Question

Use a spreadsheet to complete the table using \(f(x)=\frac{\ln x}{x}\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 5 & 10 & 10^{2} & 10^{4} & 10^{6} \\ \hline f(x) & & & & & & \\ \hline \end{array} $$ (a) Use the table to estimate the limit: \(\lim _{x \rightarrow \infty} f(x)\). (b) Use a graphing utility to estimate the relative extrema of \(f\)

Short answer

The function values are as follows: For x=1, f(x)=0; for x=5, f(x)=0.182; for x=10, f(x)=0.230; for x=\(10^{2}\), f(x)=0.046; for x=\(10^{4}\), f(x)=0.0092; and for x=\(10^{6}\), f(x)=0.00092. Through this, we can estimate the limit as x tends to infinity to be 0. Looking at the graph of the function, we can infer that our function has a relative maximum at \(x=e\), hence, it is the relative extrema of \(f(x)\).

Step-by-step solution

Evaluate Function at Given Points

Using either a calculator or spreadsheet, substitute each of the given x-values (1, 5, 10, \(10^{2}\), \(10^{4}\), and \(10^{6}\)) into the function \(f(x)=\frac{\ln x}{x}\) and calculate. Record these values in the table.

Estimate the Limit

Examine the f(x) values in the table as x tends to infinity. If the values approach a particular number, that will be the limit. If not, then the limit is either nonexistent or infinity.

Graph the Function

Using a graphing utility, plot the function \(f(x)=\frac{\ln x}{x}\).

Determine Relative Extrema

Looking at the graph, identify any points where the function reaches a local maximum or minimum. These points are your relative extrema.

Record All Results

Compile the function values, limit, and extrema from the previous steps. These constitute the complete solution for the exercise.

Key concepts

Natural Logarithm Properties

The natural logarithm, denoted as \(\ln x\), is a fundamental mathematical function with several key properties that are highly relevant for understanding and analyzing functions, such as \(f(x)=\frac{\ln x}{x}\).

Firstly, the natural logarithm is the inverse of the exponential function \(e^x\), meaning that \(\ln(e^x) = x\) and \(e^{\ln x} = x\), assuming x is positive. Secondly, one of the most critical properties of the natural logarithm is that \(\ln(ab) = \ln(a) + \ln(b)\), revealing that logarithms can be used to turn multiplication into addition, a simpler operation.

Another important property is that \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\). In addition, \(\ln(a^r) = r\ln(a)\), where r is any real number, indicates that logarithms can convert exponentiation into multiplication, which simplifies expressions and helps solve equations involving exponential growth or decay.

Importance in Calculating Limits

These properties become particularly useful when calculating limits involving natural logarithms, as they often allow complex forms to be simplified before evaluating the limit. For the function \(f(x)\) given in the exercise, we use properties of the natural logarithm to understand the behavior of the function as x approaches infinity and to simplify expressions when searching for extrema.

Relative Extrema

In calculus, relative extrema refer to the points on a curve where the function reaches a local minimum or maximum. These points provide crucial information about the function's behavior. A relative minimum is a point where the function value is lower than at any other point in its immediate vicinity, and likewise, a relative maximum is higher than nearby points.

To find these points, one typically looks for where the first derivative of the function equals zero - these are the potential critical points. A change in the sign of the first derivative as we pass through the critical point can indicate a relative extremum. However, not every critical point is a relative extremum, which requires further testing, such as the second derivative test for concavity.

Application to the Given Function

For the function \(f(x)=\frac{\ln x}{x}\), identifying the relative extrema involves examining values of x where the gradient of the function changes or simply by observing a graph for any peaks or troughs. By analyzing the graph plotted using a graphing utility, as described in Step 3 and 4 of the solution, we can conclusively determine the locations of these extrema. Estimating relative extrema provides insight into the function's behavior, useful in various applications like optimization problems.

Graphing Functions

Graphing is a vital tool in calculus and other areas of mathematics, providing a visual understanding of a function's behavior. A function's graph allows us to observe properties such as continuity, relative extrema, inflection points, and asymptotic behavior directly, which can be more challenging to discern through algebraic expression alone.

To graph a function effectively, one must identify key characteristics: the domain and range, intercepts with the axes, regions where the function increases or decreases, and any symmetries. Graphing utilities, whether computer software or graphing calculators, have become essential tools enabling students and professionals to visualize functions quickly.

Graphing the Function \(f(x)\)

For the function \(f(x)=\frac{\ln x}{x}\), when using a graphing utility, we observe how the function behaves across different domains of x. Especially, we can identify visually how \(f(x)\) changes as x approaches infinity, examining whether it approaches a certain limit. This graphical analysis complements the numerical approach taken with a spreadsheet or calculator in the exercise, thus providing an interdisciplinary method to understand the nature of the function fully.