Problem 34 Explain how the formula $$ e... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 7: Problem 34

Explain how the formula $$ e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots $$ leads to the approximation $e^{x} \approx 1+x$ if $|x|$ is small (which we derived by another method in Section 3.6).

Short Answer

Expert verified

For small $|x|$, higher power terms in the given exponential function series, $e^{x} = 1 + \frac{x}{1 !} + \frac{x^{2}}{2 !} + \frac{x^{3}}{3 !} + \cdots$, contribute less to the sum as they become relatively insignificant. Therefore, we can truncate the series after the first two terms for a simpler approximation, resulting in $e^x \approx 1 + x$, which shows how the given formula leads to the desired approximation for small $|x|$ values.

Step by step solution

Identify the given formula (power series for the exponential function)

The given formula is the power series representation of the exponential function: \[ e^{x} = 1 + \frac{x}{1 !} + \frac{x^{2}}{2 !} + \frac{x^{3}}{3 !} + \cdots \]

Recognize what "small" $|x|$ means

The term "small" $|x|$ usually means $|x|$ is much smaller than 1. In this context, it means a number that's closer to zero rather than a large number. As $x$ approaches 0, the higher power terms of the series become less significant since the result of raising a small number to a large power is an even smaller number. Therefore, we can make a reasonable approximation by taking only a few terms of the series.

Truncate the series for the small $|x|$ approximation

For small $|x|$, we know that higher power terms in the series will be smaller and contribute less to the sum. We can truncate the series after the first two terms to get a simpler approximation for small $|x|$ values: \[ e^{x} \approx 1 + \frac{x}{1!} = 1 + x \]

Compare the approximation to the desired expression

We have found the following approximation for the exponential function with small $|x|$ values: \[ e^{x} \approx 1 + x \] This is the desired expression as it shows how the original formula leads to the approximation $e^x \approx 1+x$ for small $|x|$ values.

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91影视

Explain how the formula $$ e^{x}=1+\frac{x}{1 !}+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots $$ leads to the approximation \(e^{x} \approx 1+x\) if \(|x|\) is small (which we derived by another method in Section 3.6).

Short Answer

Step by step solution

Identify the given formula (power series for the exponential function)

Recognize what "small" \(|x|\) means

Truncate the series for the small \(|x|\) approximation

Compare the approximation to the desired expression

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