Questions: The figure below to the left is a graph of f(x), and below to the right is g(x). (a) What is the average value of f(x) on 0 ≤ x ≤ 2 ? avg value = □

Solution Steps

From the graph of \( f(x) \):

• \( f(x) = 1 \) for \( 0 \leq x < 1 \)
• \( f(x) = -1 \) for \( 1 \leq x \leq 2 \)

The average value of a function \( f(x) \) over the interval \([a, b]\) is given by: \[ \text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx \] Here, \( a = 0 \) and \( b = 2 \).

Split the integral into two parts based on the piecewise definition of \( f(x) \): \[ \int_0^2 f(x) \, dx = \int_0^1 1 \, dx + \int_1^2 (-1) \, dx \]

Evaluate each integral: \[ \int_0^1 1 \, dx = [x]_0^1 = 1 - 0 = 1 \] \[ \int_1^2 (-1) \, dx = [-x]_1^2 = -2 - (-1) = -2 + 1 = -1 \]

Sum the results of the integrals: \[ \int_0^2 f(x) \, dx = 1 + (-1) = 0 \]

Calculate the average value: \[ \text{Average value} = \frac{1}{2-0} \int_0^2 f(x) \, dx = \frac{1}{2} \cdot 0 = 0 \]

Final Answer