Graph each function, considering the domain, critical...

Graph each function, considering the domain, critical points, symmetry, relative extrema, regions where the function is increasing or decreasing, inflection points, regions where the function is concave upward or concave downward, intercepts where possible, and asymptotes where applicable. (Hint: In Exercise 21, use the result of Exercise 1. In Exercises 25-27, recall from Exercise 68 in Section 3.1 on Limits that $\lim _{x \rightarrow \infty} x^{n} e^{-x}=0 . )$ $$f(x)=\frac{\ln x}{x}$$

Key Concepts

Symmetry Analysis

Symmetry analysis involves determining whether a function is symmetric about the y-axis, the origin, or exhibits any other type of symmetry. Recognizing these symmetries can simplify the graphing process because it reduces the amount of computation needed by allowing one to infer the behavior of the function on one part of the domain from its behavior on another.

Asymptotic Behavior

The examination of asymptotic behavior involves analyzing the limits of the function as the independent variable approaches infinity, negative infinity, or points of discontinuity. Identifying horizontal, vertical, or slant asymptotes helps in sketching the end behavior of the graph and in understanding the function’s behavior near the boundaries of its domain.

Intercepts

Intercepts are the points where the function crosses the coordinate axes. The x-intercepts are found by setting the function equal to zero, while the y-intercept is determined by evaluating the function at zero (if the function is defined at that point). These points are crucial for establishing reference markers when sketching the graph.

Domain Analysis

Determining the domain of a function involves identifying all the input values for which the function is defined. It is essential to examine restrictions such as division by zero or the use of logarithmic and radical expressions, which can limit the set of allowable values for the independent variable.

Increasing and Decreasing Intervals

Analyzing where a function is increasing or decreasing is accomplished by looking at the sign of its derivative. Intervals on which the derivative is positive indicate that the function is increasing, while intervals with a negative derivative signal that the function is decreasing. This information is key in understanding the overall behavior and trend of the function.

Critical Points

Critical points are the values in the domain at which the derivative of the function is zero or undefined. These points are significant because they can indicate the presence of relative maximums or minimums, serving as candidates for local extrema which are important when sketching the overall graph of the function.

Concavity and Inflection Points

The concavity of a function describes the direction in which the function curves. By using the second derivative, one can determine where the function is concave upward or concave downward. Inflection points occur where the concavity changes, which provides additional detail about the function’s shape and assists in a more precise graphing.

Transcript

Please give Ace some feedback