Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value. $$f(x)=1 / x \text { on }[1, e]$$

Short answer

Answer: The average value of the function $$f(x) = \frac{1}{x}$$ on the interval $$[1, e]$$ is $$\frac{1}{e-1}$$.

Step-by-step solution

Defining the average value formula

To find the average value of a function $$f(x)$$ over the interval [a, b], we use the following formula: $$\bar{f} = \frac{1}{b-a}\int_{a}^{b} f(x) \text{d}x$$ Here, our function is $$f(x) = \frac{1}{x}$$, and our interval is $$[1, e]$$. Therefore, $$a = 1$$ and $$b = e$$.

Calculating the definite integral

Now, we need to calculate the definite integral of the function on the given interval. This means we will find: $$\int_{1}^{e} \frac{1}{x} \text{d}x$$ To integrate this function, we will apply the power rule for integration and remember that $$\int \frac{1}{x} \text{d}x = \ln |x| + C$$. So, we will find: $$\int_{1}^{e} \frac{1}{x} \text{d}x = [\ln |x|]_{1}^{e} =\ln |e| - \ln |1| = 1-0 = 1$$.

Calculating the average value

Now that we have the definite integral, we will use the formula for average value to find the average value of our function on the given interval: $$\bar{f} = \frac{1}{e-1}\Big(\int_{1}^{e} \frac{1}{x} \text{d}x\Big) = \frac{1}{e-1}(1)=\frac{1}{e-1}$$.

Graphing the function and average value

Finally, let's sketch the graph of the given function and indicate the average value found above. The function $$f(x) = \frac{1}{x}$$ is a hyperbolic function that is positive for positive values of $$x$$. It is decreasing as $$x$$ increases. The average value of the function is a constant, so the graph of the average value will be a horizontal line at $$y=\frac{1}{e-1}$$. To find the points where the function equals the average value, we can set $$f(x)$$ equal to the average value and solve for $$x$$: $$\frac{1}{x} = \frac{1}{e-1}$$ $$x = e-1$$ So, the coordinates of the point where the function equals the average value is $$(e-1, \frac{1}{e-1})$$. You can now sketch the graph of the function $$f(x) = \frac{1}{x}$$ on the interval $$[1, e]$$, and indicate the average value $$y = \frac{1}{e-1}$$.

Key concepts

Definite Integral

The definite integral of a function provides the total accumulation of that function across a specific interval on the x-axis. When you calculate the definite integral from point a to point b, you are essentially finding the net area under the curve of the function between those points.

To calculate the definite integral, we use integral calculus, which involves the fundamental theorem of calculus. This theorem connects the concept of the derivative of a function with the concept of the integral. In practice, when we have a function such as \(f(x) = \frac{1}{x}\), and we want to calculate the definite integral across the interval [1, e], we compute \(\int_{1}^{e} \frac{1}{x} \text{d}x\) which requires us to use the properties of the natural logarithm. The solution to this particular integral is straightforward because the integral of \(1/x\) is the natural logarithm of the absolute value of x.

Integral Calculus

Integral calculus is a branch of calculus focused on the process of integration, which is essentially the inverse operation of differentiation. While derivatives measure rates of change, integrals measure the accumulated quantity.

For example, if we know the rate at which water is flowing into a tank, integration helps us figure out the total volume of water that flows in over a certain period. Integral calculus is used across numerous fields, including physics, engineering, economics, and biology, to solve problems involving areas, volumes, displacement, and more.

In the context of our average value problem, integral calculus allows us to find the total accumulation of the function \(f(x) = \frac{1}{x}\) over the interval [1, e]. This is crucial to determining the function's average value over that interval.

Natural Logarithm

The natural logarithm is a logarithm with the base of Euler's number, e, which is approximately equal to 2.71828. It's denoted as \(\ln(x)\) and is particularly important in calculus due to its unique properties involving derivatives and integrals.

One key property is that the derivative of \(\ln(x)\) is \(1/x\), and the integral of \(1/x\) is \(\ln(|x|) + C\), where C represents the constant of integration. This is directly applicable to our exercise where we find that the definite integral of \(1/x\) from 1 to e is simply the natural logarithm of e, which simplifies to 1, because \(e\) is the base of the natural logarithm.

Integration Techniques

Integration techniques are numerous methods used to evaluate integrals. These can range from straightforward methods such as the power rule to more complex strategies like substitution, integration by parts, and partial fraction decomposition.

In our exercise, we apply one of the simpler techniques to integrate \(f(x) = \frac{1}{x}\), which is a direct application of the definition of the natural logarithm. Understanding when and how to apply different integration techniques is important for solving a wide variety of problems in integral calculus. As students progress, they will encounter functions for which they need to apply more advanced techniques to find the integral.