Use the remainder term to estimate the maximum error in the following approximation on the given interval.
e Superscript x Baseline almost equals 1 plus x plus StartFraction x squared Over 2 EndFraction
​; left bracket negative three sevenths comma three sevenths right bracket
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Part 1
Select the correct choice below and fill in the answer box to complete your choice.
​(Use scientific notation. Use the multiplication symbol in the math palette as needed. Round to two decimal places as​ needed.)
A.
The maximum error is approximately

enter your response here for Mequals2.

To estimate the maximum error in the approximation \( e^x \approx 1 + x + \frac{x^2}{2} \) over the interval \(\left[-\frac{3}{7}, \frac{3}{7}\right]\), we use the remainder term from the Taylor series expansion of \( e^x \).

The Taylor series expansion of \( e^x \) around \( x = 0 \) is:
\[ e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots \]

The remainder term \( R_2(x) \) after the second term is given by:
\[ R_2(x) = \frac{e^c \cdot x^3}{6} \]
where \( c \) is some number between \( 0 \) and \( x \).

To find the maximum error, we need to consider the maximum value of \( |R_2(x)| \) on the interval \(\left[-\frac{3}{7}, \frac{3}{7}\right]\).

Since \( e^c \) is an increasing function and \( |x| \) is maximized at \( \frac{3}{7} \) in the given interval, we get:
\[ |R_2(x)| \leq \frac{e^{\frac{3}{7}} \cdot \left(\frac{3}{7}\right)^3}{6} \]

First, calculate \( e^{\frac{3}{7}} \):
\[ e^{\frac{3}{7}} \approx 1.428 \]

Now calculate the remainder term:
\[ R_2\left(\frac{3}{7}\right) = \frac{1.428 \cdot \left(\frac{3}{7}\right)^3}{6} \]
\[ \left(\frac{3}{7}\right)^3 = \frac{27}{343} \]
\[ R_2\left(\frac{3}{7}\right) = \frac{1.428 \cdot \frac{27}{343}}{6} \]
\[ R_2\left(\frac{3}{7}\right) = \frac{1.428 \cdot 27}{2058} \]
\[ R_2\left(\frac{3}{7}\right) \approx \frac{38.556}{2058} \]
\[ R_2\left(\frac{3}{7}\right) \approx 0.0187 \]

Therefore, the maximum error is approximately \( 0.0187 \).

Since we need to use scientific notation and round to two decimal places:
\[ 0.0187 \approx 1.87 \times 10^{-2} \]

So, the maximum error is approximately \( 1.87 \times 10^{-2} \).

**Would you like more details or have any questions?**

Here are five questions you might consider asking next:

1. Can you explain the Taylor series expansion in more detail?
2. How do you determine the remainder term in a Taylor series?
3. What is the significance of the interval in the error estimation?
4. How does the value of \( c \) affect the remainder term?
5. Can you show a step-by-step calculation for another interval?

**Tip:** When approximating functions using Taylor series, always consider higher-order terms to understand the accuracy of your approximation better.

Learn how to estimate the maximum error in the approximation of e^x using the Taylor series expansion. This problem explores the use of Taylor series and error estimation techniques, specifically calculating the remainder term. Suitable for undergraduate students studying calculus and mathematical analysis.

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