Problem 144 - Thomas Calculus

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: Understand the Formula for Average Value of a Function

The average value of a function \(f(x)\) on the interval \([a, b]\) is given by the formula \( \text{Average Value} = \frac{1}{b-a} \int_a^b f(x) \, dx \). In this case, \(a = 1\) and \(b = 2\).

Step 2: Set Up the Integral

We need to evaluate the integral \(\int_1^2 \frac{1}{x} \, dx \), which is part of the formula for the average value.

Step 3: Evaluate the Integral

The antiderivative of \( \frac{1}{x} \) is \( \ln|x| \). Thus, we find \( \int_1^2 \frac{1}{x} \, dx = \ln|x| \Big|_1^2 = \ln(2) - \ln(1) = \ln(2) \) because \(\ln(1) = 0\).

Step 4: Divide by the Length of the Interval

The length of the interval \([1, 2]\) is \(2 - 1 = 1\). The average value is therefore \( \frac{1}{1} \cdot \ln(2) = \ln(2) \).

Step 5: Solution Conclusion

The average value of the function \(f(x) = \frac{1}{x}\) on the interval \([1, 2]\) is \( \ln(2) \).

Key Concepts

Integral Calculus

Integral calculus is a branch of calculus that focuses on the concept of integration. Integration is the process of finding the integral of a function, which can be thought of as the reverse operation of differentiation.
It is one of the two major areas of calculus, the other being differential calculus. The primary objectives of integral calculus are:

• To find the area under a curve in a graph of a function.
• To determine the accumulation of quantities, such as distance or probability.
• To solve differential equations.

In our exercise, finding the average value of a function over an interval involves integrating the function over that interval. This means calculating the total area under the function's curve from one point to another and then dividing it by the length of the interval.
The fundamental theorem of calculus links the concept of differentiation and integration, showing that they are essentially inverse processes.

Antiderivative

An antiderivative, also known as an indefinite integral, refers to a function whose derivative results in the original function. In other words, if a function \( F(x) \) is an antiderivative of \( f(x) \), then \( F'(x) = f(x) \).
The process of finding an antiderivative is called integration.For the exercise problem, the function \( f(x) = \frac{1}{x} \) has an antiderivative of \( \ln|x| \). This means that when you differentiate \( \ln|x| \), you get \( \frac{1}{x} \).
It is crucial to find this antiderivative to evaluate the definite integral, which is used to compute the average value over the interval.

• The antiderivative allows us to compute the integral \( \int_1^2 \frac{1}{x} \, dx \).
• Understanding the power of the antiderivative helps solve various calculus problems with ease.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \) is approximately 2.718. The natural logarithm is a key function in calculus and analysis, especially when it comes to integrating functions.The function \( \ln|x| \) is particularly important because it is the antiderivative of \( \frac{1}{x} \).
In more straightforward terms, when you take the derivative of \( \ln|x| \), you arrive at \( \frac{1}{x} \), which highlights the natural relationship between these mathematical functions.

• The natural logarithm is used in our problem during the integration process to find the area under the curve of \( \frac{1}{x} \).
• This function also appears frequently in problems involving growth and decay processes, such as compound interest or population growth.
• Understanding the natural logarithm is critical for grasping more complex calculus topics and applications.