Question

Suppose that we approximate the exponential function by the first 4 terms in the Taylor series (in other words the Taylor polynomial on the interval . According to the Taylor remainder theorem, what is the maximum possible error in the approximation

Answer 1

R_4(x) = e^c(x-a)^5/5!, where c is some value between x and a.

Explanation:

According to the Taylor remainder theorem, the maximum possible error in approximating the exponential function by its first 4 terms in the Taylor series is given by the expression R_4(x) = f^(5)(c)(x-a)^5/5!, where f(x) = e^x, a is the center of the Taylor series approximation, x is the value at which we want to evaluate the approximation, and c is some value between x and a. Since the 5th derivative of e^x is itself, we have f^(5)(c) = e^c. Therefore, the maximum possible error is given by R_4(x) = e^c(x-a)^5/5!, where c is some value between x and a.