The curve in the figure below is parametrized by F(t) = (t³ – 3t) · i + (3t² – 2) · j, for -00 < t<∞. 30 20- r=(8-3t)i+(3t²-2)j 10+ -30 -20 -10 10 30 Find the curvature function k(t), and show that curvature is maximized when t = 0.

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 radius of curvature of a parabola y² - 4x = 0 at the point (4,4). 2

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Q: Find a formula for the curvature of the function P= nrt/v-nb - (an^2)/(v^2) ,where n, R, a, b,and T…

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

A:

Q: 4. At the point (0, 0) the graph of y 4.x2 has greater curvature than does y 2x2.

A: These equation represents parabola,then Radius of curvature: K=  y'' /([1+(y')²]^3/2)

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

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

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Q: Find the extreme values of the function z= &r+2ry-3r+y+1 and show that only one of the two extrema…

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

Q: 3. (a) Find the curvature (t) for the vector-valued function (t², 2t, In t), t > 0. (b) Find where…

A: Basic identities of differentiation and vector calculation are used

Q: A point moves along the intersection of the elliptic paraboloid z = x² + 3y² and the plane y = 1. At…

A: Point moves along the intersection of the elliptic paraboloid z=x2+3y2 and the plane y = 1. We have…

Q: Find the curvature of the curve r(t). r(t) = (3 + 6 cos 9t) i - (8 + 6 sin 9t)j + 6k Group of…

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: Exercise 3.2.10. Using K = (In - m²)/(EG – F²), show that the Gauss curvature of the R- sphere S²(R)…

A:

Q: Consider the curve C parametrized byx = (t^2+1)/(t^2-1) and y = (2t)/(t^2-1) for all t in (−∞, −1 )…

A: Given:

Q: Find the points on the surface 2a2 + ly² + 522 = 1 at which the tangent plane is parallel to the…

A: We will use the following method for finding the two required point.

Q: Find the center of curvature of the curve y =-(e*+e) at (0, 1). (А) (0, 2) в) (1, 1) с) (0, 3) D (0,…

A:

Q: Find the point on the curve at which the curvature is a maximum and (b) find the limit of the…

A: (a) To find the point on the curve at which the curvature is a maximum  (b) find the limit of the…

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

A: Sol

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

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.

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Q: A particle is moving in space with position function R(t) = (cos(4t), sin(4t),5 – 3t) Find the…

A:

Q: Find the curvature K of the curve at the point P. r(t) = 1/2t2i + tj + 1/3t3k,    P (1/2, 1, 1/3)

A: The curvature is given by kt=r't×r''tr't3. Given that rt=12t2i+t j+13t3k. Obtain r't as follows.…

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

A: Given: z=x2+y2

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

A: Given: z=x2+y2

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

A:

Q: 2 Find the points on the curve ax+ 2bxy + ay = c,ay = c, 0<a<b<c,whose distance from the origin is…

A:

Q: I) Find the extreme values of the function : &r+ 2ry-3x+y +1 and show that only one of the two…

A:

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

A:

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

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: 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: 8. Find the curvature of the vector-valued function r(t) = at the point t = -1.

A: given function is rt=3t2,-4t,1t find the curvature at t=-1 k=r→'t×r→''tr→'t3

Q: Find k (curvature) and t (torsion) for curve C represented by F(t)=(e^t,-e^t,(2t)^1/2)

A:

Q: 2. Let a(t) = (3t-t, 3t2, 3t + t), for te R. %3D (i) Calculate the equation of the osculating plane…

A: Introduction: The formula of curvature of and torsion is given by, κ(t)=α"(t)…

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

A: The answer is given in handwritten format --

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: ( 1 ) f α a(s)is a regular differentiable curve, then show that :- (a) kt = -(dT/ds dB/ds), (b)T =…

A: (1. (i. Let  be a differentiable regular curve. Then there exists three orthogonal unit vectors as…

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 the points on the curve of intersection of the ellipsoid x² + 4y² + 4z² = 4 and the plane x –…

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

A:

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: Find the curvature of the space curve. r(t) = (6t sin t+ 6 cos t)i + 6j + (6t cos t-6 sin t)k 1 O A.…

A: Given space curve

Q: Find the gaussian and mean curvature functions of the surface m given by the equation y3=y1y2 in R³…

A: First parameterized the equation.

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: The curvature of y = x2

A: so we have to find curvature of y =  x2. we use the formulae for curvature  k(x) =…