Question

A. Approximate f by a Taylor polynomial with degree n at the number a.

b. Use Taylor's Inequality to estimate the accuracy of the approximation f (x)∼Tₙ(x) when a lies in the given interval.
f (x) = 1/x, a=1, n=2, 0.7<

Answer 1

The task involves approximating the function f(x) = 1/x with a Taylor polynomial of degree 2 at a=1 and to estimate the accuracy within a given interval using Taylor's Inequality.

Step-by-step explanation:

The question involves approximating a function with a Taylor polynomial and estimating the accuracy of this approximation using Taylor's Inequality. Specifically, the task is to approximate the function f(x) = 1/x with a Taylor polynomial of degree n=2 at the point a=1 and to estimate the accuracy within the interval 0.7 < x < 1.3. Taylor's Inequality provides a means to estimate the remainder or error of this approximation.

The Taylor polynomial of degree 2 for f(x) = 1/x at a=1 is derived from the first three terms of the function's Taylor series expansion at a. Given the particular function, we can use common operations to find the polynomial and then apply Taylor's Inequality to estimate the error.

While we can compute values and probabilities more efficiently using calculator functions such as tcdf for Student's t-distributions or Fcdf for F-distributions, this question doesn't involve t-tests or F-tests directly but rather focuses on understanding Taylor polynomials and error estimation using Taylor's Inequality.