Problem 76 - 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: Identify the function and interval

We are given the function \( f(x) = \frac{1}{x} \) and the interval \([1, 2]\). Our goal is to find the average value of the function over this interval.

Step 2: Set up the formula for average value

The formula for the average value of a function \( f \) on the interval \([a, b]\) is given by \( \frac{1}{b-a} \int_{a}^{b} f(x) \mathrm{d}x \). Here, \( a = 1 \) and \( b = 2 \). So, \( \frac{1}{2-1} \int_{1}^{2} \frac{1}{x} \mathrm{d}x \).

Step 3: Calculate the definite integral

Next, we need to evaluate the integral \( \int_{1}^{2} \frac{1}{x} \mathrm{d}x \). The antiderivative of \( \frac{1}{x} \) is \( \ln|x| \). Thus, the integral becomes \( \left[ \ln|x| \right]_{1}^{2} \).

Step 4: Evaluate the antiderivative at the boundaries

Substitute the upper and lower limits into the antiderivative: \( \ln|2| - \ln|1| = \ln(2) - \ln(1) \). Since \( \ln(1) = 0 \), this simplifies to \( \ln(2) \).

Step 5: Find the average value

Substitute the result of the integral into the average value formula: \( \frac{1}{1}(\ln(2)) = \ln(2) \). This means the average value of \( f(x) = \frac{1}{x} \) on the interval \([1, 2]\) is \( \ln(2) \).

Key Concepts

Average Value of Function

Finding the average value of a function over an interval is a fundamental concept in calculus. It helps us understand the typical height of the function within a certain range.

To determine the average value of a function, we use the formula:

• \( \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x) \mathrm{d}x \)

This formula essentially calculates the area under the curve of the function from \( a \) to \( b \) and then averages it over the length of the interval.

• \( a \) and \( b \) are the endpoints of the interval
• \( f(x) \) is the function for which we are finding the average value

So, computing the average value involves both finding the definite integral and then dividing by the interval length. This gives a single number representing the area averaged over the specified range.

Definite Integral

In calculus, the definite integral is used to compute the area under the curve of a function between two specific points. This is an essential concept when calculating the average value of a function.

The definite integral from \( a \) to \( b \) is represented by:

• \( \int_{a}^{b} f(x) \mathrm{d}x \)

The result of this operation is a number that represents the total accumulated value of the function \( f(x) \) over the interval \([a, b]\). To solve a definite integral:

• Find the antiderivative of the function \( f(x) \).
• Evaluate this antiderivative at the bounds \( a \) and \( b \).
• Compute the difference between these two values.

This calculation helps us determine the exact area under the curve of the function over the specified interval.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative gives back the original function. This concept is vital when solving definite integrals.

For example, if we have \( f(x) = \frac{1}{x} \), its antiderivative can be evaluated as:

• \( \int \frac{1}{x} \mathrm{d}x = \ln|x| + C \)

Where \( C \) is the constant of integration. However, when evaluating a definite integral, this constant cancels out.

Finding the antiderivative involves reversing the process of differentiation. For many standard functions, the antiderivatives are well-known, but they sometimes require complex calculations for more intricate functions or irregular intervals.

Logarithmic Functions

Logarithmic functions, like \( f(x) = \frac{1}{x} \), are an essential class of functions in calculus. They appear frequently in problems involving growth and decay, among other applications.

The natural logarithm, represented as \( \ln(x) \), is particularly significant because it serves as the antiderivative of \( \frac{1}{x} \). This relationship is crucial in solving integrals involving reciprocal functions.

• Key point: \( \int \frac{1}{x} \mathrm{d}x = \ln|x| + C \)

Logarithmic functions have unique properties:

• They are only defined for \( x > 0 \).
• Their graphs rapidly increase for \( x < 1 \), but slow as \( x > 1 \).
• They have a vertical asymptote at \( x = 0 \).

Understanding the behavior of logarithmic functions helps in interpreting the results of calculus problems where they are involved, such as in determining average values on specified intervals.