The table gives the values of a continuous function 𝑓.
𝑥:20,25,30,35,40,45,50     𝑓(𝑥):42,38,31,29,35,48,60
Use the Midpoint Rule to estimate the average value of 𝑓
 on [20, 50].

To estimate the average value of the function \( f(x) \) on the interval \([20, 50]\) using the Midpoint Rule, follow these steps:

### Step 1: Identify the subintervals
Given the values of \( x \) in the table, the subintervals are:
\[ [20, 25], [25, 30], [30, 35], [35, 40], [40, 45], [45, 50] \]

### Step 2: Determine the midpoints of each subinterval
The midpoints (\( m_i \)) of the subintervals are:
\[ m_1 = \frac{20 + 25}{2} = 22.5 \]
\[ m_2 = \frac{25 + 30}{2} = 27.5 \]
\[ m_3 = \frac{30 + 35}{2} = 32.5 \]
\[ m_4 = \frac{35 + 40}{2} = 37.5 \]
\[ m_5 = \frac{40 + 45}{2} = 42.5 \]
\[ m_6 = \frac{45 + 50}{2} = 47.5 \]

### Step 3: Approximate the function values at the midpoints
Since the function values at these midpoints aren't given directly, approximate \( f(m_i) \) using the values closest to the midpoints:
- \( f(22.5) \approx f(25) = 38 \)
- \( f(27.5) \approx f(25) = 38 \)
- \( f(32.5) \approx f(30) = 31 \)
- \( f(37.5) \approx f(35) = 29 \)
- \( f(42.5) \approx f(40) = 35 \)
- \( f(47.5) \approx f(45) = 48 \)

### Step 4: Apply the Midpoint Rule
The Midpoint Rule for estimating the average value of the function over \([20, 50]\) is given by:
\[
\text{Average Value} \approx \frac{1}{b-a} \sum_{i=1}^{n} f(m_i) \cdot \Delta x
\]
where \(\Delta x\) is the width of each subinterval and \([a, b] = [20, 50]\).

Here, \(\Delta x = 5\) (since each subinterval is \(5\) units wide) and \(b - a = 30\).

\[
\text{Average Value} \approx \frac{1}{30} \cdot 5 \cdot \left[ f(22.5) + f(27.5) + f(32.5) + f(37.5) + f(42.5) + f(47.5) \right]
\]

Substituting the approximate values:
\[
\text{Average Value} \approx \frac{5}{30} \cdot \left[ 38 + 38 + 31 + 29 + 35 + 48 \right]
\]

\[
\text{Average Value} \approx \frac{1}{6} \cdot 219 = 36.5
\]

### Final Answer:
The estimated average value of \( f(x) \) on \([20, 50]\) using the Midpoint Rule is \( \boxed{36.5} \).

Would you like a detailed explanation on any part of the process?

### Related Questions:
1. How does the Trapezoidal Rule compare to the Midpoint Rule in estimating averages?
2. What would the result be if the Simpson's Rule was applied instead?
3. How do you estimate the error in the Midpoint Rule?
4. Can the Midpoint Rule be applied to non-equally spaced intervals?
5. What are some practical applications of the Midpoint Rule in real-world scenarios?

**Tip:** The Midpoint Rule tends to give more accurate approximations when the function behaves linearly within each subinterval.

Learn how to estimate the average value of a continuous function f using the Midpoint Rule on the interval [20, 50]. Follow a detailed step-by-step solution with calculated midpoints, function approximations, and the application of the Midpoint Rule formula.

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