Question

In Exercises \(41-46,\) find (a) the right end behavior model, (b) the left end behavior model, and (c) any horizontal tangents for the function if they exist. $$y=\cos ^{-1} x$$

Short answer

The right end behavior of the function \(y = \cos^{-1}x\) is \(y = 0\) as \(x \rightarrow 1\), the left end behavior is \(y = \pi\) as \(x \rightarrow -1\), and the function has horizontal tangents at \(x = -1\) and \(x = 1\).

Step-by-step solution

Establish the range of the arccosine function

The arccosine function, \(\cos^{-1}x\), is only defined in the interval [-1,1], and in this range, it decreases from \(\pi\) to 0.

Finding the right end behavior model

As x approaches 1 from the left, \(\cos^{-1}x\) approaches 0 from above. That means as we move towards the right end of the function, it will approach 0.

Finding the left end behavior model

As x approaches -1 from the right, \(\cos^{-1}x\) approaches \(\pi\) from below. That implies as we move towards the left end of the function, it will approach \(\pi\).

Identifying any horizontal tangents

Given that the function is decreasing throughout its interval of existence, there are no horizontal tangents within this interval. However, the function has horizontal touch at the end points, i.e., at \(x = -1\) and \(x = 1\).

Key concepts

Arccosine Function

Understanding the arccosine function—the inverse function of the cosine—is crucial for tackling many problems in calculus. The notation \(\cos^{-1}x\) or sometimes \(\arccos x\) represents this function, which takes the ratio of the adjacent side to the hypotenuse in a right triangle (where the ratio falls in the range [-1,1]) and gives you the corresponding angle in radians.

One key aspect of the arccosine function is that it's defined only in the domain [-1,1], which is due to the range of the cosine function being between -1 and 1. The range of arccosine is from \(0\) to \(\pi\), meaning it can only output angles that lie in the first and second quadrants. Visualizing this on a graph, you will see that the curve of \(y = \cos^{-1}x\) starts at (\(1, 0\)) and ends at (\(\-1, \pi\)), decreasing monotonically—always moving downwards as it goes from right to left.

To elaborate, at the right endpoint of the interval, the output of the arccosine function approaches zero as \(x\) approaches 1. Similarly, on the left, as \(x\) approaches \-1, the output of the function inches closer to \(\pi\). These behaviors determine the horizontal orientation at the very ends of the function's graph, which is essential when considering the concept of end behavior models.

Horizontal Tangents

Horizontal tangents are a topic of significant interest in calculus as they pertain to the points on a curve where the slope is zero. This concept is often linked with finding local maximums and minimums, or points of inflection, since the slope at these points is flat. Essentially, horizontal tangents can be detected by setting the derivative of a function to zero and solving for \(x\).

Nonetheless, with the arccosine function, the scenario is slightly different. Its slope is never truly zero within its domain; instead, it is continuously decreasing. But when considering the end behavior, horizontal tangents exist at the end points of this interval. This is because, theoretically, if you were to draw tangent lines at the points (\(1, 0\)) and (\(\-1, \pi\)), these lines would be perfectly horizontal, paralleling the \(x\-axis\) at the heights of \(\pi\) and \(0\).

To put it differently, even though the arccosine function does not have any horizontal tangent within the open interval (\(\-1, 1\)), at the boundaries of its domain, the behavior of the function mimics the existence of horizontal tangents, which are critical when considering overall trend and behavior of curves.

Limits of Trigonometric Functions

Limits form the foundation of calculus, and understanding limits of trigonometric functions is essential for studying continuity, differentiability, and overall behavior of functions near specific points. In particular, limits help us describe the behavior of a function as we approach a certain input value, regardless of the function’s actual value at that point.

For the arccosine function, which is a trigonometric function, we use limits to describe its behavior at the boundaries of its domain. As seen in the right end behavior model, as \(x\) approaches 1, the limit of \(\cos^{-1}x\) approaches 0. Conversely, as \(x\) approaches \-1, the limit of \(\cos^{-1}x\) approaches \(\pi\). These two limits are of utmost importance as they describe the terminal behavior of the arccosine function.

When you evaluate the limit of a trigonometric function, it’s also important to consider whether the limit exists and is finite, or if the function approaches infinity or negative infinity. In our case with the arccosine function, both limits at the edges of the interval are finite and significant in understanding the end behavior. Hence, these limits form a bridge connecting the intuitive visual representation of the functions' end behavior and the precise mathematical description.