What is the average value of e^x on the interval 3,9]

To find the average value of the function \( e^x \) on the interval \([3, 9]\), we use the formula for the average value of a continuous function \( f(x) \) over the interval \([a, b]\):<br /><br />\[<br />\text{Average value} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx<br />\]<br /><br />In this case, \( f(x) = e^x \), \( a = 3 \), and \( b = 9 \). So, we need to compute:<br /><br />\[<br />\text{Average value} = \frac{1}{9 - 3} \int_{3}^{9} e^x \, dx<br />\]<br /><br />First, compute the integral:<br /><br />\[<br />\int_{3}^{9} e^x \, dx = \left[ e^x \right]_{3}^{9} = e^9 - e^3<br />\]<br /><br />Now, substitute this result into the formula for the average value:<br /><br />\[<br />\text{Average value} = \frac{1}{9 - 3} (e^9 - e^3) = \frac{1}{6} (e^9 - e^3)<br />\]<br /><br />Thus, the average value of \( e^x \) on the interval \([3, 9]\) is:<br /><br />\[<br />\boxed{\frac{1}{6} (e^9 - e^3)}<br />\]