a (t) = (t – sin t, t – cos t, cos t) 1. Check that whether a is in arclenght parametrization or not. 2. Compute T, N and B. 3. Compute K and T. 4. Is a a plane curve? Validate your answer. 5. Is a a straight line? Validate your answer. 6. Is a a helix? Validate your answer. 7. Write equations of osculating, normal and rectifying planes. 8. Compute the curve 3=a+ -N. 9. Compute whether or not tanget lines to 3 = a + EN intesect orthogonally with a. K

Q: Parametrize the following two curves in R³ and find their arclength. (1) the intersection of the…

A: As per Bartleby guidelines i can only answer one question, kindly upload rest part separately. Thank…

Q: See the circular helix with equation D(t)= (-15 cost,15 sin t,8t). 1. Look for the value of t such…

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Q: Q2/Find the curvature of the curve, r= (4 cos n i+4 sin n j)

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Q: * Find the curvature .12 (ähä 2) K of the curve ř (t) = (e' cos(t), e' sin(t), 3) at t = 0. %3D

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Q: (a) Find the curvature and the radius of curvature of the curve r(t) = 3 cos ti +4 sin tj + tk at t…

A: Given r(t) = 3 cost i + 4 sint j + t k

Q: . Find an equation for the circle of curvature of the curver(t) = ti + (sin t)j at the point (pi/2,…

A: The parameterized curve are x = t , y = sin t The derivatives are                           y = sin…

Q: 5. Given that r(t) = 3i + e* sin tj + e cos tk Find equations of the normal plane and osculation…

A: We have to  find equation of normal and oscillating plane of above curve.

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Q: 1. Let C be a curve with a vector equation r(t) = (4 cos(t), -4 sin(t),-3t) where 0<t<÷. What is the…

A: Since you have asked multiple question, as per our guidelines we are supposed to answer only one…

Q: See the circular helix with equation D(t)= (-15 cost,15 sin t,8t). 1. Look for the value of t such…

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Q: D O 1.909 Question 14 Find the length of the plane curve given below. r(t) = 5cos(t)i+5 sin(t)j, [0,…

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Q: Q2/Find the curvature of the curve, r= (4 cos n i+4 sin n j) .n=(In 2t)

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Q: Find the length of r(t)=[e"(cosbt), ea"(sinbt)] from t=0 to %=D1 (a and b are constants). This plane…

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Q: I Eliminate the ar bi fary constant )- y= Ax+ B sec x 2 2 %3D I. Find the difFCrntial equation +that…

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Q: 5. Do the following tasks using Mathematica. i) Find the length L of the curve y = 1 In a, 1<¤< 2…

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Q: Find the unit tangent vector T and the curvature k for the following parameterized curve. r(t) = (3t…

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Q: Consider the following. x = sin(2t), y = -cos(2t), z = 8t; (0, 1, 47) Find the equation of the…

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Q: 10. Find the arclength of the curve given by parametric equations x = r?, y = 2/2r, z = In(x²), 1<t<…

A: Given is x=t2, y=22t, z=ln (t2), i.e.z= 2 ln t,    1&lt;t&lt;2 The main objective is to compute the…

Q: Find the length of the curve r(t) = e³t cos(t), y(t) = et sin(t) for 0 < t < 10. Give an exact…

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Q: 3) Find the radius of curvature for the parametric сurve e-t cos t y = e-t sin t.

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Q: 2. Consider the parametric equation r(t) = (1 – t°, t(1 – t)) where –1 <t < 1. (a) Plot this curve…

A: The solution is given as follows

Q: To illustrate the fact that the numbers we get for length do not depend on the way we parametrize…

A: Given,

Q: 1/B) Find the length of the cardioid 'curve: r = 1 + cose

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Q: The helix F(t) = 4cos(t)i + 2 sin(t)j + 5tk is shown below. By setting the last component to zero…

A: Given that,Equation of a helix isr→(t)=4cos(t)i^+2sin(t)j^+5tk^The projection of this curve into the…

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A: Answer

Q: 1. Find the moving trihedral and the curvature at any point of the curve R(t) = cos t î + sint ĵ+ k.

