Question

Imagine that you need to compute e^0.4 but you have no calculator or other aid to enable you to compute it exactly, only paper and pencil. You decide to use a third-degree Taylor polynomial expanded around x = 0. Use the fact that e^0.4 < e < 3 and the Error Bound for Taylor Polynomials to find an upper bound for the error in your approximation.
I error l ≤

Answer 1

Answer:

upper bound for the error, | Error |  ≤ 0.0032

Step-by-step explanation:

Given the data in the question;

$e^{0.4$ < e < 3

Using Taylor's Error bound formula

| Error | ≤ ( m / ( N + 1 )! ) $| x-a |^{N+1$

where m = $| f^{N+1 }(x) |$

so we have

| Error |  ≤ ( 3 / ( 3 + 1 )! ) $|$ -0.4 $|$⁴

| Error |  ≤ ( 3 / 4! ) $|$ -0.4 $|$⁴

| Error |  ≤ ( 3 / 24 ) $|$ -0.4 $|$⁴

| Error |  ≤ ( 0.125 ) $|$ -0.0256 $|$

| Error |  ≤ ( 0.125 ) 0.0256

| Error |  ≤ 0.0032

Therefore, upper bound for the error, | Error |  ≤ 0.0032