Let C be the curve given by r(t) = (t cos 1, 1 sin 1, 1). (a) Show that C lies on the cone x² + y² = z². (b) Sketch the cone and make a rough sketch of C on the cone.

Q: Find the arclength of the curve z 9 cos(9t), y = 9 sin(9t) with 0 <t< 36

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Q: Evaluate the arc length of the curve generated by the parametric equations below for t (-1,1) x(t)=…

A: The expression for the arc length of the parametric curve is; L=∫abx't2+y't2dt   Given: xt=3+3t2 ;…

Q: Let C be the curve given by r(t) = (t cost,t sin t,t). Sketch the cone and make a rough sketch of C…

A: Consider the given curve rt=tcost, tsint, t

Q: I. Let C be the parametric curve with parametric equations x=t(t² + 1)-¹/2 te [-3,3] ly: = 2t - 2…

A: There are two points at where the tangent line is horizontal

Q: Find parametric equations for the line tangent to the helix

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Q: Consider the parametric curve x(t) a) Sketch and identify this curve. b) Set up an integral to…

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Q: Let C be the curve given by r(t) = (t cost,t sin t,t).(a) Show that C lies on the cone x 2 + y2 = z2

A: Consider the parametric equation of the curve C : rt=tcos t,t sin t, t. To show C lies on cone…

Q: I. Let C be the parametric curve with parametric equations [x = t(t² + 1)-¹/2 te [-3,3] y = 2t - 2…

A: We have to set up the integral that gives the arc length of C.

Q: Find the length of the curve with parametric equations x = e' · cos t, y = e' sin t, with 0 <t <…

A: Given x = et.cos(t) y = et.sin(t)

Q: Let C be the curve with parametrization r(t) = cos(t)i + sin(t)j – 2tk. a) Find the parametric…

A: Let C be the curve with parametrization   rt=costi+sintj-2tk We have to find parametric equations of…

Q: 27. Show that the curve with parametric equations x = t cos t, y =t sin t, z = t lies on the cone z²…

A: To Show: i. Show that the curve with the parametric equationx=tcost, y=tsint, z=t  lies on the…

Q: Find the moving trihedral of the curve R(t) = (e^t)sin(t)i − (e^t)cos(t)j + (e^t)k at any t in the…

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Q: 3. Consider the parametric curve C given by * = te', y =t + cos t. (a) Find the equation of the…

A: Given: The parametric equation of the curve C is  x=tet, y=t+cost (a) To find: The equation of…

Q: 3. Find T (t) and a set of parametric equations for the tangent line to the helix given by ř(t) = 2…

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Q: Let C be the parametric curve defined by tan-1t y = } In(1+ t²) where t > 0. dy (a) Find and dx dx2…

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Q: Find the slope of the tangent to the parametric curve c(t) = (cos t, sin t) at the point (–1/2,…

A: We need to find the slope of the tangent to the parametric curve c(t)=(cos t, sin t ) at the point…

Q: Parameterize the curve r(t) = (2+ cos 3t, 3 – sin 3t, 4t) using arc length, s, as a parameter. Use t…

A: Given the curve rt=2+cos3t, 3-sin3t, 4t---1. Differentiating, we get vt=r't=-3sin3t,…

Q: A curve is defined by the parametric equations (t) = 8 cos(6t) and %3D y(t) 4 sin(-4t). Find . %3D

A: We have to find derivative

Q: Find the arc length of the parametric curve (sin² t)i + (cos²t)j, 0<t<5

A: Given parametric curve is r(t)=sin2ti+cos2tj,            0≤t≤π2 From the given curve,…

Q: Q1 Consider the plane curve defined by the parametric equations x = p + t sin(t),y= p² + t cos(t).…

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Q: Find parametric equations for the line that is tangent to the curve r(t) = e^t i + (sin t)j + ln (1…

A: The given curve is Differentiating with respect to t, we get Now, we find r(0) and r'(0) and

Q: Find the arclength of a parametric curve given below for 0 ≤ t ≤ π (x(t) = 3 cos t (y(t) = 3 sin t

A: Find the arc length of given parametric curve

Q: Find the slope of the tangent to the parametric curve c(t) = (cos t, sin t) at the point (-1/2,-V3).

