Suppose that you are estimating √7 using the second Taylor polynomial for √x at x=6 . Use Taylor's Theorem for Remainders to bound the error.Round your answer to six decimal places.Provide your answer below:

The Taylor series for √x centered at x = a is given by:

f(x) = √a + (1/2√a)(x - a) - (1/4a√a)(x - a)²/2! + ...

The second Taylor polynomial P2(x) for √x at x = a is:

P2(x) = √a + (1/2√a)(x - a) - (1/4a√a)(x - a)²/2!

We are given a = 6 and x = 7. Substituting these values into P2(x), we get:

P2(7) = √6 + (1/2√6)(7 - 6) - (1/4*6√6)(7 - 6)²/2!

Now, we need to find the error bound. According to Taylor's Theorem for Remainders, the error R2(x) is given by:

R2(x) = |f(x) - P2(x)| ≤ M/((n+1)!) * |x - a|^(n+1)

where M is the maximum value of the (n+1)th derivative of f on the interval between a and x. For √x, the (n+1)th derivative is a complicated expression, but it's clear that the derivatives are decreasing for x > 0. Therefore, we can take M to be the value of the third derivative at a = 6.

The third derivative of √x at x = a is -3/8a^2√a. Substituting a = 6, we get M = -3/8*6^2√6.

Substituting n = 2, a = 6, x = 7, and M = -3/8*6^2√6 into the error bound formula, we get:

R2(7) ≤ |-3/8*6^2√6|/((2+1)!) * |7 - 6|^(2+1)

Calculate the above expression to get the error bound. Round your answer to six decimal places.