Find the average rate of change for the given function. \( f(x)=x^{2}+4 x \) between \( x=0 \) and \( x=6 \)

To find the average rate of change of the function \( f(x) = x^2 + 4x \) between \( x = 0 \) and \( x = 6 \), we can use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \] where \( a = 0 \) and \( b = 6 \). ### Step 1: Calculate \( f(0) \) \[ f(0) = 0^2 + 4 \cdot 0 = 0 \] ### Step 2: Calculate \( f(6) \) \[ f(6) = 6^2 + 4 \cdot 6 = 36 + 24 = 60 \] ### Step 3: Substitute into the average rate of change formula Now we can substitute \( f(0) \) and \( f(6) \) into the formula: \[ \text{Average Rate of Change} = \frac{f(6) - f(0)}{6 - 0} = \frac{60 - 0}{6 - 0} = \frac{60}{6} = 10 \] Thus, the average rate of change of the function \( f(x) = x^2 + 4x \) between \( x = 0 \) and \( x = 6 \) is \( 10 \).