Average Value of a function: The average value of a function 
f(x) on the interval [a, b]
 is given by
Avg(f) = 
1
b − a
 
b
f(x) dx

a
.


Find the average value of 
f(x) = 

x2 − 1
x
 , 1 ≤ x ≤ 4.



Set up an integral that represents the average value of the function on the given interval.


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

\[
\text{Avg}(f) = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx
\]

For the function \( f(x) = \frac{x^2 - 1}{x} \), on the interval \( [1, 4] \), we can first simplify the function:

\[
f(x) = \frac{x^2 - 1}{x} = x - \frac{1}{x}
\]

Now, applying the formula for the average value, the integral setup becomes:

\[
\text{Avg}(f) = \frac{1}{4 - 1} \int_{1}^{4} \left( x - \frac{1}{x} \right) dx
\]

Simplifying further:

\[
\text{Avg}(f) = \frac{1}{3} \int_{1}^{4} \left( x - \frac{1}{x} \right) dx
\]

This is the integral that represents the average value of the function on the interval \( [1, 4] \).

Would you like me to solve the integral next?

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### 5 related questions:
1. How do you evaluate the integral \( \int (x - \frac{1}{x}) dx \)?
2. What are the geometric interpretations of the average value of a function?
3. How do changes in the interval \( [a, b] \) affect the average value of a function?
4. Can we apply this average value formula to any continuous function?
5. How does the average value differ for polynomial versus rational functions?

**Tip:** Always simplify the given function if possible before setting up the integral for the average value.

Find the average value of f(x) = (x^2 − 1) / x on the interval [1, 4]. Set up an integral that represents the average value of the function on the given interval.

Learn how to calculate the average value of the function f(x) = (x^2 − 1) / x on the interval [1, 4]. This problem involves setting up a definite integral and applying the formula for the average value of a function. Suitable for high school calculus students.

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