Explanation Formula used: The graphs of the form y = a sin ( b x + c ) or y = a cos ( b x + c ) has the displacement − c b . Calculation: Consider the function y = 2 csc ( 2 x + π 4 ) . If the sign before π 4 is changed in the above function, then the function becomes y = 2 csc ( 2 x − π 4 ) . Since the period for csc x is 2 π , the period of y = 2 csc ( 2 x − π 4 ) will be 2 π 2 = π . Thus, one-fourth of the period will be, 1 4 ( π ) = π 4 . Since csc x = ( sin x ) − 1 , the curve y = 2 csc ( 2 x − π 4 ) will have the same displacement as that of y = 2 sin ( 2 x − π 4 ) . By the above mentioned formula, the displacement of y = 2 sin ( 2 x − π 4 ) is − ( − π 4 ) 2 = π 8 . That is, the curve y = 2 csc ( 2 x − π 4 ) starts repeating π units to the right of π 8 . Since the period is π , the one complete cycle ends at π 8 + π = 9 π 8 . Thus, the two complete cycles ends at π 8 + 2 π = 17 π 8 . That is, two cycles occur between x = π 8 and x = 17 π 8 . The important values for the first cycle are at π 8 , π 8 + π 4 , π 8 + 2 ( π 4 ) , π 8 + 3 ( π 4 ) and π 8 + 4 ( π 4 ) . That is, at π 8 , 3 π 8 , 5 π 8 , 7 π 8 and 9 π 8 . Substitute x = π 8 in the equation y = 2 csc ( 2 x − π 4 ) . y = 2 csc ( 2 x − π 4 ) = 2 csc ( 2 ( π 8 ) − π 4 ) = 2 csc ( 0 ) = Undefined That is, x = π 8 is an asymptote of the curve y = 2 csc ( 2 x − π 4 )

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

11th Edition

ISBN: 9780137554843

Author: Allyn Washington, Richard Evans

Publisher: PEARSON+

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Chapter 10.4, Problem 2E

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To sketch: At least two cycles of graph of the function y=2csc(2x+π4), if the sign before π4 is changed.

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Chapter 10 Solutions

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

Ch. 10.1 - For the graph of y = −6 sin x, set up a table of...Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - In Exercises 3–6, complete the following table for...Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - In Exercises 7–22, give the amplitude and sketch...Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E

Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - In Exercises 7–22, give the amplitude and sketch...Ch. 10.1 - In Exercises 7–22, give the amplitude and sketch...Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - In Exercises 27–32, solve the given problems. 32....Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.2 - Prob. 1PE Ch. 10.2 - Prob. 2PE Ch. 10.2 - Prob. 3PE Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - In Exercises 3–22, find the amplitude and period...Ch. 10.2 - Prob. 5E Ch. 10.2 - In Exercises 3–22, find the amplitude and period...Ch. 10.2 - 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Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.2 - Prob. 45E Ch. 10.2 - Prob. 46E Ch. 10.3 - Prob. 1PE Ch. 10.3 - Prob. 2PE Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Prob. 10E Ch. 10.3 - Prob. 11E Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - Prob. 19E Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - 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At least two cycles of graph of the function y = 2 csc ( 2 x + π 4 ) , if the sign before π 4 is changed.

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To sketch: At least two cycles of graph of the function y=2csc(2x+π4), if the sign before π4 is changed.

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Chapter 10 Solutions