Resolution to Quiz 15 - Analytic Geometry and Calculus III | MATH 2400, Quizzes of Analytical Geometry and Calculus

Download Resolution to Quiz 15 - Analytic Geometry and Calculus III | MATH 2400 and more Quizzes Analytical Geometry and Calculus in PDF only on Docsity! Quiz 15 13.2.44: y′(t) = 〈cos t, sin t〉 ⇒ y(t) = 〈sin t,− cos t〉+ C 〈1,−1〉 = y(0) = 〈0,−1〉+ C ⇒ C = 〈1, 0〉 y(t) = 〈sin t + 1,− cos t〉. 13.3.24: r′(t) = 〈−3 cos2 t sin t, 3 sin2 t cos t〉 s = ∫ t 0 ‖r′(u)‖ du = ∫ t 0 √ 9 cos4 u sin2 u + 9 sin4 u cos2 u du = ∫ t 0 3 cos u sin u √ cos2 u + sin2 u du = ∫ t 0 3 cos u sin u du = [ 3 2 sin2 u ]t 0 = 3 2 sin2 t. sin t = √ 2 3 s⇒ cos t = √ 1− 2 3 s r(s) = 〈( 1− 2 3 s ) 3 2 , ( 2 3 s ) 3 2 〉 , 0 ≤ s ≤ 3 2 . HB: If a runner’s path is modeled by the vector valued function r(s) = 〈x(s), y(s)〉, which is parameterized by a unit speed parametrization, what distance does the runner travel from s = a to s = b, a < b? Since the function is parameterized by a unit speed parametrization, ignoring units, the distance traveled equals the time spent traveling. Thus, the runner travels b−a units.