Problem 13 - Calculus Single Variable

Question

In each of Exercises \(13-16,\) a function \(f\) and an interval \(I=[a, b]\) are given. Find a number \(c\) in \((a, b)\) for which \(f(c)\) is the average value of \(f\) on \(I\). $$ f(x)=1 / x \quad I=[1,3] $$

Step-by-Step Solution

Answer

The number \( c \) is \( \frac{2}{\ln(3)} \) within \((1, 3)\).

Step 1: Understanding the Problem

The objective is to find a value \( c \) in the open interval \((1, 3)\) such that the value of the function \( f(c) \) equals the average value of the function \( f(x) = \frac{1}{x} \) over the closed interval \([1, 3]\).

Step 2: Finding the Average Value of the Function

The average value of a function \( f \) over an interval \([a, b]\) is given by \( \frac{1}{b-a} \int_a^b f(x) \, dx \). For this problem, \( f(x) = \frac{1}{x} \) and \([a, b] = [1, 3]\), so we find \( \frac{1}{3-1} \int_1^3 \frac{1}{x} \, dx \).

Step 3: Calculating the Definite Integral

Calculate \( \int_1^3 \frac{1}{x} \, dx \). The antiderivative of \( \frac{1}{x} \) is \( \ln|x| \), so evaluate it from 1 to 3: \( \ln(3) - \ln(1) = \ln(3) \) as \( \ln(1) = 0 \).

Step 4: Finding the Average Value

Substitute the result of the integral into the formula for the average value: \( \frac{1}{2}(\ln(3)) \). Thus, \( \ln(3)/2 \) is the average value of \( f(x) \) over \([1, 3]\).

Step 5: Finding the Value of c

Set up the equation \( f(c) = \frac{1}{c} = \ln(3)/2 \) to find \( c \). Solve for \( c \) by rearranging: \( c = \frac{2}{\ln(3)} \). Ensure the value of \( c \) lies in the interval \((1, 3)\).

Step 6: Verification

The calculated \( c = \frac{2}{\ln(3)} \approx 1.8205 \) lies within the interval \((1, 3)\), verifying that it is a valid solution.

Key Concepts

Average value of a function

The average value of a function over a specific interval provides an excellent summary of the function's behavior across that interval. It's akin to finding the 'center of gravity' for the function in terms of its functional values. To compute this average, particularly for continuous functions, we use integration. The formula is given by:

• \( \frac{1}{b-a} \int_a^b f(x) \, dx \)

Where \(a\) and \(b\) are the bounds of the interval and \(f(x)\) is our function. Essentially, you're accumulating all the values of the function over the interval and then 'averaging' them out by dividing by the interval's length \(b-a\). This concept is crucial, especially when dealing with functions that vary irregularly over different intervals, as it smooths out the fluctuations to a single representative value.

Definite integral

The definite integral is a fundamental concept in calculus that provides a powerful tool for finding areas under curves, among other things. When you calculate a definite integral, you are finding the net area covered by a function between two points, usually within a closed interval \([a, b]\). To calculate a definite integral, you need to find the antiderivative of the function, which is also known as the `indefinite integral`.The general procedure involves:

• Finding the antiderivative of the function \( F(x) \)
• Evaluating \( F(x) \) at the upper limit \( b \) and subtracting the value found by evaluating it at the lower limit \( a \).

Mathematically, this is represented as: \( \int_a^b f(x) \, dx = F(b) - F(a) \). In our problem, the function \( \frac{1}{x} \) had its antiderivative found as \( \ln|x| \), simplifying our definite integral in the given interval to \( \ln(3) - \ln(1) = \ln(3) \).

Logarithmic function

Among the many types of functions, logarithmic functions have unique and interesting properties. The logarithmic function, denoted as \( \ln(x) \), is the inverse of the exponential function. This particular function considers the power to which a fixed number, \( e \), must be raised to produce a given number \( x \).Key properties include:

• The domain of \( \ln(x) \) is \( x > 0 \), since you cannot take the log of a non-positive number in the real number system.
• \( \ln(1) = 0 \), as any number to the power of 0 is 1.
• It is a continuous and increasing function.

Logarithmic functions are particularly useful in scenarios involving growth and decay, financial models, and in this context, they serve as antiderivatives, which makes them essential for integration tasks.

Interval notation

Interval notation is a mathematical shorthand used to express a set of numbers located between two endpoints on a number line. It's incredibly useful in calculus and other disciplines to clearly denote the domain and range of functions.Basics of interval notation include:

• Saturated intervals use brackets \([ \)] indicating inclusive ends.
• Open intervals, like \((a, b)\), indicate the endpoints are not included.
• Half-open or half-closed intervals, denoted as \([a, b)\) or \((a, b]\), include one endpoint but not the other.

In our particular problem, the interval [1,3] was used for the purpose of finding an average value while the point \(c\) as sought in the open interval \((1, 3)\). Understanding the distinctions between these notations is crucial, as it can entirely change the scope of solutions or domains in a given problem.