Chapter 7.1, Problem 9QR

To determine

To state: All the values of x (if any) at which the function $\sec \left(1+\sqrt{1-{\sin }^{2}x}\right)\text{on}\left[-\infty ,\infty \right]$ changes sign on the given interval then sketch a number line graph if the interval, and indicate the sign of the function on each subinterval.

Answer to Problem 9QR

The function changes sign at $0.9633+k\pi \text{or}2.1783+k\pi$ and the line graph is:

  

Explanation of Solution

Given information:

The given function is $\sec \left(1+\sqrt{1-{\sin }^{2}x}\right)\text{on}\left[-\infty ,\infty \right]$ .

Now consider the given function,

  $\sec \left(1+\sqrt{1-{\sin }^{2}x}\right)\text{on}\left[-\infty ,\infty \right]$

First, find he solution of the equation $\sec \left(1+\sqrt{1-{\sin }^{2}x}\right)=0$ , then use the obtained values to get the partitioning interval.

  $\begin{array}{c}\sec \left(1+\sqrt{1-{\sin }^{2}x}\right)=\frac{1}{\cos \left(1+\left|\cos x\right|\right)}\\ x\approx 0.9633+k\pi ,k\in Z\end{array}$

Now test one point on each subinterval:

For $x=0$ ,

  $\sec \left(1+\sqrt{1-{\sin }^{2}0}\right)\approx -2.4030<0$

For $x=\pm 1$ ,

  $\sec \left(1+\sqrt{1-{\sin }^{2}1}\right)\approx 32.7984>0$

The sign alternates over successive subintervals. Therefore, the function changes sign at $0.9633+k\pi \text{or}2.1783+k\pi$ , where k is an integer.

The line graph is: