Math Problem Statement
Solution
To find the average value of the function on the interval , we use the formula for the average value of a function on a closed interval :
Here, , , and .
So, the average value will be:
Now, let's compute the integral.
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The integral of is straightforward:
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The integral of is computed using substitution. Let , then , or . This gives:
Thus, the integral becomes:
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:
First, compute each part:
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For :
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For : Using logarithmic properties:
Now, let's add these together and then divide by to get the average value. Let me calculate this manually.The average value of the function on the interval 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:
- How does changing the interval affect the average value of the function?
- How can we compute the average value if involves more complex functions like trigonometric or exponential terms?
- What is the geometric interpretation of the average value of a function?
- Can the average value of a function be negative, and under what conditions?
- How does substitution help in simplifying certain integrals?
Tip: When solving definite integrals that involve logarithms, always simplify using logarithmic properties, like .
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Math Problem Analysis
Mathematical Concepts
Average Value of Function
Definite Integrals
Substitution Method
Logarithmic Properties
Formulas
Average value of f(x) = (1 / (b - a)) ∫[a, b] f(x) dx
Integral of 5x is (5x^2 / 2)
Integral of 1 / (4x + 3) is (1/4) ln|4x + 3|
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Undergraduate Calculus