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Q: Find the length of the curve x(t) = e" cos(t), y(t) = e" sin(t) for 0 <t < 3. Give an exact answer,…

A: Given curve: x(t) = e^(5t) cos(t) y(t)= e^(5t) sin(t)

Q: Q1: Consider the curve in E' parameterized by C:(t) = 1- Find the arc-length of the curve r(t)? 2-…

A: Given equation of the curve is rt=cos5t, sin5t, 12t

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Q: 1. Given the parametrized curve (helix) a(9) = (a cos , a sin, b), seR, a(s) C C wodd where c² = a²…

A: For your demand i solve your question no c and d

Q: Q2/Find the curvature of the curve, r= (4 cos ni+4 sin n j) .n=(In 2t)

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Q: Find an equation of the plane tangent to the following surface at the given point. 13 cos (xyz) =- 2…

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Q: * Q7 The curvature for the curve R(cos t +t sin t)i + (sint - t cos t)j is a) 1 b) t2 +1 c) (cos t -…

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Q: Consider the curve C (a helix) parametrized by r(t) = (cos t, sin t, t), 0<t<4T. a) Compute the…

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Q: 6. Consider the same curve from the last problem, C parametrized by T(t) = (2 cos t, –1,2 sin t)…

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Q: Consider the curve ´9 F(t) := 27 cos(t), 27 cos sin(t) 10 t, 27sin 10 sin(t) + 162 cos 10 0<t<T. -…

A: Given that, r(t)=27cost,27cos9π10sint-162sin9π10t,27sin9π10sint+162cos9π10t Take the derivative for…

Q: 2. For each of the following, sketch the parametrized curves. Be sure to indicate the direction of…

A: Firstly for solving subpart (a), substitute the value of t = y in f(t) = t.And simplify to get the…

Q: 5. Find the length of the curve r = e", - n<0 < n given in polar coordinates. V5 sinh(2z) b)V5…

A: The arc length of polar curve r=fθ between θ=a and θ=b is given by the formula L=∫abr2+drdθ2dθ We…

Q: 2. The Cornu (or Euler) spiral, shown below, is defined by r(s) = (f* cos (u²) du, f *cos (u²) du,…

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Q: q (s) = ( a (9) Show that q (s) = cos (ø (u)) du, sin (ø (u)) du So is a unit speed curve with k± dø…

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Q: (a) Find a parametrization y of the curve shown below, where 0 < 0 < n. (b) Find -dz. (2z – 1)3 (You…

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Q: Consider the following. x = sin(3t), y = -cos(3t), z = 6t; (0, 1, 2n) Find the equation of the…

A: Given :- The given curve x = sin 3t ,  y =-cos 3t  , z =6t   ;  0,1,2π To determine :- The equation…

Q: 2. the area of the curve r-a cos 2 0

A: The curve contains two petals,

Q: 16. Let C be the curve given by r(t) (t cos(t), t sin(t), t). Show that C lies on the cone x2 + y? =…

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Q: 1. The line parametrized by x 7, y = 5t, z = 6+t is parallel to the x-axis. 2. The parametric curve…

A: (1)

Q: Find an arc length parametrization of the curve r(t) = (1/1+t^2,-1,t/1+t^2)

A: Find an arc length parametrization of the curve r(t) = (1/1+t^2,-1,t/1+t^2)

Q: 10. Find the curvature K of the given curve. r(1) = ti +2sin(t)j+2cos (1)k

A: Curvature κ of the given curve r→(t) is T→'(t)r→'(t) where T→(t) = r→'(t)r→'(t).

Q: Find the length of the curve #(t) = et cos(t), y = e* sin(t) for 0 <t < 5. %3D %3D Give an exact…

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Q: Find an equation for the circle of curvature of the curve r(t) = t i + (sin t)j at the point (pai/2,…

A: The parameterized curve are x = t, y = sin t.The derivatives are,

Q: 1. Consider the circle in the ry-plane given by the parametrization x = 2 cos t, y = 1 +2 sin t, 0…

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