Problem 54 - Calculus of a Single Variable

Question

Graphical Reasoning Consider the function $$f(x)=\tan (\sin \pi x)$$ (a) Use a graphing utility to graph the function. (b) Identify any symmetry of the graph. (c) Is the function periodic? If so, what is the period? (d) Identify any extrema on \((-1,1)\) . (e) Use a graphing utility to determine the concavity of the graph on \((0,1) .\)

Step-by-Step Solution

Answer

After evaluating with a graphing utility, the function \(f(x) = \tan (\sin \pi x)\) can be observed to have no symmetry, is periodic with a period of \(1\), and does not have extremas on the interval \((-1,1)\). The function is concave up on the interval \((0,1)\).

Step 1: Plot the Function

Use a graphing utility to plot the function \(f(x)=\tan (\sin \pi x)\). Use a good range for \(x\) to cover at least a couple of periods of the function.

Step 2: Identify Symmetry

Study the graph and identify any axis or origin symmetry. Origin symmetry suggests an odd function, while symmetry about the y-axis suggests an even function. If there's neither, the function is neither odd nor even.

Step 3: Identify Periodicity

Identify if the function is periodic by seeing if the pattern of the graph repeats after a certain interval on the x-axis. If it does, the function is periodic and the length of this interval is the period.

Step 4: Identify Extrema

Locate the highest (maxima) and lowest (minima) points of the function on the interval \((-1,1)\).

Step 5: Determine Concavity

Use the graphing utility to determine the concavity of the graph on the interval \((0,1)\). If the function opens upwards it is concave up, and if it opens downwards it is concave down.

Key Concepts

Trigonometric Function Periodicity

Understanding the periodic nature of trigonometric functions is crucial when it comes to graphing. Periodicity refers to the repeating pattern of a function over a set interval. For a function such as f(x) = \tan (\text{sin} \( \( \pi x \) \)), we observe its behavior regarding how it repeats itself. Specifically, because the sine function has a period of 2\pi, any function using \sin \pi x as its argument will have periodic attributes. That means after every specific interval of x, called the period, the function will exhibit the same set of values as before.

However, due to the tangent function's unique characteristics, such as it being undefined at certain points where the sine function equates to 1 or -1, the period may be affected. So, although the sinusoidal component would suggest a period of 2, we have to factor in these undefined points when determining the true period of the entire function.

Function Symmetry

Symmetry in functions can greatly simplify the process of graphing and understanding their behavior. When a function is symmetric about the y-axis, we say it's even, while symmetry about the origin signifies an odd function. For the function f(x) = \tan (\text{sin} \( \( \pi x \) \)), assessing symmetry involves looking for these visual cues within the graph. Odd symmetry implies that f(-x) = -f(x), while even symmetry implies that f(-x) = f(x).

If no symmetry is apparent, then the function does not exhibit origin or axis symmetry, meaning it's neither odd nor even. Noting this characteristic helps in predicting the behavior of the function for negative x values, based solely on its behavior for positive x values, and vice versa.

Identifying Extrema

Extrema refer to the highest (maximum) and lowest (minimum) points of a function within a certain interval. To identify extrema on the interval (-1,1) for the function f(x) = \tan (\text{sin} \( \( \pi x \) \)), we look for the peaks and troughs of the graph. These points represent the maximum and minimum values of the function, respectively.

Extrema are important features in understanding a function's overall shape and can provide insights into the function's rate of change. For example, local maximums occur where the graph changes from increasing to decreasing, and local minimums where it changes from decreasing to increasing. Detecting these points involves the examination of first and second derivatives as well, but with the aid of a graphing utility, it becomes a visual task of pinpointing the highest and lowest points on the graph within the given interval.

Concavity of Graphs

The concavity of a graph refers to the direction in which a function curves. When graphing, we determine whether a function is 'concave up', resembling the shape of a bowl that can hold water, or 'concave down', shaped like an arch or upside down bowl. To discern concavity on the interval (0,1) for the function f(x) = \tan (\text{sin} \( \( \pi x \) \)), one can visually inspect the curvature of the graph or use calculus methods involving the second derivative.

The concavity helps to understand the rate of change of the function's slope: concave up graphs indicate increasing slopes, while concave down graphs suggest decreasing slopes. This property is essential for predicting function behavior and graphing accurately, especially when first learning how to draw complex functions such as trigonometric ones.