(6) 1. Sketch and identify (a) x=tcost,y=tsint, z=5 zz+y2=z XXXXXXXXXXWrite an equation in the indicated coordinate system. Also write a parametric equation and a vector equation for each of the following equation. (i) x2 + y2 = 25 in R3; cylindrical coordinates, spherical coordinates (ii) x2 + y2 = 100 in R2; polar coordinates XXXXXXXXXXx = t cost, y = taint is a parametric equation of a curve in R2. Write a polar equation of this curve. Answer: r = XXXXXXXXXXFind the length of the polar curve r = 6, for 0 at the point (1,1,1). Answer XXXXXXXXXXEvaluate g'4 (cos 2ti + t23 + e-'k)dt. Answer: (1, 4, 1 - c-*/ XXXXXXXXXXFind an equation of the tangent line to the curve f(t) = cos ti + c-' sin t3 + at (1,0,1). Also find the unit tangent vector f(t) at this point. Answer: F(t) 1— t, t, 1 — t > XXXXXXXXXXFind and sketch the domain of the function (x, y) = y XXXXXXXXXXFind the limit, if exists, or show that the limit does not exist.+y. - 16y4 (a) lim Answer: (b) lim —. Answer: XXXXXXXXXX1D) LT' XXXXXXXXXXx - 2y + y3 x0 (c) (.. v).( lim0.0) 3x2 + y2' Answer: 0 (d) (..„..)illoomo x2 + 2y2 + 3y z • Answer: DNE (e) lim (x2 + y2) sin 1 ( ) Answer: 0

(6) 1. Sketch and identify (a) x=tcost,y=tsint, z=5 zz+y2=z XXXXXXXXXXWrite an equation in the indicated coordinate system. Also write a parametric equation and a vector equation for each of the...

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(6) 1. Sketch and identify (a) x=tcost,y=tsint, z=5 zz+y2=z XXXXXXXXXXWrite an equation in the indicated coordinate system. Also write a parametric equation and a vector equation for each of the following equation. (i) x2 + y2 = 25 in R3; cylindrical coordinates, spherical coordinates (ii) x2 + y2 = 100 in R2; polar coordinates XXXXXXXXXXx = t cost, y = taint is a parametric equation of a curve in R2. Write a polar equation of this curve. Answer: r = XXXXXXXXXXFind the length of the polar curve r = 6<, where 0 < 9 < 2r XXXXXXXXXXFind the length of the curve qt) =< t,8sin t, 8cost >, for 0 < t < 4r. Answer: 4%,/r5tr XXXXXXXXXXFind the curvature of rat) =< t, t2, t3 > at the point (1,1,1). Answer XXXXXXXXXXEvaluate g'4 (cos 2ti + t23 + e-'k)dt. Answer: (1, 4, 1 - c-*/ XXXXXXXXXXFind an equation of the tangent line to the curve f(t) = cos ti + c-' sin t3 + at (1,0,1). Also find the unit tangent vector f(t) at this point. Answer: F(t) 1— t, t, 1 — t > XXXXXXXXXXFind and sketch the domain of the function (x, y) = y XXXXXXXXXXFind the limit, if exists, or show that the limit does not exist.
+y. - 16y4 (a) lim Answer: (b) lim —. Answer: XXXXXXXXXX1D) LT' XXXXXXXXXXx - 2y + y3 x0 (c) (.. v).( lim0.0) 3x2 + y2' Answer: 0 (d) (..„..)illoomo x2 + 2y2 + 3y z • Answer: DNE (e) lim (x2 + y2) sin 1 ( ) Answer: 0

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David answered on Dec 26 2021

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(6) 1. Sketch and identify (a) x=tcost,y=tsint, z=5 zz+y2=z XXXXXXXXXXWrite an equation in the indicated coordinate system. Also write a parametric equation and a vector equation for each of the...

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