(a) Explanation Given data: The expression of the curve is, r ( t ) = ( 3 cos t ) i + ( 3 sin t ) j + t k , 0 ≤ t ≤ 4 π (1) Formula used: Write a general expression to calculate constant vector v . v = d r d t (2) Here, d r d t is the value of change in plane curve. Write a general expression to calculate the unit tangent vector. T = v | v | (3) Write a general expression to calculate the curvature. κ = 1 | v | | d T d t | (4) Calculation: Substitute the equation (1) in equation (2) to find v . v = d d t ( ( 3 cos t ) i + ( 3 sin t ) j + t k ) = 3 ( − sin t ) i + 3 ( cos t ) j + ( 1 ) k = − 3 sin t i + 3 cos t j + k Take the magnitude of above equation to find | v | . | v | = | − 3 sin t i + 3 cos t j + k | = ( − 3 sin t ) 2 + ( 3 cos t ) 2 + ( 1 ) 2 = 9 sin 2 t + 9 cos 2 t + 1 = 9 ( sin 2 t + cos 2 t ) + 1 Simplify the above equation to find | v | . | v | = 9 ( 1 ) + 1 { ∵ sin 2 θ + cos 2 θ = 1 } = 9 + 1 = 10 Substitute − 3 sin t i + 3 cos t j + k for v , and 10 for | v | in equation (3) to find T . T = − 3 sin t i + 3 cos t j + k 10 = 1 10 ( − 3 sin t i + 3 cos t j + k ) Differentiate the above equation with respect to t (b) To determine Find the total curvature for the given parabola functions.

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 13.4, Problem 20E

(a)

To determine

Find the total curvature of the portion of the helix for the given curve.

(b)

To determine

Find the total curvature for the given parabola functions.

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Chapter 13.4, Problem 19E

Chapter 13.4, Problem 21E

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Chapter 13 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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