For which the tangent line to the graph of x t t sin

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), for which thetangent line to the graph ofx(t) =t-sin 2t ,y(t) =t+ sin 2t ,is vertical.1.t=π6,5π6correct2.t=7π12,11π123.t=5π12,7π124.t=π3,2π35.t=π4,3π46.t=π12,5π12Explanation:After differentiating, we see thatx(t) = 1-2 cos 2t ,y(t) = 1 + 2 cos 2t .Thusdydx=y(t)x(t)=1 + 2 cos 2t1-2 cos 2t.Now the tangent line will be vertical whenx(t) = 1-2 cos 2t= 0,hence when cos 2t= 1/2.Consequently, fortin (0, π) the tangentline will be vertical whent=π6,5π6.00710.0pointsWhich integral represents the arc length ofthe astroid shown inxyand given parametrically byx(t) = cos3t ,y(t) = sin3t .1.I= 3integraldisplayπ0|costsint|dt2.I= 3integraldisplay2π0costsint dt3.I=integraldisplay2π0|costsint|dt4.I=integraldisplay2π0costsint dt5.I= 3integraldisplay2π0|costsint|dtcorrect
choice (hac762) – HW Quest Week 11 – cepparo – (53850)46.I=integraldisplayπ0|costsint|dtExplanation:The arc length of the parametric curve(x(t), y(t)),atbis given by the integralI=integraldisplaybaradicalBig(x(t))2+ (y(t))2dt .But whenx(t) = cos3t ,y(t) = sin3twe see thatx(t) =-3 sintcos2t ,y(t) = 3 costsin2t ,in which caseradicalBig(x(t))2+ (y(t))2=radicalBig(3 costsint)2(cos2t+ sin2t)=|3 costsint|.Consequently,I= 3integraldisplay2π0|costsint|dt.keywords:arc length, parametric curve, as-troid trig functions, definite integral1.B only2.A only3.A and B only4.all of them5.C only6.B and C only7.none of them8.A and C onlycorrectExplanation:To convert from Cartesian coordinates topolar coordinates we use the relations:x=rcosθ ,y=rsinθ ,so thatr2=x2+y2,tanθ=yx.For the pointP(2,23) in Cartesian co-ordinates, therefore, one choice ofrandθisr= 4 andθ=π/3, but there are equivalentsolutions forr <0 as well as values ofθdif-fering by integer multiples ofπ. For the givenchoices we thus see thatA.TRUE:-4 cos(4π/3) = 2,-4 sin(4π/3) = 23.B.FALSE: differs fromπ/6 by 2π.C.TRUE: solution noted already.
00810.0pointsWhich, if any, ofA.(-4,4π/3),B.(4,13π/6),C.(4, π/3),are polar coordinates for the point given inCartesian coordinates byP(2,23)?00910.0pointsA pointPis given in Cartesian coordinatesbyP(1,-1). Find polar coordinates (r, θ) ofthis point withr <0 and 0θ <2π.1.parenleftBig-2,7π4parenrightBig
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