Chapter 4.6, Problem 42E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Analyzing the Graph of a Trigonometric Function In Exercises 35-42, analyze and sketch a graph of the function over the given interval. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results.Function Interval g ( x ) = x cot x − 2 π < x < 2 π

To determine

To graph: The function g(x)=xcotx

Explanation

Given: The function g(x)=xcotx over the interval âˆ’2Ï€â‰¤xâ‰¤2Ï€.

Graph: The function is not defined at the points âˆ’2Ï€,âˆ’Ï€,0,Ï€,2Ï€ domain of the function has been provided as (âˆ’2Ï€,âˆ’Ï€)âˆª(âˆ’Ï€,0)âˆª(0,Ï€)âˆª(Ï€,2Ï€).

Now find the x and y intercepts by equating g(x) and x to zero respectively to obtain:

The x-intercepts are (âˆ’3Ï€2,0),(âˆ’Ï€2,0),(Ï€2,0),(3Ï€2,0) and no y-intercept.

The function has four vertical asymptotes as x=âˆ’2Ï€,x=2Ï€,x=âˆ’Ï€,x=Ï€ because the cotangent function is not defined at these points.

Now, differentiate the function with respect to x and equate it to zero to obtain the critical points.

cotxâˆ’xcsc2x=0x=0

This point does not lie in the domain of the function and thus the function has no critical points.

This gives four test intervals (âˆ’2Ï€,âˆ’Ï€),(âˆ’Ï€,0),(0,Ï€),(Ï€,2Ï€).

Let âˆ’3Ï€2âˆˆ(âˆ’2Ï€,âˆ’Ï€).

f'(âˆ’3Ï€2)=cot(âˆ’3Ï€2)âˆ’(âˆ’3Ï€2)csc2(âˆ’3Ï€2)>0

The function is increasing on this interval.

Let âˆ’Ï€2âˆˆ(âˆ’Ï€,0)

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