Problem 134 - Thomas Calculus in SI Units

Question

Find the average value of \(f(x)=1 / x\) on [1,2].

Step-by-Step Solution

Answer

The average value is \(\ln 2\).

Step 1: Set Up the Average Value Formula

The average value of a function on the interval \([a, b]\) is given by the formula \(\frac{1}{b-a} \int_a^b f(x) \, dx\). For this problem, \(a = 1\), \(b = 2\), and \(f(x) = \frac{1}{x}\). Thus, the average value is \(\frac{1}{2-1} \int_1^2 \frac{1}{x} \, dx\).

Step 2: Evaluate the Definite Integral

Integrate \(\frac{1}{x}\) from 1 to 2. The integral \(\int \frac{1}{x} \, dx\) is \(\ln|x| + C\). Therefore, \(\int_1^2 \frac{1}{x} \, dx = [\ln|x|]_1^2 = \ln 2 - \ln 1\).

Step 3: Simplify the Natural Logarithm Expression

Recall that \(\ln 1 = 0\), so \(\ln 2 - \ln 1 = \ln 2\).

Step 4: Calculate the Average Value

Since \(\frac{1}{2-1} = 1\), the average value is \(\ln 2\).

Key Concepts

Average Value of a Function

The average value of a function over a given interval provides a single value that represents the function's overall behavior on that interval. It's like finding the function's "center" on that stretch.
To calculate this, you use the formula:

• \( \text{Average value of } f(x) = \frac{1}{b-a} \int_a^b f(x) \, dx \)

This formula is derived from integrating the function over a specific interval and scaling it by dividing by the length of the interval.
For example, when you need to find the average value of the function \( f(x) = \frac{1}{x} \) over the interval \([1, 2]\), you start by setting \(a = 1\) and \(b = 2\). Inserting into the formula gives:
\( \frac{1}{2-1} \int_1^2 \frac{1}{x} \, dx \), which simplifies to \( \int_1^2 \frac{1}{x} \, dx \).
The average value shows us what constant value the function equals if we averaged out all its values between \(x = 1\) and \(x = 2\).

Definite Integral

The definite integral is a fundamental concept in calculus used to calculate areas under curves, among other things. When you see an expression like \( \int_a^b f(x) \, dx \), it indicates the accumulation of the function \(f(x)\) from \(x = a\) to \(x = b\).
For example, integrating the function \(f(x) = \frac{1}{x}\) over the interval \([1, 2]\) means you want to find the total area under the curve of \( \frac{1}{x} \) from \(x = 1\) to \(x = 2\).

• The integral of \( \frac{1}{x} \) is \( \ln|x| + C \) (where C is the constant of integration, ignored in definite integrals since it cancels out).
• For this function, evaluate it from 1 to 2: \( \ln 2 - \ln 1 \), simplifying to \( \ln 2 \) because \( \ln 1 = 0 \).

This process transforms a function into a single numerical answer representing accumulated quantity, like area.

Natural Logarithm

The natural logarithm, denoted by \( \ln\), is the logarithm to the base \(e\), where \(e \) is an important mathematical constant approximately equal to 2.71828. The natural logarithm is the inverse function of exponential functions.
It has special properties that make it particularly useful in calculus, such as simplifying multiplicative expressions and handling growth processes.

• For example, \( \ln(1) = 0 \), since \( e^0 = 1 \).
• The expression \( \ln(2) \) appears frequently in calculus problems since it represents where the exponential growth doubles from the starting point.

In the context of finding the average value in the example exercise, simplifying \( \ln 2 - \ln 1 \) to \( \ln 2 \) showcased using natural logarithm properties to streamline the calculation. Understanding these properties allows smoother transitioning through more complex calculus operations.