Find the equation of the osculating, normal, and rectifying planes for a (t) ( cos t, – sin t, cos t) at t= n. COs

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

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Q: 4. Find the moving trihedral of the curve f(t) at the indicated value: †(t) = cos? t î + sin² t ĵ…

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Q: The parametric form of the tangent line to the image of f(t) = (2t², ?,t +2) at t = 1 is L(t) =

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Q: Let P(cos t, sin t) be a point on the unit circle x2 + y2 = 1 in the first quadrant. Show that t is…

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Q: Describe the shape of the curve x = 3 cos t, y = 3 sin t .

A: Given:

Q: Find the equation of the plane perpendicular to the curve r(t) = (sin(2t), t, cos(2t)) at the point…

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Q: Find the exact length of the curve (b) x = e' cos t, y= e' sint, 0<t<T

A: Length of the curve for parametric equation is given by:

Q: Find the length of the curve. r(t) = cos(6t) i + sin(6t)j + 6In(cos(t)) k, 0 <t< n/4

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Q: Consider the curve C shown in the figure, parameterized by: C: r(t) = (1 - cos(t), 1- sin(t), 1 +…

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Q: Given the function y = f(x) represented by the parametric equations x = - 1 and y = t + 2, determine…

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Q: Find the lengths of the curves x = cos t, y = t + sin t, 0 … t … pi

A: Solution:Given   x= cos t,      y= t+sin t,   0≤t≤πFormula:Lenght=∫dxdx2+dydt2dt1+cos 2x=2 cos2…

Q: Find the parametric equations of the tangent line LT to the curve C:r (t) = 2 cos(t)i + sin(t)j +…

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Q: Evaluate [x sin(z)ds where C is the curve given by the equations x= cos(t),y= sin(t), z =t, 0<t<

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

A: Differentiating under the sign of integral.

Q: the length of the parametric curve a =t+cos(t), y =t – sin(t), 0 < t < 2n ?

A: The length of the curve formula is

Q: Consider the parametric curve C that is defined by C: x=t^3, y=t^2+t, −1≤t≤3. (a). Find the…

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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: Find the length of the curve r(t) = (cos 3t, sin 3t, 2t3/2),0 sts 3. %3D

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Q: Consider the parametric curve given by x = t2 + 5, y= e* est + 2t. 3t Find the slope of the tangent…

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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: 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: Find the equation of the tangent plane R(a, b) = to the surface at the point that corresponds to a…

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Q: Find the exact length of the curve. 43. x = t sin t , y = t cos t , 0 ≤ t ≤ 1

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

A: Use the formula for the length of curve in parametric form.

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: Let C be the curve parameterized by Y: [0, b] R³, y(s) = (12 cos(s²), 12 sin(s), 9 s²). Find b>0…

A: The given problem is to find the value of b for which the given length of curve of the given…

Q: Find a rectangular-coordinate equation for the curve by eliminating the parameter. x = 2 cos(t), y =…

A: Given parametric equations are

Q: Find the lengths of the curves x = cos t, y = t + sin t, 0≤ t ≤ π

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Q: Find the length of the curve. r(t) = cos(6t) i + sin(6t)j + 6In(cos(t)) k, 0stsa/4

A: In this question we have to find the length of the curve.

Q: Find the length of the following curve using methods for parametric curves. x = 4 sin t y = 4 cos t…

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Q: Find an equation of the tangent to the curve at the point corresponding to the given value of the…

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Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Find an equation for the projection of…

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Q: Find the exact length of the curve. y = In(cos(x)), TC 0 < x<.

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Q: Find parametric equations for the line tangent to the helix r(t) =(sqrt(2)cos t)i + (sqrt(2) sin t)j…

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Q: Consider the following. x = sin(t), y = csc(t), 0 < t < A/2 %3D %3D (a) Eliminate the parameter to…

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Q: 8. Given the vector r(t) = (sint,cos?, In(cos?)), find the equation of the tangent line in…

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Q: Find length of the curve described by the parametric equations x = T sin t, y = r cost, e <t< T. А.…

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Q: Let Sketch the curve and find the length of this curve. r(t) = (t, 3 cost, 3 sin t), 0≤ t ≤ 2π.

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Q: Find an equation of the tangent(s) to the curve with parametric equations x = 1 + √t, y = e^(t^2)…

A: Given parametric equation of curve x=1+t   y=et2we want to find equation of tangent at  ( 2, e)

Q: Consider the following. x = sin(6t),  y = −cos(6t),  z = 24t;    (0, 1, 4?) (a) Find the equation…

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Q: Consider the parametric curve x = sint + cos t, = cos² t 0 <t < 2n п Find an equation for the…

A: We find dx/dt and dy/dt from the given equations

Q: Find the length of the curve. r(t) = cos(5t) i+ sin(5t)j + 5ln(cos(t)) k, 0<t< n/4

A: r(t)=cos(5t)i+sin(5t)j+5ln(cos(t))kWhere, 0≤t≤π/4

Q: Consider the parametric curve C: x=t³ + 1, y = t² – t,0 <t < 3. Find the equation of the tangent(s)…

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Q: Find parametric equations for the line that is tangent to the given curve at the given parameter…

A: Consider, r(t)=lnt4i+t-4t+8j+t lnt4kWe first evaluate r(t) at t=t0=4r(4)=ln44i+4-44+8j+4 ln44k…

Q: The parametric form of the tangent line to the image of f(t) = (3t2,,t – 2) at t = 2 is L(t) =

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Q: Find the length of the following​ three-dimensional curve. r(t)=cos t, sin t, 3t​, for 0≤t≤2

A: Given that rt =cost,sint,3t  for 0≤t≤2 find the length

Q: Find an equation of the tangent(s) to the curve with parametric equations E = t cos t, y =t sin t nt…

A: follow next step

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…