Prove that any polar curve r= f θ with the propertythat the angle ψ between the radial line and the tangentline is a constant must be of the form r = Cekθ where cand k are constants.

Q: Let F(r, y) = (ye,re"") and let C be the closed plane curve given in polar coordinates by r =1+ecos,…

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Q: Let o(0,ø) = (cos 0 sin ø, sin 0 sin ó, cos 6) be a parametrization of the unit sphere. Show that…

A: Given, σθ,ϕ=cosθsinϕ,sinθsinϕ,cosϕ ,0<t<π2...1γt=σln cotπ4-t2,π2-t...2 The equation (1) can be…

Q: Which of this is a parametric representation of the unit circle? * O x=sin t, y=cos t O x=3sin t,…

A: The solution of the problem is given in the next step.

Q: Find the length of the following two-dimensional curve. r(t) = (cos t, sin , for 0sts5

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Q: Consider the curve given by the parametric equations = 3 cos 0 and y = 3 sin 6, 0<0<T. Which of the…

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Q: Consider the parametric curve defined by x=3+5t and y=1+3t Over which values of does the curve has a…

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Q: Find the length of the curve given in polar form as r= e,- <o<a a)2/2 sinh(7) b)sinh(") c)2 sinh(7)…

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Q: the xy-plane, centered at the origin, oriented clockwise so that the point (5,0) corresponds to t =…

A: we have to find parametrization of circle.

Q: Consider the polar curves C1 : r = −3 sin(2θ) and C2 : r = 3 sin θ. 1. What is the equation of the…

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Q: Convert the following parametric curve into Cartesian form: x = 3 sin t, y = 4sin t

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Q: r = 2 sen 0, 0 = "/6 15.

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Q: Consider the polar curves Cı :r = 3 sin 20 and C2 : r = 3 cos 0, and let R be the shaded region as…

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Q: Eliminate the parameter t from the following. x = 4 sin t, y = 4 cos(t) Sketch the graph of the…

A: The given question has been solved in detail

Q: A Cartesian equation for the parametric curve x=24sin?e, y=21cos20, n ses 31/2 is given by O…

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Q: Given the line segment from the point (13, 7) when t = 0 to the point ( – 1, – 9) at t = 1, find the…

A: When t=0 then point (13,7) and t=1 then point (-1,-9)

Q: A curve is defined by the parametric equations z (t) = e-* and y (t) = e*. What is in terms of t ? A…

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Q: Sketch the plane curve r(t) = a cos3 ti + a sin3 tj and find its length over the given interval [0,…

A: The curve is r(t)=a cos3ti + a sin3tj   Interval is 0, 2π

Q: Find the length of the curve. r(t) = (3t, 3 cos(t), 3 sin(t)), -3 sts 3 %3D

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Q: Prove that the curve y:(-,) → R³ is regular with factor H and determine whether the image y (t) = (2…

A: To determine : the      included in a plane in   or not. Here,…

Q: Let C be the parametric curve defined by z= tan" 120 a. Find and . Do NOT eliminate the parameter.…

A: From the given problem: x=tan-1ty=12ln1+t2,t≥0 We know that, the chain rule: dydx=dydt×dtdx

Q: Determine the exact length of the curve for the parametric equation x = tsint and y = tcost for 0 ≤…

A: To find the exact length (arc length) of the curve for the parametric equation x=t sin(t), y=t…

Q: Find the length of the following two-dimensional curve. r(t): = (cos t, sin t) , for 0stsT The…

A: The formula for finding the length of the curve is explained in step 2 with a general observation…

Q: Consider the parametric curve defned by r=3+5t and y=1+ Over which values of y does t curve has a…

A: Given problem:- Consider the parametric curve defect by   x = 3 + 5t and y = 1 + 3t   Over which…

Q: Find the length of the parametric curve defined over the given interval. A) B) 3 3 2 C) D) 1

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Q: Given the cardioid r= 3(1+cosθ) 0 ≤ θ ≤ 2π Determine in wich points the cardiod has vertical…

A: Differentiate the equations r=31+cosθ and x=rcosθ with respect to θ, to find drdθ and dxdθ.…

Q: Find the length of the following two-dimensional curve. π 3 r(t) = (3 cos ³t, 3 sin ³t), for Osts- 4

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Q: Show that every vertical line in the xy-plane has a polar equation of the form r = a sec u.

A: The relation between polar coordinates r,u and rectangular coordinates x,y is x=rcosu…

Q: Find the length of the following two-dimensional curve. r(t) = (2 cos t,2 sin t), for 0stsT

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Q: 9. Find the exact length of the parametric curve for the given interval. x = 4t 2 - 3, y = 3t, 0 St5…

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Q: (a) Find the slope of the tangent line to the given polar curve at the point specified by the value…

A: We have to find the slope of the given polar curve at the point specified.

Q: Find the radius of curvature of the curve generated by the position vector r (t) = (-40 sin(t),–9t,…

A: Given, r(t)=-40sint,-9t,40cost r'(t)=-40cost,-9,-40sint r'(t)=-40cost2+(-9)2+-40sint2=41…

Q: Find T(t) and then find a set of parametric equations for the tangent line to the helix given by…

A: Consider the equation of helix given by r(t). Find the value for t , derivative at t and modulus of…

Q: Find all points on the polar curve r = 1 − (1/√2)sin(θ) where there is a horizontal tangent.

A: Here we have to find all points on the polar curve r= 1- (1/√2)sin(θ) where there is a horizontal…

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: Identify the symmetries of the curves in. Then sketch the curves in the xy-plane. r = 1 + 2 sin θ

A: Explanation:  Given that equation of the curve:                             Check the symmetries…

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: Consider the polar curves Cı : r = 3 sin 20 and C2 : r = 3 cos 0, and let R be the shaded region as…

A: It is known that sin2x=2sinxcosx. Using the zero factor property, it can be written that: if ab=0,…

Q: Find the length of the following three-dimensional curve. r(t)= (sin t, cos t, 2t), for 0sts4

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Q: Find the length of the curve C defined parametrically by C: x(1) = e' cos t, y(t) = e' sin t, 0 <t <…

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Q: Determine the Cartesian equation of the tangent line to C1 at the point where 0 = π/2. Given: polar…

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Q: The parametric representation of a curve is given as 3 3 r(t) = (V i+ + k. 12 12 (a) Eliminate the…

A: We are given that r(t)=32ti+t312+1tj-312t3k. From this, we get x=32t, y=t312+1t, and z=312t3. From…

Q: If the parametric curve starts at (5, 10) when t = 0 and ends at (3, 8) at t = 1, then find a, b, c,…

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Q: Which of the following is the length of the polar curve r = 0 , 0<0 <1? V2 – In (v2+ 1) 2 V3 – In…

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Q: Find T(t) and a set of parametric equations for the tangent line to the helix given by 7(t) = 2…

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Q: 2. Determine the ry-coordinates of the points where the following parametric equations will have…

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Q: Sketch the curve r= cos 2θ

A: Symmetric test for polar coordinates r=f(θ): If f(θ)=f(-θ) then graph is symmetric about polar…