator for this part either, since WeBWorK can do the calculations for you.
The error (positive or negative) of the first approximation is

$T_1(101) - \sqrt{101} =$
The error (positive or negative) of the second approximation is

$T_2(101) - \sqrt{101} =$

You can earn partial credit on this problem.