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 f(x)f(x) over the interval [a,b][a, b], the formula is:

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

In this case, we are given f(x)=84xf(x) = 8 - 4x and the interval is [0,2][0, 2].

Step 1: Set up the integral

Average Value=12002(84x)dx\text{Average Value} = \frac{1}{2 - 0} \int_0^2 (8 - 4x) \, dx

Step 2: Calculate the integral

First, we need to compute the integral of f(x)=84xf(x) = 8 - 4x:

(84x)dx=8x2x2\int (8 - 4x) \, dx = 8x - 2x^2

Step 3: Evaluate the integral over the limits

Now, we substitute the upper and lower limits (2 and 0):

02(84x)dx=[8x2x2]02\int_0^2 (8 - 4x) \, dx = \left[ 8x - 2x^2 \right]_0^2

Evaluating this:

(8(2)2(2)2)(8(0)2(0)2)=(168)(0)=8\left( 8(2) - 2(2)^2 \right) - \left( 8(0) - 2(0)^2 \right) = (16 - 8) - (0) = 8

Step 4: Calculate the average value

Now, divide by the length of the interval (which is 20=22 - 0 = 2):

Average Value=12×8=4\text{Average Value} = \frac{1}{2} \times 8 = 4

Final Answer:

The average value of the function f(x)=84xf(x) = 8 - 4x over the interval [0,2][0, 2] is 4.


Would you like more details about the steps or have any other questions? Here are some related questions for further exploration:

  1. How would the average value change if the interval were [1,3][1, 3]?
  2. What if the function was f(x)=5x22x+3f(x) = 5x^2 - 2x + 3? How would we calculate its average value over an interval?
  3. How can you interpret the average value of a function geometrically?
  4. What is the importance of the average value of a function in real-world applications like physics or economics?
  5. 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