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The problem asks for the average rate of change of \( f(x) = \frac{1}{x} \) over the interval \([3, 10]\).

The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by the formula:

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

In this case, \( f(x) = \frac{1}{x} \), \( a = 3 \), and \( b = 10 \).

First, let's compute the values of \( f(3) \) and \( f(10) \):

\[
f(3) = \frac{1}{3}, \quad f(10) = \frac{1}{10}
\]

Now, applying the formula for the average rate of change:

\[
\text{Average rate of change} = \frac{f(10) - f(3)}{10 - 3} = \frac{\frac{1}{10} - \frac{1}{3}}{7}
\]

Next, we simplify this expression.The average rate of change of \( f(x) = \frac{1}{x} \) over the interval \([3, 10]\) is \( -\frac{1}{30} \).

Would you like more details or have any questions about this?

Here are some related questions:
1. What is the geometric interpretation of the average rate of change?
2. How would the average rate of change differ for other intervals of \( f(x) = \frac{1}{x} \)?
3. What is the instantaneous rate of change of \( f(x) = \frac{1}{x} \) at \( x = 3 \)?
4. How do you compute the average rate of change for a more complex function?
5. What happens if the function is not continuous or differentiable over the interval?

**Tip:** The average rate of change provides a way to measure the overall slope between two points, while the derivative gives the instantaneous rate of change at any specific point.

Compute the average rate of change f(x) = 1/x on the interval [3,10].

This problem demonstrates how to compute the average rate of change of the function f(x) = 1/x over the interval [3, 10]. Learn how to apply the average rate of change formula to functions in calculus. Suitable for students in Grades 9-12.

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