Problem 90 - University Calculus: Early Transcendentals

Question

The function \((\sin x)^{\tan x}\). a. Graph \(f(x)=(\sin x)^{\tan x}\) on the interval \(-7 \leq x \leq 7 .\) How do you account for the gaps in the graph? How wide are the gaps? b. Now graph \(f\) on the interval \(0 \leq x \leq \pi .\) The function is not defined at \(x=\pi / 2,\) but the graph has no break at this point. What is going on? What value does the graph appear to give for \(f\) at \(x=\pi / 2 ?\) (Hint: Use I'Hópital's Rule to find lim \(f^{-}\) as \(\left.x \rightarrow(\pi / 2)^{-} \text {and } x \rightarrow(\pi / 2)^{+} .\right)\) c. Continuing with the graphs in part (b), find max \(f\) and min \(f\) as accurately as you can and estimate the values of \(x\) at which they are taken on.

Step-by-Step Solution

Answer

Gaps occur at multiples of \\(\pi\\); the graph approaches 1 at \\(\pi/2\\); maxima/minima near specific rational points.

Step 1: Understanding the Function's Domain

For the function \(f(x)=(\sin x)^{\tan x}\), the sine function \(\sin x\) has values between \(-1\) and \(1\). As such, \(\tan x\) must be rational multiples such that the base \(\sin x\) remains non-negative, i.e., \(\sin x > 0\), which occurs at \(0 < x < \pi\), excluding multiples of \(\pi\). In the interval \((−7, 7)\), the function shows undefined regions where \(\sin x\) becomes zero.

Step 2: Observing Gaps in the Graph

In the interval \([-7, 7]\), the function \(f(x)\) will have gaps where \(\sin x = 0\), specifically at \(x = n\pi\) (where \(n \text{ is an integer}\). The gaps occur at integer multiples of \(\pi\) such as \(-6\pi, -5\pi, \dots, 1\pi, 2\pi\), etc. The graph is undefined at these points, creating gaps in intervals around these points.

Step 3: Graphing from 0 to π and Analyzing x=π/2

In the interval \(0 \leq x \leq \pi\), \(f(x)\) is undefined at \((\pi/2)\) since \((\sin x)^{\tan x}\) tries to compute \(1^{\infty}\) form. However, the function appears to show continuance without a break due to the limit behavior which we investigate next with L'Hôpital's Rule.

Step 4: Apply L'Hôpital's Rule at x=π/2

To find \(\lim_{x \to \pi/2} (\sin x)^{\tan x}\), consider \(y = (\sin x)^{\tan x}\), so \(\ln y = \tan x \ln(\sin x)\). As \(x \to \pi/2\), \(\tan x \ln(\sin x) \to 0 \cdot (-\infty)\) which needs L'Hôpital's Rule. Set \(u = 1/\cos x \, v = \ln(\sin x), \), transform to recognizable form for L'Hôpital: \(\lim_{x \to \pi/2}v/u\). With derivatives, you find it evaluates to zero, setting \( f(x) ightarrow 1\) at \(x=\pi/2\).

Step 5: Finding Maxima and Minima on (0, π)

Observing the behavior on graphs, \(f(x)\) maximizes near \(x=\pi/2\) but precisely, calculations show no 'global' max; whereas minima occur when \(\sin x\) is lowest, at \(x=\pi/4\) or \(x=3\pi/4\) close values due to periodicity and properties, the function approaches an approximate value depending on rational closeness for the 'valid' input distances.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a handy tool used in calculus for finding limits, especially when you encounter indeterminate forms like \(0/0\) or \(\infty/\infty\). In essence, it states that if you have a limit of a fraction where both the numerator and the denominator approach zero or infinity, you can differentiate both the top and the bottom separately to try removing the indeterminate form.

This can be particularly useful when dealing with the trigonometric function \((\sin x)^{\tan x}\). When you attempt to evaluate this function at certain points, such as \(x = \pi / 2\), you notice it results in an indeterminate expression like \(1^{\infty}\). Using L'Hôpital's Rule, you transform the expression into something more manageable. In the case of \((\sin x)^{\tan x}\), you rewrite the function in terms of a \ logarithmic change, turning it into \(\tan x \ln(\sin x)\) and then take the limit as \(x\) approaches \(\pi / 2\).

By doing this, you bring clarity to the function's continuity, finding out that the expression actually approaches \(1\) instead of being undefined, thus preserving the function's graph continuity at \(x = \pi / 2\). This process elegantly showcases how L'Hôpital's Rule aids us in resolving limits involving more complex trigonometric and exponential functions.

Function Domain

Understanding the function domain is crucial for analyzing any mathematical equation or function. A function's domain is the set of all possible input values (typically \(x\) values) for which the function is defined.

For the function \(f(x) = (\sin x)^{\tan x}\), the domain is particularly important. The sine function, \(\sin x\), oscillates between \(-1\) and \(1\). However, for \((\sin x)^{\tan x}\) to be real-valued and defined, \(\sin x\) needs to be greater than zero because negative or zero values would make the function undefined or complex at certain points.

The restricted domain for this is primarily where \(0 < x < \pi\), excluding points where \(\sin x = 0\) such as \(0, \pi, 2\pi,\) etc. These exclusions create gaps in the graph of the function, occurring at intervals of \(\pi\), which correspond to integer multiples of \(\pi\). This understanding is essential when graphing the function, as it highlights where the function will not provide real values.

Exponential Functions

Exponential functions are characterized by their variable being an exponent. In the function \(f(x) = (\sin x)^{\tan x}\), the \(\tan x\) acts as the exponent, resulting in a more complex exponential function since the base is bounded and oscillatory.

While typical exponential functions involve constants as the base, like \(e^x\), having \(\sin x\) complicates things due to its trigonometric nature. This function exhibits unique behaviors mirroring those seen in trigonometric graphs, but compounded through exponentiation which can lead to unexpected patterns and points of interest at various \(x\) values.

When you graph this function or analyze it over different intervals, specific features emerge, like indeterminate forms or continous behavior at points generally thought undefined. Understanding the interplay between the exponential component and the trigonometric base helps us unravel the function's overall behavior, making it possible to find limits and ranges across specific intervals.