   Chapter 3.5, Problem 38E

Chapter
Section
Textbook Problem

# 1-40 Use the guidelines of this section to sketch the curve. y = csc x − 2 sin x ,    0 < x < π

To determine

To sketch:

The curve of y

Explanation

1) Concept:

i) A domain is the set of x values that satisfy the function.

ii) To find x-intercept, put y=0, and to find y-intercept, put x=0 in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behaviour of function. Thus, if f-x=fx, then it is an even function, so it has y-axis symmetry. If f-x=-fx, then it is an odd function, so it has x-axis symmetry. And if f-x-fxfx, then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical, and slant asymptote, follow the rules.

v) A function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

vi) The number f(c) is a local maximum value of f  if fcf(x) when x is near c and is a local minimum value of f if fc f(x) when x is near c.

vii) If f''x>0, the function is concave up and if f''x<0, the function is concave down in that particular interval. And if f''x=0, give the values of inflection points

2) Given:

y=cscx-2sinx, 0<x<π

3) Calculations:

Here, first find the domain of the given function and the x & y intercepts. Next, check the symmetry, asymptotes, intervals of increase and decrease, local maximum and minimum values, concavity, and points of inflection. Using these, sketch the curve

A. Domain:

fx is defined for all values of the given interval

So Domain is 0<x<π

B. Intercepts:

For y intercept plug x=0  in the given function and solve y.

y=csc0-2sin0=

The y-intercept is 0,

For x intercept, plug y=0 in the original function and solve x.

0=cscx-2sinx

2sinx=cscx

sin2x=12

sinx=±12

x=π4, 3π4

The x-intercepts are π4, 0, 3π4, 0

C. Symmetry:

For f-x replace x by (-x)

f-x= csc(-x)-2sin(-x)

f-x= -cscx+2sinx

f-x= -(cscx-2sinx)

f-x=-f(x)

f(x) is odd function

So f(x) has rotational symmetry about origin

D. Asymptote:

a) Horizontal asymptotes

The given function is bounded so there are no horizontal asymptotes

b) Vertical asymptotes

limnπcscx-2sinx=

limn0cscx-2sinx=

So there are vertical asymptotes at x=0, π

E

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