(a) Find the derivative of r(t) = (4 + t)i + tej + sin(5t) k. (b) Find the unit tangent vector at the point t = 0. Solution (a) According to the theorem that states if r(t) = (f(t), g(t), h(t)) = f(t)i + g(t)j + h(t) k, where f, g, and h are differentiable functions, then r'(t) = (f'(t), g'(t), h'(t)) = f'(t)i + g'(t)j + h'(t) we differentiate each component of r. r'(t) = (b) Since r(0) = T(0) = r'(0) Ir'(0)| and r'(0)j + 5k, the unit tangent vector at the point (4, 0, 0) is j + 5k

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Question
(a) Find the derivative of r(t) = (4 + t)i + te tj + sin(5t) k.
(b) Find the unit tangent vector at the point t = 0.
Solution
(a) According to the theorem that states
if r(t) = (f(t), g(t), h(t)) = f(t)i + g(t)j + h(t) k, where f, g, and h are differentiable functions, then r'(t) = (f'(t), g'(t), h'(t)) = f'(t)i + g'(t)j +h'(t)
we differentiate each component of r.
r' (t) =
(b) Since r(0)
T(0)
=
r'(0)
Ir'(0)|
and r'(0) = j + 5k, the unit tangent vector at the point (4, 0, 0) is
j + 5k
Transcribed Image Text:(a) Find the derivative of r(t) = (4 + t)i + te tj + sin(5t) k. (b) Find the unit tangent vector at the point t = 0. Solution (a) According to the theorem that states if r(t) = (f(t), g(t), h(t)) = f(t)i + g(t)j + h(t) k, where f, g, and h are differentiable functions, then r'(t) = (f'(t), g'(t), h'(t)) = f'(t)i + g'(t)j +h'(t) we differentiate each component of r. r' (t) = (b) Since r(0) T(0) = r'(0) Ir'(0)| and r'(0) = j + 5k, the unit tangent vector at the point (4, 0, 0) is j + 5k
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