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To solve the problem, we need to calculate the average value of the function \( f(x) = 4x + 3 \) on the interval \([0, 2]\).

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

\[
\text{Average Value} = \frac{1}{b - a} \int_a^b f(x) \, dx
\]

In this case, \( a = 0 \), \( b = 2 \), and \( f(x) = 4x + 3 \). Plugging in the values:

\[
\text{Average Value} = \frac{1}{2 - 0} \int_0^2 (4x + 3) \, dx = \frac{1}{2} \int_0^2 (4x + 3) \, dx
\]

Now, let's compute the integral of \( 4x + 3 \):

\[
\int (4x + 3) \, dx = 2x^2 + 3x
\]

Evaluating this from 0 to 2:

\[
\left[ 2x^2 + 3x \right]_0^2 = (2(2)^2 + 3(2)) - (2(0)^2 + 3(0)) = (8 + 6) - 0 = 14
\]

Now, dividing by 2:

\[
\text{Average Value} = \frac{14}{2} = 7
\]

Thus, the average value of the function is \( \boxed{7} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you calculate the average value of a function over a different interval?
2. What does the average value of a linear function represent geometrically?
3. How does the average value formula apply to non-linear functions?
4. Can you find the average value of a function using numerical integration methods?
5. What happens to the average value if the function is shifted vertically?

**Tip**: For linear functions, the average value is always the midpoint between the function's values at the endpoints.

The average value of the function f(x) = 4x + 3 on [0, 2] is

Learn how to calculate the average value of the function f(x) = 4x + 3 on the interval [0, 2]. This solution involves calculus concepts such as integration and applying the average value formula. Suitable for advanced high school students in Grades 11-12.

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