Let C be the space curve r(t) = (3 sin(t) – 1,4 sin(t) + 1), 0 < t < . The length of C is

Q: Two particles travel along the space curves r,(t) = (t, t²,t³) and r2(t) (1+ 2t, 1 + 6t, 1+ 14t),

A: Given data: The first curve is r1(t)=&lt;t, t2, t3&gt;. The second curve is r2(t)=&lt;1+2t, 1+6t,…

Q: Sketch the space curve and find its length over the given interval r(t) = ⟨8 cos t, 8 sin t, t⟩,…

A: Given Finding the derivative with respect to t:

Q: Let M be the space curve given with the vector valued function R(t) = (1 + 2 sin(2πt))i + (ln t)J +…

A: The equation of tangent line of M at t=2 with  the vector valued function Rt=1+2sin2πti+ln tj+cos…

Q: Let C be the parametric curve defined by (x(t) = t2 – t – 2, 1 y(t) = t³ – t² + t. Determine if C is…

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Q: Let C be a smooth curve traced by Ř(t) with R(1) = (2, –1, 1) and R' (1) = (1, –2, –1). Let g(t)…

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Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(at) î + at j+ sin(rt) k. %3D…

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Q: Find T, N, and k for the space curve r(t) = - 2e cos t)i- (2e sin t) j+ 2k.

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Q: Determine whether the space curve given by r(t) = (t, t 3 , t 2 + 1) intersects the xy-plane, and if…

A: Given the space curve,  r(t)=t,t3,t2+1

Q: Find T, N, and k for the space curves r(t) = (6 sin 2t)i + (6 cos 2t)j + 5tk

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Q: Find the curvature of the space curve. r(t) = -10 i + (t + 4) j +(In(cos t) +9) k %3D OK= sect K =…

A: Formula of radius of curvature:      k=r→'(t)×r→''(t)r→'(t)3 First we need to find r→'(t),r→''(t),…

Q: Find T, N, and k for the space curve, where t>0. r(t) = (8 cos t+ 8t sin t)i + (8 sin t- 8t cos t)j+…

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Q: Let C be the curve in the plane with parametrization r(t) = cos ti+ sin' tj for t e [0, 47]. Compute…

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Q: Find T, N, and k for the space curve, where t> 0. r(t) = ( cos t+t sin t)i + ( sin t-t cos t)j + 4k

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Q: Let C be a plane curve given by the parametric equations = f(t), y= g(t), a <t<b. Suppose that C has…

A: Slope of tangent

Q: Let r (t) = sin t i + etj + ln (t + 1) k be a space curve. Which of the following vectors is…

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Q: Let C be the curve with parametrization r(t) = cos(t)i+ sin(t)j – 2tk. (b) Compute the Frenet frame…

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Q: Let r(t) = sin t i + et j + In (t + 1) k %3D be a space curve. Which of the following vectors is…

A: See the attachment.

Q: Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the…

A: Given the curve     rt=2sin(t),2cos(t),4sin2(t) and the point P1,3,1.

Q: Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the…

A: Given that rt=3cost,3sint,9 P32,32.9 To find the parametric equation for the line tangent to the…

Q: 2. Let L be the line tangent to the space curve represented by r(t) = (t² – In(t))i+(e²-t – t)j+…

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Q: Let C be the space curve defined by Ř(t) = (2cos t,2 sin t, t). O Find the moving trihedral of C at…

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Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t2j+ 9k, 0 <ts v7.…

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Q: Find T, N, and k for the space curve r(t) = i+Tj, t> 0. T(t) = (O i+ (Oi

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Q: = Let Co be the curve formed by the intersection of the surfaces x₁ = 2x² and X3 3x1x2, and let C be…

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Q: Let be a curve defined by the vector function (t)= 6ti + çin 2t -cos 2+ k 3- and Find the moving…

A: Let C1 be a curve defined by the vector function R→(t)=6ti→+sin2tj→-cos2tk→. Find the moving…

Q: Find T, N, and k for the space curve, where t> 0. r(t) = (2 cos t+ 2t sin t)i + (2 sint- 2t cos t)j…

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Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

A: given that r(t)=(sintcost,cos2t,sint) we need to show that the curve lines on the surface of the…

Q: F(x, y) = 2xyi + (y² – x²)j (x2 + y2)2 and C is any positively oriented simple closed curve that…

A: Given  F(x,y) = 2xyx2+y22i+y2-x2x2+y22j To Evaluate  ∫CF(x,y)dr

Q: Give the length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in…

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

Q: 3. Sketch the space curve, r(t) = 8 cos ti +8 sin t j+tk , and find its length over the interval []…

A: Given: r(t)=8cost+8sint+tk    [0,pi/2]

Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(at) î + at j + sin(at) k. Find…

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Q: Let C be a plane curve given by the parametric equations x = f(t), y= g(t), a <t <b. Suppose that C…

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Q: Find the equation of the tangent line to the space curve r(t) = 2cos (t)i + 2 sin(t)j + tk at the…

A: The curve is given by: r(t)=2cos(t)i+2sin(t)j+tk

Q: Let r (t ) sin t i + etj + ln (t + 1) k be a space curve. Which of the following vectors is…

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Q: Let R be the region enclosed by the loop of the t², y = t³ – 3t. parametric curve x =

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Q: (b) Consider F(x, y) = -i+ j, where (x, y) # (0, 0). x²+y2 x2+y2- Without using Green's Theorem,…

A: Consider the provided function,

Q: Find T, N, and B for the given space curve. r(1)= (6+t)i + (2 + In (sec t))j-3k, - T/2 <t < T/2 O A.…

A: Find r'(t) by differentiating r(t) with respect to t. r(t)=6+ti+2+lnsec tj-3kr'(t)=i+sec t·tan tsec…

Q: Suppose that C is the graph of a smooth vector-valued function r(s) parametrized by arc length with…

A: Introduction: The tangent vector to a vector valued function rt is formulated by the equation r't…

Q: 12. Let T be the line tangent to the space curve represented by r(t) = (t³ + e=t )i+ (ln(t) + 2t )j+…

A: First we find the equation of tangent in Cartesian form, and then check whether the tangent…

Q: Let C be the space curve given with the vector valued function R(t) = (1 + 2 sin(2πt))i + (ln t)J +…

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Q: 8 Find T, N, and k for the space curve r(t) = Gi+gj, t> 0 8

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Q: Suppose r(t) is a vector function in R3 Show that if r(t)is perpendicular to the tangent vector…

A: Suppose that the position vector r→t be, r→t=ft,gt,ht…

Q: Find T, N, and k for the space curve r(t) =i+ i, t>0_

A: In this question, we find the unit tangent and unit normal along with the curvature of given r(t)

Q: Let r(t)=⟨sin(t),cos(t),9sin(t)+3cos(2t)⟩. Find the projection of r(t)r(t) onto the xz-plane for…

A: Set y=0 as the answer is asked for xz plane.

Q: Sketch the space curve. Function r(t) = (5 sin t, 3t, 5 cos t) 1.0 0.5 0.0 0.0 -0.5 -1.0 -5 N 0.5…

A: Please see the below picture for detailed solution.

Q: A smooth curve C is defined by some vector function R(t) with R (7) 2 = = (π,0,−2) and R' (t) = (2,…

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Q: Sketch the plane curve. r(t) = 2ti-tj, [0, 2] y -1 y 2 1 Find its length over the given interval. 2…

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Q: Show that the vector-valued function r(t) = e−t cos ti + e−t sin tj + e−t k  lies on the cone z2 =…

A: Given vector valued function is r(t)=e-tcosti+e-tsintj+e-tk.