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.