Show that the circular helix r(t) = (a cos t, a sin t, bt), where a and b are positive constants, has constant curvature and constant torsion. [Use the result of Exercise 63(d).]

Q: Find the curvature of y = sin(3T) at a = =.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Consider the helix r(t) = (cos(1t), sin(1t), −1t). Compute, at t = 픔: A. The unit tangent vector T =…

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Q: find the curvature of r(t)=⟨3t2, ln(t), t ln(t)⟩ at the point (3, 0, 0).

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Q: Express the curvature of a twice-differentiable curve r = ƒ(u) in the polar coordinate plane in…

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Q: Find the radius of curvature at origin at (0,0) y - x = x2 + 2xy + y2

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Q: Find the line integral with respect to arc length (8x + 9y)ds, where C is the line segment in the…

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Q: 2.Let a be a unit speed curve lying on a sphere center c and radius a > 0. Find the minimum value of…

A: Given That : Let α be the unit speed  curve lying on a sphere center c and radius a>0. To Find :…

Q: Find a vector tangent to the curve of intersection of the surfaces given by x² + y² + z² = 1 and 2²…

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Q: Find the angle of inclination of the tangent plane to the ellipsoid                           x2/ 12…

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Q: Find the curvature of the curve C: r (t) = (e, e', 2t) at t = 0

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Q: Find the curvature of r(t)=[e*,e2",e 3t] at t=0.

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Q: Obtain the curvature of x= k (0 – sin 0); y=k(1+ cos0) in terms of '0'.

A: This is the question from application of advanced Differential Calculus.

Q: Find the curvature of the plane curve with equation y (x, f (x)) on the curve, by using the…

A: We will find out the required value of curvature .

Q: at a unit-speed regular parametrised curve 3: (-7, 7)→ R² with T, N) has constant curvature k> 0.…

A: Sol

Q: Let V be a connected neighborhood of a point p of a surface S, and assume that the parallel…

A: Let the two points are p and q p, q∈S And the two curves are: α:I→S β:I→Sα0=p=β0αt1=q=βt2 And the…

Q: Exercise 10. Show that the curvature of a planar curve which satisfies the equation y = f(r) is…

A: To Show- The curvature of a planar curve, which satisfies the equation y=fx is given by…

Q: Let the the position vecor be R(t) = (-t,2 cos(2t),2 sin(2t)). Compute the curvature x(t).

A: The position vector Rt=-t,2cos2t,2sin2t. We have curvature k(t).

Q: Exercise 7. Show that the curvature of a circle of radius r is , and the curvature of the line is…

A: Show that the curvature of a circle of radius r is 1r and the curvature of the line is zero.

Q: 1) T(t) - (cost + 1 sint)i +(int-t cost) J +r'k %3D + rk

A: The arc rate at which the tangent changes direction as the point P moves along the curve is called…

Q: Show that curvature at an inflection point of a plane curve y = f (x) is zero.

A: We have to show that curvature at an inflection point of a plane curve y=fx is zero. We know that if…

Q: b.) show that brnormal of the muolute the unit *-ズ+(c-s)で g =dG) |k-s) K| K* where K is the…

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Q: Find the radius of curvature of 6y = x³ at the point (2,).

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Q: 4) Find the families of lines of curvature of the surface z = r + y? .

A: Given: z=x2+y2

Q: find the points on the surface + z² = 1 that have their tangent plane perpendicular to the vector…

A: For tangent plane of the surface f(x,y,z)=k at a given point normal vector to tangent plane  will be…

Q: Find the center of curvature of y = x3 at the point (1, 1) A. (-4, 8/3) C. (6, -2/3) B. (4, -8/3)…

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Q: Find the center of curvature of the curve y-Inx at (1, 0).

A: We Know that Centre of curvature for y=f(x) is κ=|f′′(x)|/(1+f′(x)3/2  -----(1)

Q: find the curvature of the plane curve at the point indicated.  y = tn, t = 1

A: Given y=tn

Q: Find the line integral with respect to arc length / Q = (0,3). (2x + 7y)ds, where C is the line…

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Q: Find the y coordinate of the center of curvature of th parabola at (2,2).

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Q: Find the center of curvature of the curve y=Inx at (1, 0). A) (4, 3) (в) (3,4) c) (3, -2) D (-2, 3)

A: Ans D

Q: Find the curvature of a superellipse defined by the parametric equations…

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Q: Find the line integral with respect to arc length ∫C(6x+4y)ds, where C is the line segment in the…

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Q: ) Calculate the curvature of the position vector f(t) = sin tāx + 2cos t āy + V3 sin tāz is a…

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Q: find the curvature of y=x^4 at (1,1).

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Q: Find a parametrization for the curve y = Vx with terminal point (0, 0) using the angle 0 in the…

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Q: Find the curvature of the plane curve y = -t at the point t = 2. к(2) -

A: Given:y = -t4differentiating with respect to t, we get:            y' =     d(-t4)dt…

Q: Find the curvature of the cardioid r=a(1+cosθ) at θ=0.

A: Given equation of cardioid is Derivatives of polar curve are

Q: Find the curvature and the radius of curvature at the point t = 0. x = e' cos(t), y = e' sin(t), z =…

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Q: 15. Show that the ellipse x = acos t, y = b sin t, a > b> 0, has its largest curvature on its maior…

A: If a curve is defined in parametric form by x(t), y(t), then, the curvature at point M(x, y) is…

Q: Find the curvature, torsion and tangent plane for the curve C : x = t sin(2t); y = t cos(2t); z = 2t…

A: We have to find the curvature, torsion, and tangent plane for the givne curve.

Q: Find a plane curve a(s) endowed with a unit-speed prametrization by SE (-0, 0) such that its signed…

A: The curvature of the curve αs is given by: αs=∫s0scos(θ(u))du, ∫s0ssin(θ(u))du-----(1) in which…

Q: Show that the Gauas and mean curvatures on , X = (u + v,u –- v,uv) at u = 1, v = 1 are K = 116 and H…

A: Given, X=u+v,u-v,uv u=1,v=1 , k=116 and H=182 We know that, E=xu,xuF=xu,xvG=xv,xv So, we need to…

Q: Show that for a plane curve the torsion T = 0.

A: We have to shown that, For a plane curve the torsion T=0

Q: x= 0, x= π &x= π

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Q: 4. Find the curvature of r(t) = (tv2, e', e-t) at the point (0,1,1).

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Q: Determine the radius of the curvature of the plane curve given by y In sec x at the || point (0,0).

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Q: Find the radius of curvature of 6y = x3 at the point (2,). %3D

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Q: Show that the ellipse x = a cos t, y = b sin t, a> b>0, has its largest curvature on its major…

A: The curvature for a parametrized curve c(t) = x(t)i + y(t)j is :

Q: Consider the plane curve parametrized by F(1) -i+ (In(com(1)J, Find curvature s(4).

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