Show that the regular curve a = a(s) is a straight line if and only if its curvature k is dentically equals zero .Also, show that if a = a(s) is a circle with non- zero radius a hen its curvature is constant. Finally,show that a = - is identically equals zero. a(s)is a plane curve iff its torsion

Q: Find the curvature k of the curve C : 16 =1 at the point (0, 3)

A: 1. convert the equation into parametric form 2. use formula for curvature written below

Q: Find the curvature of the parabola given byy = x −(1/ 4)x2at x = 2. Sketch the circle of curvature…

A:

Q: At what point does the curve have maximum curvature? What happens to the curvature as x→∞ 30. y=1n x…

A: As per our guidelines, we can answer only one question in a single request. Kindly re-post the other…

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 ordinate of the center of curvature of the curve y² = 4x at point (2, 4) from the x-axis.

A: Given option are wrong, option c should be '-16'.

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

A: Given ,

Q: What is the curvature of the path of the motion at t=1

A: Since velocity is given we need to calculate  r''(t) only at t=1

Q: Show that the parabola y = ax2, a ≠ 0, has its largest curvature at its vertex and has no minimum…

A: Using the relation,

Q: (b) Find the circle of curvature at (0, 1)of the curve y = x3 + 2x² + x +1

A:

Q: Determine the maximum curvature for the graph of f(x) = 6 In (3x). The maximum curvature is at x =

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 K of the curve at the point P. r(t) = 2ti + 5t´j, P(-2, 5) K =

A:

Q: 4. Does there exist a simple closed unit-speed plane curve with curvature less than or equal to 2…

A: In a simple closed unit-speed plane curve, curvature =1r According to the question, Area of the…

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: ANSWER:

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

A:

Q: The curvature of any line with slope m in 2-space at any point on it is TTL

A:

Q: a) Find a plane curve whose signed curvature is given by k,(s) where a is a positive constant and s>…

A: The given signed curvature is ks(s)=12as

Q: 22. Find an equation for the circle of curvature of the curve r(r) = (2 In )i- [t + (1/]j, esIse, at…

A: The formula for the curvature is given by  κ(t)=r⇀'(t)×r⇀''(t)r⇀'(t)3 In order to use this formula,…

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: Q2 Find the radius of curvature of y5 =x at the point (x,v.). Hence, show that the radius of %3D…

A: The curve is y5=x3

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

A:

Q: Find the curvature K of the curve at the point . r(t) = 2ti + 4tj, P(-2, 4) K =

A:

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: Suppose I → R³ is a regular curve -c3 with zero curvature anywhere zero, and then the H-curve holds…

A: A given regular curve with zero curvature is -c3 We change in terms of t, then the regular curve is…

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 curvature k of the circle C: (T – 2)² + (y + 3)² = 25 at the point (5, 1).

A:

Q: 3. Find the radius and center of curvature of the parabola y=x²-4x+4 at any point (x,y)on the curve.

A:

Q: Find and sketch the circle of curvature at the point on the curve y = x2 where the tangent line has…

A:

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

A: Consider the provided equation, y=sin2x,   x=π4 Find the curvature and radius of curvature of the…

Q: Find the curvature of the curve r(t) = (3 + t, – 4+ 2t, – 2 – t) at the point t = - 8.

A:

Q: Find the curvature x for the curve y = x - K = x² at the point x = 2.

A:

Q: Find the curvature of the curve r (t) = (- 5+ 4t, 1+ 3t, 5t) at the point t - 7.

A:

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 the curve r(t) = 3ti + 2t2j at the point P(−3, 2) .

A: rt=3ti+2t2j Comparing with rt=xi+yj, we get x=3t & y=2t2 Now put x=-3 & y=2, we get -3=3t…

Q: 1. Find the points on the parabola y 8x at which the radius of 125 curvature is 16

A:

Q: 4) Find the families of lines of curvature of the surface : =+ y² .

A: Given: z=x2+y2

Q: Find the curvature k of the circle C:(x – 2)2 + (y +3) = 25 - at the point (5, 1).

A: circle equation is x-22+y+32=25 and point 5,1find curvature k

Q: The curvature of y =x is

A: In this question we have to find the curvature of the given function which we can find out by using…

Q: Find the curvature K of the curve C x² 1 16 9. at the point (0, 3)

A:

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: The curvature of y=x? is:

A: The curvature of a curve is defined as the amount by which a curve deviates from being a straight…

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: Given the curve f (t)= (2t2 - 3, 3 - t, 3 + 2t?). Find the equation of the line parallel to the…

A: Consider the equation as shown below: Here, curve ft =2t2-3, 3-t, 3+2t2 Here, the object is to find…

Q: Show that the parabola y = ax2, a  0, has its largest curvature at its vertex and has no minimum…

A:

Q: Find the curvature k of the circle C : (x – 2)² + (y+ 3)² 16 at the point (2, 1).

A: Use the formula of curvature to solve this problem.

Q: (a) Prove that curves a, ß: I → R² are congruent if k̟ = Kg and they have the same speed.

A: Part (a):   If any two continuous function defined on the interval I, and the two curves are…

Q: 2) Find and sketch the circle of curvature at the point on the curve y = x² where the tangent line…

A:

Q: Find the curvature of the parabola f(x)= ax +bx+c at its vertex 2a ’

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: Determine the maximum curvature for the graph of f(x) = 9 In (8x). The maximum curvature is at x=

A:

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: Find the curvature k of the curve C: 16 + =1 at the point (0, 3)

A: To find the curvature of the curve at the point (0,3).

Q: Find the curvature K of the circle C: (x - 2) + (y+3)² = 16 %3D at the point (2, 1).

A: Given: Equation of circle C: (x-2)2+(y+3)2=16 ⇒ (x-2)2+(y-(-3))2=42 We know that circle equation is…

Q: Find the curvature of the plane curve y = 4t³ at the point t = 2.

A:

Q: The curvature of any line with slope m in 2-space at any point on it is 0

A:

Q: Find the curvature of the parabola y = x² at the origin.

A: Given :-   parabola y=f(x) = x2  To find:-  curvature of parabola at origin(0,0)

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

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

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: Determine the maximum curvature for the graph of f(x) = 7 In (2x). The maximum curvature is at x=

A: We are given function f(x)=7ln(2x) of the graph Formula for curvature is κ(x)=|f′′(x)(1+(f′(x))2)32|…

Q: 2) Find and sketch the circle of curvature at the point on the curve y x where the tangent line has…

A:

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 :