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To estimate \( h'(2) \) using the table, we can approximate it by calculating the difference quotient, which is a way to approximate the derivative using nearby points. Specifically, we'll use the values of \( h(q) \) near \( q = 2 \).

The general form of the difference quotient for estimating \( h'(2) \) is:
\[
h'(2) \approx \frac{h(q_2) - h(q_1)}{q_2 - q_1}
\]
We can use points closest to \( q = 2 \), which are \( q = 1.5 \) and \( q = 2.5 \).

Using the values from the table:
- \( h(1.5) = 201 \)
- \( h(2.5) = 1484 \)

Thus, the difference quotient becomes:
\[
h'(2) \approx \frac{1484 - 201}{2.5 - 1.5} = \frac{1283}{1} = 1283
\]

So, \( h'(2) \approx 1283 \).

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:
1. How would the estimate change if we used a different interval around \( q = 2 \)?
2. What is the significance of using the average rate of change to approximate derivatives?
3. How does the accuracy of the approximation depend on the choice of points?
4. Can this method be extended to higher derivatives, and if so, how?
5. How would you interpret the result \( h'(2) \) in the context of a real-world problem?

**Tip:** Using smaller intervals around the point of interest usually gives a more accurate estimate of the derivative.

Estimate h'(2) using the table of values for h(q) and q.

Learn how to estimate h'(2) using a difference quotient from a table of values for h(q). In this example, we approximate the derivative at q = 2 by calculating the average rate of change between h(1.5) and h(2.5). Suitable for high school calculus students.

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