/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Estimate the indicated value wit... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 10

Estimate the indicated value without using a calculator. \(e^{-0.00046}\)

Short Answer

Expert verified
\(e^{-0.00046} \approx 0.99954\)

Step by step solution

01

Recall the Taylor series expansion of the exponential function

The Taylor series expansion of the exponential function \(e^x\) around \(x = 0\) is given by: \[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dotsb\]
02

Substitute the value of x

Now, we need to estimate \(e^{-0.00046}\), so we will substitute \(x = -0.00046\) into the Taylor series expansion: \[e^{-0.00046} = 1 - 0.00046 + \frac{(-0.00046)^2}{2!} + \frac{(-0.00046)^3}{3!} + \dotsb\]
03

Truncate the series to get an approximate value

Since the value of \(x = -0.00046\) is very small, the higher order terms in the series will become negligible. Therefore, we can truncate the series after the second or third term to get an approximate value. First, truncate the series after the 2nd term: \[e^{-0.00046} \approx 1 - 0.00046 = 0.99954\] Next, truncate the series after the 3rd term: \[e^{-0.00046} \approx 1 - 0.00046 + \frac{(-0.00046)^2}{2!} = 0.99954 + \frac{0.0000002116}{2} = 0.99954 + 0.0000001058 = 0.9995401058\] As we can see, the approximate value is very close even when we truncate the series after the 2nd term. Thus, the final approximation is: \[e^{-0.00046} \approx 0.99954\]

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