Final answer:
The maximum possible error when approximating e^x with its 4th Taylor polynomial on the interval [-0.5,0.5] can be found using the formula R_4(x) <= e^0.5 * |x-0|^5 / 5!, which yields an error less than 0.01374.
Step-by-step explanation:
When approximating e^x with its 4th Taylor polynomial centered at a=0, we need to estimate the remainder or error, that results from truncating the series. The remainder Rn(x) for the Taylor series of a function f centered at a is given by the Lagrange form of the remainder: Rn(x) = f(n+1)(z) * (x-a)n+1 / (n+1)! where z is some number between a and x. Since we are interested in the interval [-0.5, 0.5] and e^x is an increasing function, the maximum value of e^x on this interval occurs at x=0.5. For the 4th Taylor polynomial, n=4, which means we need the 5th derivative of e^x. However, the derivatives of e^x are all e^x, hence the 5th derivative evaluated at any point is just e raised to that point.
The maximum error occurs when z is chosen to maximize e^z, which on the interval [-0.5,0.5], will be e^0.5. Thus, the error bound is calculated using: R4(x) <= e^0.5 * |x-0|5 / 5! = e^0.5 / 120. Note that the inequality is because we are considering an upper bound. To find the max error within the desired interval, we evaluate this expression at x=0.5: Max error <= e^0.5 / 120 The value e^0.5 (with x being 0.5) is approximately 1.64872, and the upper bound for the error in using the 4th Taylor polynomial to approximate e^x on the interval [-0.5, 0.5] is thus: Max error <= 1.64872 / 120 < 0.01374. This value provides a bound on the maximum possible error when using the 4th Taylor polynomial to approximate e^x on the given interval.