Question

Find the average value of the function over the given interval and all values of \(x\) in the interval for which the function equals its average value. $$ f(x)=\frac{1}{2 x}, \quad[1,4] $$

Short answer

The average value of the function over the interval [1, 4] is \(\frac{1}{6} ln|4|\). The x-values at which the function equals its average value are solutions to the equation \( \frac{1}{2x} = \frac{1}{6} ln|4| \) and they lie in the interval [1, 4].

Step-by-step solution

Find the Average Value of the Function

To find the average value \(f_{avg}\) of the function, integrate \(f(x)\) over the interval [1, 4] and then divide by the length of the interval. The formula for average value of a function over an interval \([a,b]\) is given by: \[ f_{avg} = \frac{1}{b-a} \cdot \int_{a}^{b} f(x) dx \] Substituting the given function and values of \(a\) and \(b\) into the formula: \[ f_{avg} = \frac{1}{4-1} \cdot \int_{1}^{4} \frac{1}{2x} dx \]

Solve the Integral

The integral of \(\frac{1}{2x}\) can be found using the basic rules of integral calculus. The integral of \(\frac{1}{x}\) is \(ln|x|\). Therefore the integral of \(\frac{1}{2x}\) is \(\frac{1}{2}ln|x|\). Find the definite integral from 1 to 4: \[ f_{avg} = \frac{1}{3} \left[ \frac{1}{2}ln|4| - \frac{1}{2}ln|1| \right] \]

Compute the Average Value

Evaluate the expression to find \(f_{avg}\). Since \(ln|1|\) is 0, it simplifies to: \[ f_{avg} = \frac{1}{3} \cdot \frac{1}{2} ln|4| = \frac{1}{6} ln|4| \]

Solve for x-values Where the Function Equals Its Average Value

Set the function \(f(x)\) equal to the average value \(f_{avg}\), and solve for \(x\). This gives: \[ \frac{1}{2x} = \frac{1}{6} ln|4| \] From this, solve for \(x\) keeping in mind that the solutions must be within the interval [1, 4].

Final solutions

The last step would be to solve the equation set in Step 4, and round off the solution as required. The final x-values found are considered the solution of this exercise assuming they lie within the interval [1,4]

Key concepts

Definite Integral

In integral calculus, the definite integral is used to calculate the exact area under a curve between two points on the x-axis. To find it, you use the limits of integration, which in this exercise are 1 and 4, as given in the interval \[1,4\]. The general notation for the definite integral of a function \(f(x)\) from \(a\) to \(b\) is illustrated as \[ \int_{a}^{b} f(x) \, dx\], which represents a sum of infinitesimally small areas under the curve between \(a\) and \(b\).

Understanding how to calculate a definite integral is crucial for various applications, such as finding the average value of a function over an interval, as demonstrated in the textbook exercise. The definite integral of the function \(f(x)=\frac{1}{2x}\) from 1 to 4 is necessary for calculating the average value of the function over that interval. It is the continuous equivalent of summing discrete data and plays a fundamental role in areas such as physics, engineering, and economics.

Integral Calculus

The branch of mathematics that deals with integrals and their properties is called integral calculus. It is used to find quantities like areas, volumes, and the sum of infinitely many small factors that add up to a whole. There are two main types of integrals: indefinite and definite. An indefinite integral, also known as an antiderivative, represents a family of functions that can produce the original function when differentiated. An indefinite integral is written without upper and lower limits and is denoted by \[ \int f(x) \, dx\].

In contrast, a definite integral has limits—representing the boundaries of integration—and retains a specific value or area. Integral calculus is more than a tool for computation; it's also crucial for understanding and describing the physical world. Within the context of this exercise, integral calculus is the underlying mechanism that enables the determination of the function's average value over a specified interval.

Natural Logarithm

The natural logarithm is a logarithm to the base \(e\), where \(e\) is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm of a number \(x\) is typically represented by \(ln(x)\) and is the power to which \(e\) must be raised to obtain the number \(x\). It is a fundamental concept in mathematics, particularly important in the field of calculus.

In the context of this exercise, when you integrate a function like \(\frac{1}{x}\), the natural logarithm emerges naturally as the antiderivative, resulting in the expression \(ln|x|\). This characteristic makes the natural logarithm particularly useful when dealing with growth processes and compound interest, as well as in calculus for integrating and differentiating functions that model these and similar phenomena. When calculating the average value of \(f(x) = \frac{1}{2x}\), the integral's evaluation involves taking the natural logarithm of the limits of the integration interval.