Given f(x) = 1/x+11, find the average rate of change of f (x) on the interval [3,3 + h] your answer will be a expression involving h

To find the average rate of change of the function \( f(x) = \frac{1}{x} + 11 \) on the interval \([3, 3 + h]\), we use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(3 + h) - f(3)}{(3 + h) - 3}
\]

### Step 1: Find \( f(3 + h) \)
Substitute \( 3 + h \) into the function:

\[
f(3 + h) = \frac{1}{3 + h} + 11
\]

### Step 2: Find \( f(3) \)
Substitute \( 3 \) into the function:

\[
f(3) = \frac{1}{3} + 11
\]

### Step 3: Calculate the difference in function values
Now subtract \( f(3) \) from \( f(3 + h) \):

\[
f(3 + h) - f(3) = \left( \frac{1}{3 + h} + 11 \right) - \left( \frac{1}{3} + 11 \right)
\]
The 11's cancel out, so you're left with:

\[
f(3 + h) - f(3) = \frac{1}{3 + h} - \frac{1}{3}
\]

### Step 4: Simplify the difference
To simplify \( \frac{1}{3 + h} - \frac{1}{3} \), we combine the fractions:

\[
\frac{1}{3 + h} - \frac{1}{3} = \frac{3 - (3 + h)}{3(3 + h)} = \frac{-h}{3(3 + h)}
\]

### Step 5: Divide by \( h \)
Now, divide the result by \( h \) to find the average rate of change:

\[
\text{Average Rate of Change} = \frac{\frac{-h}{3(3 + h)}}{h} = \frac{-1}{3(3 + h)}
\]

### Final Answer:
The average rate of change of \( f(x) \) on the interval \([3, 3 + h]\) is:

\[
\boxed{\frac{-1}{3(3 + h)}}
\]

Would you like further details or clarifications?

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**Related Questions:**
1. How do you find the instantaneous rate of change of \( f(x) \) at \( x = 3 \)?
2. What is the derivative of \( f(x) = \frac{1}{x} + 11 \)?
3. Can you generalize the average rate of change for any interval \( [a, a+h] \)?
4. What happens to the average rate of change as \( h \to 0 \)?
5. How would the result change if the function was \( f(x) = \frac{2}{x} + 11 \)?

**Tip:** The average rate of change is closely related to the concept of the derivative, which gives the instantaneous rate of change at a specific point.

Given f(x) = 1/x + 11, find the average rate of change of f(x) on the interval [3, 3 + h].

Learn how to find the average rate of change of the function f(x) = 1/x + 11 over the interval [3, 3 + h]. This problem includes key algebraic concepts and detailed step-by-step solutions, suitable for high school students in Grades 10-12.

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