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To address the problem in the uploaded image, we need to use Taylor's Remainder Theorem to find an upper bound for the approximation made in problem 1. To proceed, it would be helpful to see problem 1. However, I can provide a general explanation of how to use Taylor's Remainder Theorem for finding an upper bound.

### Taylor's Remainder Theorem

Taylor's Remainder Theorem states that for a function \( f \) that is \((n+1)\)-times differentiable on an interval containing \( a \) and \( x \), the remainder \( R_n(x) \) in the Taylor series expansion of \( f \) around \( a \) is given by:

\[ R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x - a)^{n+1} \]

for some \( \xi \) between \( a \) and \( x \).

### Steps to Find the Upper Bound

1. **Identify the Degree of the Taylor Polynomial:** Determine the degree \( n \) of the Taylor polynomial used in the approximation from problem 1.

2. **Determine the \((n+1)\)th Derivative:** Find the \((n+1)\)th derivative of the function \( f \), denoted \( f^{(n+1)}(x) \).

3. **Find the Maximum Value of the \((n+1)\)th Derivative:** Determine the maximum value of \( |f^{(n+1)}(\xi)| \) on the interval between \( a \) and \( x \). This maximum value is denoted by \( M \).

4. **Calculate the Remainder:** Use the formula for the remainder to find the upper bound:

\[ |R_n(x)| \leq \frac{M}{(n+1)!} |x - a|^{n+1} \]

### Example

If problem 1 involved approximating \( f(x) \) using a Taylor polynomial of degree \( n \) around \( a \), and we need to find the upper bound for the approximation error at \( x \), the steps would be:

1. **Identify \( n \):** Suppose the approximation uses a polynomial of degree 2 (i.e., a quadratic approximation), so \( n = 2 \).

2. **Find \( f^{(3)}(x) \):** Compute the third derivative of \( f \), \( f^{(3)}(x) \).

3. **Determine \( M \):** Find the maximum value of \( |f^{(3)}(\xi)| \) on the interval from \( a \) to \( x \).

4. **Compute the Bound:**

\[ |R_2(x)| \leq \frac{M}{3!} |x - a|^3 \]

This provides the upper bound for the error.

If you provide the details of problem 1 or the function and interval involved, I can give a more specific solution.

Would you like to provide more details, or do you have any questions about the process? Here are some follow-up questions that might help:

1. What function \( f(x) \) was approximated in problem 1?
2. What was the degree of the Taylor polynomial used in problem 1?
3. Around what point \( a \) was the Taylor polynomial expanded?
4. What is the interval for \( x \) in the approximation?
5. Do you need help finding the specific derivatives of the function?

**Tip:** When using Taylor's Remainder Theorem, carefully consider the interval over which you are maximizing the \((n+1)\)th derivative to ensure accurate error bounds.

Use Taylor's Remainder Theorem to find an upper bound for the approximation made in problem 1.

Learn how to apply Taylor's Remainder Theorem to find an upper bound for approximation errors. This example guides through the process of using the remainder formula to determine the error bound in a Taylor series expansion. Suitable for university-level calculus students.

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