Find the average rate of change for 
f(x) = 1/x from x = 6 to x = 8.
 (Round your answer to two decimal places.)

The average rate of change of a function \( f(x) \) between two points \( x = a \) and \( x = b \) is given by the formula:

\[
\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}
\]

In this case, \( f(x) = \frac{1}{x} \), \( a = 6 \), and \( b = 8 \). Let's calculate:

1. \( f(6) = \frac{1}{6} \)
2. \( f(8) = \frac{1}{8} \)

Now, use the formula:

\[
\text{Average rate of change} = \frac{f(8) - f(6)}{8 - 6} = \frac{\frac{1}{8} - \frac{1}{6}}{2}
\]

We need to subtract the fractions and simplify:

\[
\frac{1}{8} - \frac{1}{6} = \frac{6 - 8}{48} = \frac{-2}{48} = \frac{-1}{24}
\]

So, the average rate of change is:

\[
\frac{\frac{-1}{24}}{2} = \frac{-1}{48}
\]

Finally, rounding to two decimal places:

\[
\text{Average rate of change} \approx -0.02
\]

The average rate of change is approximately \(-0.02\).

Would you like further details on the process or have any questions?

Here are five related questions:
1. What is the average rate of change formula for a quadratic function?
2. How does the average rate of change relate to the derivative?
3. What is the geometric interpretation of the average rate of change?
4. Can you find the instantaneous rate of change of \( f(x) = 1/x \) at \( x = 6 \)?
5. How would the average rate of change change if we used a larger interval?

**Tip:** The average rate of change gives the slope of the secant line between two points on a curve, helping to understand the function's behavior over an interval.

Find the average rate of change for f(x) = 1/x from x = 6 to x = 8. (Round your answer to two decimal places.)

Calculate the average rate of change for the function f(x) = 1/x from x = 6 to x = 8. This problem involves understanding the concept of average rate of change and using the appropriate formula, suitable for high school calculus students.

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