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The task involves using a difference quotient to estimate the derivative of a function at \(x = 1\). The difference quotient is given by:

\[
f'(x) \approx \frac{f(x+h) - f(x)}{h}
\]

where \(h\) is a small number.

### Part (a)
To determine which pair of \(x\)-values provides the best estimate of the derivative at \(x = 1\), you would ideally choose \(h\) values that are small and symmetric around 1. From the options provided:

- \(1\) and \(1.999\) (which gives \(h = 0.999\))
- \(1\) and \(1.001\) (which gives \(h = 0.001\))
- \(0\) and \(2\) (which gives \(h = 1\) but is symmetric around 1)
- \(1\) and \(2\) (which gives \(h = 1\))

The best estimate will likely come from using \(h\) values that are smaller and more symmetric around \(1\). Therefore, the pair \(1\) and \(1.001\) would provide the most precise estimate since \(h = 0.001\) is very small, minimizing the approximation error.

### Part (b)
For the function \( f(x) = 5^x \), we are to estimate \( f'(1) \) using the difference quotient with \(h\) being very small, as chosen above (\(h = 0.001\)). The derivative estimation at \(x = 1\) is:

\[
f'(1) \approx \frac{f(1.001) - f(1)}{0.001}
\]

Calculating:

\[
f'(1) \approx \frac{5^{1.001} - 5^1}{0.001}
\]

Let's compute this value.It seems like I can’t do more advanced data analysis right now. Please try again later. If you have access to a calculator, you can perform the calculation yourself using the formula:

\[
f'(1) \approx \frac{5^{1.001} - 5^1}{0.001}
\]

Feel free to reach out if you need help with the calculation or have any other questions.

Here are some follow-up points and a tip you might consider:

1. What does the difference quotient tell us about the behavior of the function near a point?
2. How does changing the value of \( h \) affect the accuracy of the derivative estimate?
3. What would happen if we used a negative \( h \) for the difference quotient?
4. Can this method be applied to estimate derivatives at points other than \( x = 1 \)?
5. What are some limitations of using the difference quotient for derivative estimation?

**Tip:** When using the difference quotient, ensure that \( h \) is small enough to provide a good approximation but not so small that numerical precision issues occur in the calculation.

When using a difference quotient to estimate the derivative of a function at x = 1, some choices of the x-values to input are expected to give better estimates than others. Which of the following pairs of x-values should you use for the difference? (a) 1 and 1.999, 1 and 1.001, 0 and 2, or 1 and 2 (b) Using your answer to part (a), for the function f(x) = 5^x, estimate f'(1). Round your answer to one decimal place.

Learn how to estimate the derivative of the function f(x) = 5^x at x = 1 using the difference quotient method. This problem involves choosing the optimal x-values for the best estimation and calculating the derivative f'(1). Suitable for high school and early college students.

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