/*! 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 2 Estimate the indicated value wit... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 2

Estimate the indicated value without using a calculator. \(\ln 1.0007\)

Short Answer

Expert verified
The estimated value of \(\ln(1.0007)\) without using a calculator is approximately \(0.000699865\), using the first 3 terms of the Taylor series.

Step by step solution

01

Recalling properties of logarithm

Recall that the natural logarithm of a number can be expanded using the Taylor series: \[\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\] Since the value of \(x\) in our case is quite small (0.0007), we can consider only a few terms of the Taylor series to estimate the value of the logarithm.
02

Calculate the required terms

Let's consider the first 3 terms of the Taylor series to estimate the value of the logarithm: \[\ln (1 + 0.0007) \approx 0.0007 - \frac{0.0007^2}{2} + \frac{0.0007^3}{3}\]
03

Calculate the result

Perform the calculations, and we get: \[\ln (1 + 0.0007) \approx 0.0007 - \frac{0.00000049}{2} + \frac{0.000000000343}{3}\] \[\ln(1.0007) \approx 0.0007 - 0.000000245 + 0.000000000114\] By adding these three values, we get: \[\ln(1.0007) \approx 0.000699865\] So, without using a calculator, the approximate value of \(\ln(1.0007)\) is \(0.000699865\).

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