Explanation Given information: The interval is ( − ∞ , ∞ ) . Show the function as follows: r ( θ ) = 2 θ − cos 2 θ + 2 (1) Calculation: Calculate the first derivative of the function r ′ ( θ ) as follows: r ′ ( θ ) = d d x ( 2 θ − cos 2 θ + 2 ) = 2 − ( − 2 sin θ cos θ ) + 0 = 2 + 2 sin θ cos θ (2) Refer Equation (2). The value of the sin 2 θ is always positive for any value of t in the interval ( − ∞ , ∞ ) . The value of the r ′ ( θ ) is always positive and greater than zero in the interval ( − ∞ , ∞ ) . Thus, the function is increasing in the interval ( − ∞ , ∞ ) . Calculate the value of the function r ( θ ) at r = − 2 π as follows: Substitute − 2 π for θ in Equation (1)

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 4.2, Problem 26E

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Show that the function r(θ) has exactly one zero in the interval (−∞,∞).

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Chapter 4.2, Problem 25E

Chapter 4.2, Problem 27E

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Chapter 4 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 4.1 - In Exercises 16, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 3E Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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Show that the function r ( θ ) has exactly one zero in the interval ( − ∞ , ∞ ) .

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Show that the function r(θ) has exactly one zero in the interval (−∞,∞).

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