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Q: A curve is defined by the parametric equations sin z dz, CoS Z dz, t 2 1. y= X= Find the length of…

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Q: Consider the curve C in R³ defined by the parametric equation c(t) V2 v2 T = (cos(t), sin(t), t).…

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Q: Let 3 x = cos' t, y = sin³ t, z = 7; 0 < t < r. Find the arc length of the parametric curve.

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Q: Find the arclength of the curve r = 5 cos(6t), y = 5 sin(6t) with 0 <t<. %3D

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Q: where C is the curve r(t) = (sin(t), cos(t), sin(2t)), 0 ≤ t ≤ 2

A: Given C is the curve                        r(t)=sin(t), cos(t), sin(2t), 0≤t≤2π and…

Q: B. Find the length of the circle of radius r defined parametrically by x = r(v1 – sin t) and y =…

A: We need to find the length of the circle which is parametrically defined as below.

Q: Let C1 be the curve with parametric equations x = cos t, y = sin t – t, where t E [0, 5). (a) Find…

A: This question can be solved using the concept of equation of tangent and length of curve when the…

Q: Find an equation of the tangent(s) to the curve with parametric equations x = t cost, y = tsin t…

A: Given- x=tcost , y=tsint To find - To find the equation of tangent to the curve with given…

Q: Find the length of the arc of the curve having the parametric equations x = t³3 and y = from t = 0…

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Q: Consider the parametric curve x(t) = a cos t, y(t) = b sin t, 0 <t<n for a # 0, b # 0 a) Sketch and…

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Q: Let C1 be the curve with parametric equations x = Cos t, y = sin t – t, where t e [0, ). (a) Find…

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Q: Let C1 be the curve with parametric equations x = cost, y = sint – t, where t e [0, 5]. (a) Find the…

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Q: Let C be the curve represented by R(t) = (cos 2t, 2√3t, 5 — sin 2t). Find the coordinates of point Q…

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Q: Find the length of the curve C defined parametrically by C: r(t) = e' cos t, y(t) = e sin t, 0<t< T

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Q: Consider the curve given by the parametric equations x (t) = sin t – t cos t and п dy 3 dx y (t) cos…

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Q: Find the length of the parametric curve I = cos(t) y = 1 + sin(t) from t = 0 to t =

A: Given- x=cost and y=1+sint To Find- The length of the parametric curve from t=0 to t=π. Identity…

Q: Show that the curve with parametric equations x = t cos t, y = t sin t, z =t lies on the cone z? =…

A: We have to check if x,y,z parametric equations lies on the given cone.

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: Find the length of the curve C defined parametrically by C: r(t) = y(t) = e' sin t, 0<t<n e cost,…

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Q: A curve is defined by the parametric equations T = sin t, y = 1 – cos t, 0<t< 27.

A: Given parametric equations is x = sin t; y = 1 - cos t, 0 ≤t≤2π b) Eliminate t and determine what…

Q: 1. Sketch the parametric curve x = (cos t) (1+ cos t), y= (sin t) (1 + cos t) , 0 < t < 27 .

A: we have been given the parametric curve x=(cos t)(1+cos t) and y=(sin t)(1+cos t) where 0≤t≤2π

Q: Consider the curve, y with parameter (, and function f in 3 dimensional Euclidean space with…

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Q: 3. Find the exact length of the parametric curve x = sin-1t, y = In /1 – t2, 0<t<

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Q: Find the length of the curve C defined parametrically by C:x(t) = e' cos t, y(t) = e' sint, 0<tS#…

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Q: Consider the parametric curve C defined by x = x (t) and %3D y = y (t). (a) How do you find the…

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