Final answer:
The question covers the third-degree Taylor polynomial for f(x) = x cos(x) at b = 1, finding an upper bound for the error, and the minimal value of the error on a given interval. Calculations are required for a complete answer.
Step-by-step explanation:
You have asked for help with a calculus problem involving Taylor polynomials and estimation of error. Specifically, you're asking about the third-degree Taylor polynomial T3(x) for the function f(x) = x cos(x) at b = 1, an upper bound for the error on a specific interval, and the smallest value of the error on that interval.
For part (a), we would need to calculate the derivatives of f(x) at x=1 and substitute these into the formula for a third-degree Taylor polynomial. However, without further calculation, I can't provide the polynomial here.
For part (b), we would use Taylor's Remainder Theorem to find an upper bound for the error. We would need to find the fourth derivative of f(x), evaluate it on the interval (T - 0.1, T + 0.1), and select the maximum value to estimate the error. Again, I cannot provide a numerical value without additional work.
For part (c), to find the smallest value of the error, we would evaluate |T3(x) - f(x)| on the interval [T - 0.1, T + 0.1], either by looking at critical points of the error function or evaluating it at the ends of the interval.