Mathematics: Find the average value of the function g(x) = over the interval [1, e].

Find the average value of the function g(x) =

Ответ:

hunterbrod71

21.01.2022

Mathematics

$\displaystyle average=\frac{1}{2(e-1)}$

Step-by-step explanation:

Average Value of a Function

Given a function g(x), we can compute the average value of g in a given interval (a,b) with the equation:

$\displaystyle average=\frac{1}{b-a}\int_{a}^{b} g(x)dx$

We use the given data

$\displaystyle average=\frac{1}{e-1}\int_{1}^{e} \frac{lnx}{x}dx$

We now compute the indefinite integral with a u-substitution

$\displaystyle I=\int \frac{lnx}{x}dx$

We'll use the substitution u=lnx, du=dx/x. Then

$\displaystyle I=\int u.du$

Integrating

$\displaystyle I=\frac{u^2}{2}$

Since u=lnx

$\displaystyle I=\frac{ln^2x}{2}$

The average value is

$\displaystyle average=\frac{1}{e-1}\left|\frac{ln^2x}{2} \right|_1^e$

$\displaystyle average=\frac{1}{e-1}\left(\frac{ln^2e}{2}-\frac{ln^21}{2} \right )$

Since lne=1, and ln1=0

$\displaystyle average=\frac{1}{e-1}\left(\frac{1}{2}-0 \right )$

$\displaystyle average=\frac{1}{2(e-1)}$

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Ответ:

lilpeepxliltracy

05.04.2023

Mathematics

a) 11550

b)35%

Step-by-step explanation:

Hi, to answer this question, first, we have to multiply the stadium capacity by the percentage in decimal form (75/100)

a)15400 x (75/100) = 11,550

For b), we know that 15400 is the 100% of the capacity, so, for 20790 the percentage will be:

15400/100= 20790/x

Solving for x

x= 20790 / 154

x= 135%

135%-100% = 35%

The percentage increase in the number of seats will be 35%

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Find the average value of the function g(x) = over the interval [1, e].

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