At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
The average value have of the function h on the given interval is [tex]\frac{1}{2} \ln^2 (2)[/tex].
How to find the Average Value of function ?
The given function is [tex]h (u) = \frac{\ln (u)}{u}[/tex] on the interval (1, 5).
The average value of function (a, b) is found by using the formula:
[tex]\frac{1}{(b -a)} \displaystyle \int^{b}_{a} f(x) dx[/tex]
Here the function is [tex]h (u) = \frac{\ln (u)}{u}[/tex] and a = 1, b = 5
Now,
[tex]\dfrac{1}{(5 - 1)} \displaystyle \int^{5}_{1} \frac{\ln (u)}{u}\ du[/tex]
[tex]= \dfrac{1}{4} \displaystyle \int^{5}_{1} \frac{\ln (u)}{u}\ du[/tex]
Substitute [tex]\ln (u) = t, dt = \frac{1}{u}\ du[/tex]
[tex]\dfrac{1}{4} \displaystyle \int^{4}_{1} \frac{\ln (u)}{u} du = \frac{1}{4} \int^{4}_{1} t (dt)[/tex]
[tex]= \frac{1}{4} \left [\frac{t^2}{2} \right ]^{4}_{1}[/tex]
[tex]= \frac{1}{8} \left [\{\ln (u)\}^2 \right ]^{4}_{1}[/tex]
[tex]= \frac{1}{8} [\{\ln (4) \}^2 - \{\ln (1) \}^2 ][/tex]
[tex]= \frac{1}{8} [\{\ln (4)\}^2 - 0][/tex]
[tex]= \frac{1}{8} \ln^2 (4)[/tex]
[tex]= \frac{2}{4} [2 \ln (2)]^2[/tex]
[tex]= \frac{1}{2} \ln^2 (2)[/tex]
Thus from the above conclusion we can say that The average value have of the function h on the given interval is [tex]\frac{1}{2} \ln^2 (2)[/tex].
Learn more about the Integrational Average value please click here : https://brainly.com/question/27419605
#SPJ4
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
