a) Find a plane curve whose signed curvature is given by k,(s) where a is a positive constant and s> 0. 2as

Q: Find the curvature of r(t) = at t = ,.

A: Given: r→t=3cost, 3sint, t Calculation: Compute the Curvature at t=π2 as follows. It is known that…

Q: Calculate a curvature of the curve r(t) → = ⟨t,t2,t3⟩ at point (4,16,64)

A:

Q: 04. Show that the Gauas and mean curvatures on, X-(u+v,u-v, uv) at u=1, v=1 are K=1/16 and H= JI.…

A: Given: 1) X=(u+v,u-v,uv) at u=1,v=1. 2) The paraboloid is xy=aZ.

Q: Find the curvature and radius of curvature of the plane curve at the given value of x.

A: Given that: The equation is y = e-x4 and the point x = 8 .

Q: Find the curvature of the plane curve at the pointindicated. y = e^t , t = 3

A: y=et,  t=3 Differentiating with respect to t, we get dydt=et Again differentiating with respect to…

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 curvature and radius of curvature of the plane curve at the given value of x. y = xn, x =…

A: Given ,

Q: Find the curvature and radius of curvature of the plane curve at the given value of x.

A: We have,   It is known that radius of curvature is given as, Here K is curvature of the curve.…

Q: Q1. Find the curvature of the line whose parametric equations are given as: x = -2+t, y =3+ 2t, z…

A:

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 the curvature of the curve given by r(t) = 2ti + t2j −( 1/ 3) t3k.

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Q: Let P be a point on a regular surface S and let f and g be two curves on S passing through P and…

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Q: Find the curvature of the curve r(t) = eti + 4tj at the point P (1, 0) .

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Q: Show that the regular curve a = a(s) is a straight line if and only if its curvature k is dentically…

A: WE HAVEDB/DS = τNWHERE AT ANY POINT P[X,Y,Z] ON THE CURVEB IS UNIT VECTOR CALLED BINORMAL TO THE…

Q: Show that along the plane curve x = #1(t)e, +2(t)e2 the curvature is 4.30. [(a)2(5)213/2 a0 + 3

A: The curvature K at a point of a curve defines the speed of rotation of the tangent of the curve at…

Q: 21. r(t) = t³ j + t² k %3D

A: Given  r(t)=t3j+t2k

Q: Find the curvature and radius of curvature of the plane curve at the given value of x. y = 9x 4…

A: Given curve is, y=9x-4x Differentiating this with respect to x we get,  dydx=9-4ddx(1x)⇒dydx=9+4x2…

Q: If a curve has the property that the position vector r(t) is always perpendicular to the tangent…

A: given that, a curve has the property that the position vector r(t) is always perpendicular to the…

Q: 3. Find the curvature of f(x) = e" at point (0, 1). Then find the equation of the osculating circle…

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Q: Find the curvature and radius of curvature of the plane curve at the given value of x. y = e X/4, x…

A: Given y =e^(-x/4)  ,x=8 Now we have to find curvature and radius of curvature at x=8

Q: 4 Find the curvature k of the plane curve y = x +- at x = 1. K =

A: Given curve is , y= x + 4x  at x=1

Q: Q2 Find the radius of curvature of yv5 = x' at the point (x, y,). Hence, show that the radius of…

A: Use formula of radius of curvature

Q: 4. Let r(s) be a unit speed curve with a curvature of zeroc fo. prove that the image r lies in a…

A:

Q: |x'y" -y': y'a - y'= es K(t) = (x2 + у'?

A:

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: 8. Find a plane curve a(s) endowed with a unit-speed prametrization by S E (-∞, 0) such that its…

A: Question: Find a plane curve αs endowed with a unit-speed parametrization by s∈-∞,∞ such that its…

Q: 2-Show that the regular curve a = a(s) is a straight line if and only if its curvature k is…

A: Here we have to show that a regular curve is a straight line if and only if, its curvature is…

Q: Find the currvature and radius of curvature of the plane curve at the given value of x. y%3D 5x+ 7.…

A:

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

A:

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

A: Given y=tn

Q: Find the curvature and radius of curvature of the plane curve at the given value of x. y =e-x/4, x =…

A: The given curve y=e-x4  ,  x=8

Q: Find the curvature of r(t) = √√7ti + etj + e-k at point P(0, 1, 1).

A:

Q: At what point does the curve have maximum curvature? y = 4ex (х, у) %3D What happens to the…

A:

Q: 1. Find the equation of the circle of curvature for the following curves: 1.1 x+y =x + y +x at…

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Q: Find the curvature of r(t) = at the point (1,0,0)

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Q: Consider the curve given by the equation y2+x3=0. Find its curvature at the point (−1,1).

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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: 15. y= 2x +3 at x = 27

A:

Q: Find the radii of curvature and torsion at a point of the curve r? + y2 = a², x² – y? = az. %3D %3D

A: Given curve is x2+y2=a2x2-y2=az

Q: 2. Show that if a : [a, b] → R" is a regular parameterization of a curve then the curvature at α(t)…

A: Let,  is a regular parameterization of a curve To show the curvature at  is

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 of the curve. r(t) =<2t, - 3sin t, 3cost)

A:

Q: Find the curvature and radius of curvature of the plane curve at the given value of x. 32 y = 9x, x…

A: Given that Y= 9x - 32/x ; x= 4. And we find curvature and curvature of the plane at give x value.…

Q: Find the curvature and radius of curvature of the plane curve at the given value of x. y = x3, x = 2

A: Consider the following equation of curve: y=x3 Differentiate y with respect to x: dydx=ddxx3y'=3x2…

Q: 10.) At what point does the curve have maximum Curvature? y=In(x). K(x)= f"|

A: firstly find f, f' and f'' . And substitute all these values in given curvature formula.  After that…

Q: Let r (t,t,t) be a curve in 3-dimensional space having unit tangent and norma V2 vectors u and n.…

A: We will use formula of curvature of a curve to solve it.

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: Consider the paraboloid paremterized by o(u, v) = (u, v, u² + v²). 1. Find the second fundamental…

A: Consider the given equation of circle, γt=sint,cost,1 The formula for normal curvature is defined…

Q: Find the radius of curvature at any point x = - of the curve

A: Given,             y=4sin2x-sin4x at  x=π4

Q: .3 Find the radius of curvature of y=x' at x=1,

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Q: Find the curvature K of the plane curve y 3 at x = 3. 2х + K =

A:

Q: 8) If curvature of a space curve at (2,3) is 8 and curvature at (7,1) t=7 is 1, what can be said…

A: Given that curvature of a space curve at 2,3 is 8 and curvature at 7,1 is 1.

Q: Show that the ellipse x = a cos t, y = b sin t, a&gt; b&gt;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: Example: Find the curvature of the curve, where a&b> o r= (acos ti + asintj+ btk)

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Q: What is the radius of curvature of the polar curve r = 02 at 0=3/2?

A:

Q: 1) Calculate the curvature of the position vector 7(t) = sin tax + %3D 2cos tay + V3 sin tāz is a…

A: In this question we have to find curvature and radius of curvature.

Q: At what point does the curve have maximum curvature? y = 7 In(x) (x, y) = What happens to the…

A: Given,              y = 7 ln(x)

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

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

A: