https://file.aiquickdraw.com/mathbot/file/1726788967452e5fvta8u.jpg

To find the average value of the function \( f(x) = 5x + \frac{1}{4x + 3} \) on the interval \([6, 10]\), we use the formula for the average value of a function on a closed interval \([a, b]\):

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

Here, \( a = 6 \), \( b = 10 \), and \( f(x) = 5x + \frac{1}{4x + 3} \).

So, the average value will be:

\[
\text{Average value} = \frac{1}{10 - 6} \int_6^{10} \left( 5x + \frac{1}{4x + 3} \right) dx
\]

Now, let's compute the integral.

1. The integral of \( 5x \) is straightforward:
   \[
   \int 5x \, dx = \frac{5x^2}{2}
   \]

2. The integral of \( \frac{1}{4x + 3} \) is computed using substitution. Let \( u = 4x + 3 \), then \( du = 4 dx \), or \( dx = \frac{du}{4} \). This gives:
   \[
   \int \frac{1}{4x + 3} dx = \frac{1}{4} \ln|4x + 3|
   \]

Thus, the integral becomes:

\[
\int_6^{10} \left( 5x + \frac{1}{4x + 3} \right) dx = \left[ \frac{5x^2}{2} + \frac{1}{4} \ln|4x + 3| \right]_6^{10}
\]

Let me calculate this expression.It seems there was an issue with the calculation. Let me break the steps down and manually handle the evaluation for the integral:

We have the integral to compute:

\[
\int_6^{10} \left( 5x + \frac{1}{4x + 3} \right) dx = \left[ \frac{5x^2}{2} + \frac{1}{4} \ln|4x + 3| \right]_6^{10}
\]

First, compute each part:

1. For \( \frac{5x^2}{2} \):
   \[
   \frac{5(10)^2}{2} - \frac{5(6)^2}{2} = \frac{5 \times 100}{2} - \frac{5 \times 36}{2} = 250 - 90 = 160
   \]

2. For \( \frac{1}{4} \ln|4x + 3| \):
   \[
   \frac{1}{4} \ln|4(10) + 3| - \frac{1}{4} \ln|4(6) + 3| = \frac{1}{4} \ln(43) - \frac{1}{4} \ln(27)
   \]
   Using logarithmic properties:
   \[
   \frac{1}{4} \ln \left( \frac{43}{27} \right)
   \]

Now, let's add these together and then divide by \( 10 - 6 = 4 \) to get the average value. Let me calculate this manually.The average value of the function \( f(x) = 5x + \frac{1}{4x + 3} \) on the interval \([6, 10]\) is approximately **40.03**.

Would you like a further breakdown of the steps or have any other questions? 

Here are some related questions you might explore:

1. How does changing the interval affect the average value of the function?
2. How can we compute the average value if \( f(x) \) involves more complex functions like trigonometric or exponential terms?
3. What is the geometric interpretation of the average value of a function?
4. Can the average value of a function be negative, and under what conditions?
5. How does substitution help in simplifying certain integrals?

**Tip:** When solving definite integrals that involve logarithms, always simplify using logarithmic properties, like \( \ln \frac{a}{b} = \ln(a) - \ln(b) \).

Find the average value of f(x) = 5x + 1/(4x + 3) on [6, 10].

Learn how to find the average value of the function f(x) = 5x + 1/(4x + 3) over the interval [6, 10]. This problem involves definite integrals, substitution, and logarithmic properties. Suitable for undergraduate calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation