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To find the average value of a function \( f(x) \) on a given interval, we use the following formula:

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

Where:
- \( a \) and \( b \) are the bounds of the interval,
- \( f(x) \) is the function being evaluated.

Given the function \( f(x) = 4x^2 - 3x + 5 \) and the interval \([-3, 5]\), we can now compute the average value.

### Steps:
1. **Find the integral**: We need to compute the integral of \( f(x) = 4x^2 - 3x + 5 \) from \( x = -3 \) to \( x = 5 \).
2. **Divide by the length of the interval**: After finding the integral, divide the result by \( b - a = 5 - (-3) = 8 \).

Let me calculate this now. sympy import symbols, integrate

# Define the variable and the function
x = symbols('x')
f = 4*x**2 - 3*x + 5

# Define the limits of integration
a, b = -3, 5

# Calculate the integral
integral_value = integrate(f, (x, a, b))

# Find the average value
average_value = integral_value / (b - a)
average_valueThe average value of the function \( f(x) = 4x^2 - 3x + 5 \) on the interval \([-3, 5]\) is \( \frac{82}{3} \).

Would you like more details on this solution, or do you have any other questions? 

Here are some related questions you might find helpful:

1. How do you evaluate the integral of a polynomial function?
2. What does the average value of a function represent in the context of calculus?
3. How do you compute the definite integral of a quadratic function?
4. What is the significance of the interval in finding the average value?
5. Can the average value formula be applied to other types of functions, like trigonometric ones?

Tip: When finding the average value, remember that it's related to the area under the curve, normalized by the width of the interval.

Find the average value of the function f(x) = 4x^2 - 3x + 5 on the interval [-3, 5].

Learn how to find the average value of the function f(x) = 4x^2 - 3x + 5 on the interval [-3, 5] using calculus techniques. Explore detailed steps involving definite integration and the application of the average value formula. Perfect for high school calculus students.

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