Find the average value of the function f(x) = |x| - 1 over [-1, 3].
Understand the Problem
The question is asking to find the average value of the function f(x) = |x| - 1 over the interval [-1, 3]. This involves applying the formula for the average value of a function over a defined interval. The approach will involve integrating the function over the interval and then dividing by the width of the interval.
Answer
The average value is $\frac{1}{2}$.
Answer for screen readers
The average value of the function $f(x) = |x| - 1$ over the interval $[-1, 3]$ is $\frac{1}{2}$.
Steps to Solve
- Determine the average value formula
The average value of a function $f(x)$ over the interval $[a, b]$ is given by the formula:
$$ \text{Average value} = \frac{1}{b-a} \int_a^b f(x) , dx $$
For this problem, we have $a = -1$ and $b = 3$.
- Calculate the width of the interval
First, we find the width of the interval:
$$ b - a = 3 - (-1) = 4 $$
- Set up the integral for the function
Next, we need to integrate the function $f(x) = |x| - 1$. The function has two cases in the interval $[-1, 3]$:
- For $x \in [-1, 0]$: $f(x) = -x - 1$
- For $x \in [0, 3]$: $f(x) = x - 1$
Now, break the integral into the two parts:
$$ \int_{-1}^3 f(x) , dx = \int_{-1}^0 (-x - 1) , dx + \int_{0}^3 (x - 1) , dx $$
- Calculate the first integral
Calculate the first part:
$$ \int_{-1}^0 (-x - 1) , dx = \left[ -\frac{x^2}{2} - x \right]_{-1}^0 $$
Evaluating this, we get:
$$ \left[ 0 - 0 \right] - \left[ -\frac{1}{2} + 1 \right] = 0 + \frac{1}{2} = \frac{1}{2} $$
- Calculate the second integral
Calculate the second part:
$$ \int_{0}^3 (x - 1) , dx = \left[ \frac{x^2}{2} - x \right]_{0}^{3} $$
Evaluating this, we get:
$$ \left[ \frac{9}{2} - 3 \right] - \left[ 0 - 0 \right] = \frac{9}{2} - 3 = \frac{3}{2} $$
- Combine the integrals
Now combine the results of both integrals:
$$ \int_{-1}^3 f(x) , dx = \frac{1}{2} + \frac{3}{2} = 2 $$
- Calculate the average value
Finally, plug this value into the average value formula:
$$ \text{Average value} = \frac{1}{4} \cdot 2 = \frac{1}{2} $$
The average value of the function $f(x) = |x| - 1$ over the interval $[-1, 3]$ is $\frac{1}{2}$.
More Information
The average value of a function provides insight into the overall behavior of the function over a specific interval. It essentially gives a single number that represents the "typical" value of the function within that interval.
Tips
- Forgetting to consider the piecewise nature of the absolute value function while integrating.
- Not evaluating the definite integrals correctly.
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