The average value of a function $$ f $$ over the interval $$ [-1,2] $$ is -4 , and the average value of $$ f $$ over the interval $$ [2,7] $$ is 8 . What is the average value of $$ f $$ over the interval $$ [-1,7] $$ ? (A) 2 (C) $$ \frac{1}{2} $$ (D) 14

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UpStudy AI · Accepted Answer

To determine the average value of the function $$ f $$ over the interval $$ [-1,7] $$, we'll use the information provided about the average values over the subintervals $$ [-1,2] $$ and $$ [2,7] $$.

1. **Calculate the Integral Over Each Subinterval:**

- **For $$ [-1,2] $$:**
     - **Average Value ($$ A_1 $$)**: $$-4$$
     - **Length of Interval**: $$2 - (-1) = 3$$
     - **Integral**: $$ \int_{-1}^{2} f(x) \, dx = A_1 \times \text{Length} = -4 \times 3 = -12 $$

- **For $$ [2,7] $$:**
     - **Average Value ($$ A_2 $$)**: $$8$$
     - **Length of Interval**: $$7 - 2 = 5$$
     - **Integral**: $$ \int_{2}^{7} f(x) \, dx = A_2 \times \text{Length} = 8 \times 5 = 40 $$

2. **Compute the Total Integral Over $$ [-1,7] $$:**
   
   $$
   \int_{-1}^{7} f(x) \, dx = \int_{-1}^{2} f(x) \, dx + \int_{2}^{7} f(x) \, dx = -12 + 40 = 28
   $$

3. **Determine the Average Value Over $$ [-1,7] $$:**
   
   - **Total Length of Interval**: $$7 - (-1) = 8$$
   - **Average Value ($$ A $$)**:
     $$
     A = \frac{\text{Total Integral}}{\text{Total Length}} = \frac{28}{8} = 3.5
     $$
     $$
     3.5 = \frac{7}{2}
     $$

**Conclusion:**

The average value of $$ f $$ over the interval $$ [-1,7] $$ is $$ \frac{7}{2} $$ or **3.5**.

**Answer:** $$ \frac{7}{2} $$
The average value of $$ f $$ over the interval $$ [-1,7] $$ is $$ \frac{7}{2} $$ or 3.5. **Answer:** $$ \frac{7}{2} $$

The average value of a function over the interval is -4 , and the average value of over the
interval is 8 . What is the average value of over the interval ?
(A) 2
©
(D) 14

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The average value of a function over the interval is -4 , and the average value of over the interval is 8 . What is the average value of over the interval ? (A) 2 © (D) 14