Chapter 4.3, Problem 46E

(a)

To determine

To calculate: The right end behavior model.

Answer to Problem 46E

Limit does not exist.

Explanation of Solution

Given Information: The function is $y={\cos }^{-1}\left(x\right)$ .

Calculation:

The graph of the function looks like this,

  

As it is clear from the graph above, $x\to \infty ,y$ does not exist.

The domain of the given function is $\left[-1,1\right]$ and therefore it is not defined for $x=\infty$ .

Limit does not exist.

(b)

To determine

To calculate: The left end behavior model.

Answer to Problem 46E

Limit does not exist.

Explanation of Solution

Given Information: The function is $y={\cos }^{-1}\left(x\right)$ .

Calculation:

The graph of the function looks like this,

  

As it is clear from the graph above, $x\to -\infty ,y$ does not exist.

The domain of the given function is $\left[-1,1\right]$ and therefore it is not defined for $x\to -\infty$ .

Limit does not exist.

(c)

To determine

To calculate: To find if there are any horizontal tangents.

Answer to Problem 46E

The function has no horizontal tangent.

Explanation of Solution

Given Information: The function is $y={\cos }^{-1}\left(x\right)$ .

Calculation:

The graph of the function looks like this,

  

The horizontal tangent is obtained by equating derivative of the function to 0.

The derivative of the function is,

  $\begin{array}{c}y={\cos }^{-1}\left(x\right)\\ \frac{dy}{dx}=-\frac{1}{\sqrt{1-x^2}}\end{array}$

Equate the derivative to 0,

  $\begin{array}{c}\frac{dy}{dx}=0\\ -\frac{1}{\sqrt{1-x^2}}=0\\ -1=0\end{array}$

The above condition is not possible for any value of x, hence the function has no horizontal tangent.

The function has no horizontal tangent.