Questions: Find the average value of the function f(x) = x / 2 over the interval [0,3]. Enter the exact answer. Average value of f(x) = Find c such that f(c) equals the average value of the function over [0,3]. Enter the exact answer. r = Number

Solution Steps

To find the average value of the function \( f(x) = \frac{x}{2} \) over the interval \([0, 3]\), we use the formula for the average value of a function over an interval \([a, b]\), which is given by:

\[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]

1. Calculate the definite integral of \( f(x) = \frac{x}{2} \) from 0 to 3.
2. Divide the result by the length of the interval, which is \( 3 - 0 = 3 \).

To find \( c \) such that \( f(c) \) equals the average value, solve the equation \( f(c) = \text{Average value} \).

We need to find the definite integral of the function \( f(x) = \frac{x}{2} \) over the interval \([0, 3]\):

\[ \int_{0}^{3} \frac{x}{2} \, dx = \left[ \frac{x^2}{4} \right]_{0}^{3} = \frac{3^2}{4} - \frac{0^2}{4} = \frac{9}{4} \]

The average value of the function over the interval \([0, 3]\) is given by:

\[ \text{Average value} = \frac{1}{3 - 0} \int_{0}^{3} f(x) \, dx = \frac{1}{3} \cdot \frac{9}{4} = \frac{9}{12} = \frac{3}{4} \]

We need to find \( c \) such that:

\[ f(c) = \frac{3}{4} \]

Substituting \( f(c) = \frac{c}{2} \):

\[ \frac{c}{2} = \frac{3}{4} \]

Multiplying both sides by 2:

\[ c = \frac{3}{2} \]

Final Answer