Chapter 4.6, Problem 8E

(a)

To determine

Find any x-intercepts.

Answer to Problem 8E

None

Explanation of Solution

Given information: $f(x)=\csc x$

Formula Used: Graphic Utility = For such equations, a graphing utility can often be used to investigate possible solutions. When a graphing utility is used to solve an equation, usually approximate solutions are obtained.

Calculation:

x-intercepts of $f(x)=\csc x$ : None

  

(b)

To determine

Find any y-intercepts.

Answer to Problem 8E

None

Explanation of Solution

Given information: $f(x)=\csc x$

Formula Used: Graphic Utility = For such equations, a graphing utility can often be used to investigate possible solutions. When a graphing utility is used to solve an equation, usually approximate solutions are obtained.

Calculation:

y-intercepts of $f(x)=\csc x$ : None

  

(c)

To determine

Find the interval y=f(x) increasing and y=f(x) decreasing.

Answer to Problem 8E

Increasing in the interval= ( $-2\pi ,-\frac{3\pi }{2}$ )U( $-\pi ,-\frac{\pi }{2}$ )U( $0,\frac{\pi }{2}$ )U( $\pi ,\frac{3\pi }{2}$ )

Decreasing in the interval= ( $-2\pi ,-\frac{3\pi }{2}$ )U( $-\pi ,-\frac{\pi }{2}$ )U( $0,\frac{\pi }{2}$ )U( $\pi ,\frac{3\pi }{2}$ )

Explanation of Solution

Given information: $f(x)=\csc x$

Formula Used: Graphic Utility = For such equations, a graphing utility can often be used to investigate possible solutions. When a graphing utility is used to solve an equation, usually approximate solutions are obtained.

Calculation:

Increasing in the interval= ( $-2\pi ,-\frac{3\pi }{2}$ )U( $-\pi ,-\frac{\pi }{2}$ )U( $0,\frac{\pi }{2}$ )U( $\pi ,\frac{3\pi }{2}$ )

Decreasing in the interval= ( $-2\pi ,-\frac{3\pi }{2}$ )U( $-\pi ,-\frac{\pi }{2}$ )U( $0,\frac{\pi }{2}$ )U( $\pi ,\frac{3\pi }{2}$ )

  

(d)

To determine

Find all relative extrema

Answer to Problem 8E

Relative maxima = 8

Relative minima = 8

Explanation of Solution

Given information: $f(x)=\csc x$

Formula Used: Graphic Utility = For such equations, a graphing utility can often be used to investigate possible solutions. When a graphing utility is used to solve an equation, usually approximate solutions are obtained.

Calculation:

Relative maxima = 8

Relative minima = 8

  

e-asymptotes of $f(x)=\cot x$ : $x=\frac{\pi }{2}+n\pi$

(e)

To determine

Find all vertical asymptotes

Answer to Problem 8E

  $-\frac{\pi }{2},\frac{\pi }{2}$

Explanation of Solution

Given information: $f(x)=\csc x$

Formula Used: Graphic Utility = For such equations, a graphing utility can often be used to investigate possible solutions. When a graphing utility is used to solve an equation, usually approximate solutions are obtained.

Calculation:

The vertical asymptotes of cot x occur at $-\frac{\pi }{2},\frac{\pi }{2}$