(BC only) Taylor polynomials and error bounds

Learning objectives

Construct the nth-degree Taylor polynomial of a function about a center, and use it to approximate a value of the function near that center

Apply the alternating series error bound, that when the alternating series test hypotheses hold the error in a partial sum is no greater than the magnitude of the first omitted term

Use an error bound to determine how many terms are required to guarantee a stated accuracy, and to state an interval within which the exact value must lie

Apply the Lagrange error bound, that the remainder after the nth-degree Taylor polynomial is at most M|x − a|ⁿ⁺¹/(n + 1)!, where M is an upper bound for the magnitude of the (n + 1)th derivative on the interval between the center and x, and justify the value chosen for M

Explain why a Taylor approximation becomes less accurate as x moves away from the center and more accurate as the degree increases, and compare the errors of polynomials of different degrees