Math Problem Statement
Find the average value of f(x)=8−4x over the interval [0,2] .
Solution
To find the average value of a function over the interval , the formula is:
In this case, we are given and the interval is .
Step 1: Set up the integral
Step 2: Calculate the integral
First, we need to compute the integral of :
Step 3: Evaluate the integral over the limits
Now, we substitute the upper and lower limits (2 and 0):
Evaluating this:
Step 4: Calculate the average value
Now, divide by the length of the interval (which is ):
Final Answer:
The average value of the function over the interval is 4.
Would you like more details about the steps or have any other questions? Here are some related questions for further exploration:
- How would the average value change if the interval were ?
- What if the function was ? How would we calculate its average value over an interval?
- How can you interpret the average value of a function geometrically?
- What is the importance of the average value of a function in real-world applications like physics or economics?
- How would you find the average value of a function if the function were defined piecewise?
Tip: When working with average values, always remember that you're essentially finding the constant height of a rectangle that has the same area as the region under the curve!
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Average Value of a Function
Formulas
Average Value = (1 / (b - a)) * ∫[a, b] f(x) dx
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12