Explanation Given information: The smooth curve is, r ( t ) = e t i + ( sin t ) j + ln ( 1 − t ) k , t = 0 Formula used: The formula to calculate the parametric equations for a line through P 0 ( x 0 , y 0 , z 0 ) parallel to nonzero vector v = v 1 i + v 2 j + v 3 k is, x = x 0 + t v 1 (1) y = y 0 + t v 2 (2) z = z 0 + t v 3 (3) Write the expression to calculate the velocity vector. v = d r d t (4) Calculation: Substitute the given curve function in equation (4) to find v . v = d d t ( e t i + ( sin t ) j + ln ( 1 − t ) k ) = e t i + ( cos t ) j + ( − 1 1 − t ) k = e t i + ( cos t ) j − ( 1 1 − t ) k Substitute 0 for t in above equation to find v ( 0 )

Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137399185

Author: Joel Hass, Christopher Heil

Publisher: PEARSON+

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Chapter 13, Problem 27PE

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Find the parametric equations for the line that is tangent to the given curve.

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Chapter 13 Solutions

Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

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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