Find the average rate of change of f(x)=11
x over the interval [15,16].
Write your answer as an integer, fraction, or decimal rounded to the nearest tenth. Simplify any fractions.

The function given is \( f(x) = \frac{11}{x} \), and we are asked to find the average rate of change over the interval \([15, 16]\).

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

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

For this problem:
- \( a = 15 \)
- \( b = 16 \)

Now, calculate \( f(15) \) and \( f(16) \):

\[
f(15) = \frac{11}{15}, \quad f(16) = \frac{11}{16}
\]

Now, plug these values into the formula:

\[
\text{Average rate of change} = \frac{\frac{11}{16} - \frac{11}{15}}{16 - 15}
\]

This simplifies to:

\[
\text{Average rate of change} = \frac{\frac{11}{16} - \frac{11}{15}}{1}
\]

We need to subtract the two fractions. To do this, find a common denominator for \( \frac{11}{16} \) and \( \frac{11}{15} \). The least common denominator of 16 and 15 is 240. Convert both fractions:

\[
\frac{11}{16} = \frac{11 \times 15}{16 \times 15} = \frac{165}{240}
\]
\[
\frac{11}{15} = \frac{11 \times 16}{15 \times 16} = \frac{176}{240}
\]

Now subtract:

\[
\frac{165}{240} - \frac{176}{240} = \frac{165 - 176}{240} = \frac{-11}{240}
\]

So, the average rate of change is:

\[
\frac{-11}{240}
\]

To express this as a decimal, divide:

\[
\frac{-11}{240} \approx -0.0458
\]

Rounded to the nearest tenth, the average rate of change is approximately:

\[
\boxed{-0.05}
\]

Would you like further details or clarification?

Here are 5 related questions to expand this topic:
1. What is the meaning of the average rate of change for a constant function?
2. How does the average rate of change differ from the instantaneous rate of change?
3. What would the average rate of change be if the interval was \( [1, 10] \)?
4. How does the average rate of change behave as the interval increases for this function?
5. Can we interpret the average rate of change as the slope of a secant line? Why or why not?

**Tip:** For functions involving fractions, always simplify them before subtracting or adding to avoid complex calculations.

Find the average rate of change of f(x)=11/x over the interval [15,16]. Write your answer as an integer, fraction, or decimal rounded to the nearest tenth. Simplify any fractions.

Learn how to find the average rate of change for the function f(x)=11/x over the interval [15,16]. This solution involves calculating the difference in function values over the interval and simplifying the result. The final answer is approximately -0.05, rounded to the nearest tenth.

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