Problem 65 - Essential Calculus

Question

Find the average value of the function over the given interval. $$ f(x)=\frac{\ln x}{x}, \quad[1, e] $$

Step-by-Step Solution

Answer

The average value of \(f(x)=\frac{\ln x}{x}\) over the interval [1, e] is \(\frac{0.5}{e - 1}\).

Step 1: Get the formula

The formula for finding the average value of a function, \(f(x)\), from \(a\) to \(b\) is: \(\frac{1}{b - a} \int_{a}^{b} f(x) dx\).

Step 2: Substitute the limits into the formula

Now, the function \(f(x)=\frac{\ln x}{x}\) is given over the interval [1, e]. Therefore, the average value can be calculated as \(\frac{1}{e - 1} \int_{1}^{e} \frac{\ln x}{x} dx\).

Step 3: Evaluate the integral

The integral \(\int_{1}^{e} \frac{\ln x}{x} dx\) can be evaluated using the substitution method. Let \(u = \ln x\), then \(du = \frac{1}{x} dx\). So the integral becomes \(\int_{0}^{1} u du\), which equals to \(0.5*1^{2}-0.5*0^{2} = 0.5\).

Step 4: Compute the average value

Substitute the value of integral back into the formula of average value. We get \(\frac{1}{e - 1}*0.5 = \frac{0.5}{e - 1}\). Therefore, the average value of the function over the interval [1, e] is \(\frac{0.5}{e - 1}\).