1) Calculate the curvature of the position vector F(t) = sin t ax+ 2cos tay + V3 sin tāz is a circle, also find its radius.

Q: Let r= (3v/2t, 3ť, /2t³) be a curve in 3-dimensional space having unit tangent and normal vectors u…

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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: 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: Show that the ellipse x = a cos t, y = b sin t, a 7 b 7 0, has its largest curvature on its major…

A:

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: Exercise 8. Show that the sum of the angles of a geodesic triangle on a surface of positive…

A: We will use the Gauss–Bonnet theorem to prove the given statement.

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

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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: 3. Find the curvature of the plane curve (t, t") at t = 1.

A: see below the answer and calculation as per bertleby guideline I have to solve only 1 question. For…

Q: 1) Calculate the curvature of the position vector r(t) = sin tāx + 2costāy + V3 sin tāz is a circle,…

A:

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

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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: 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: 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: 4) Find the curvature of y = cos x at the point (÷,²). Then find the equation of the osculating…

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Q: 15. Find the radius of curvature at 0 = -n of the parametric equations x = 3(0 + cos 0), y = 3(1 –…

A: For the parametric curve x=x(θ), y=y(θ)Curvature K=|x'y''-y'x''|[(x')2+(y')2]32Radius of…

Q: Find curvature of the parametric curve r(t) = (t, t3 – t, 3) at t = 1. Show your work.

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Q: 15. Find the radius of curvature at 0 = ÷n of the parametric equations x = 3(0 + cos 0), y = 3(1 —…

A: Given two parametric equations, x=3(θ+cosθ),y=3(1-cosθ) To find the radius of curvature at θ=π3

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: In Exercises 1–8, find the curve’s unit tangent vector. Also, find the length of the indicated…

A: You will say to solve the problem number 8.(8.) We will find the curve’s unit tangent vector. And…

Q: If r(t) = , then unit tangent is a principal unit normal is a and its curvature is a vector scalar

A: Explained below

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

A:

Q: Chapter 2 - Exercises 1) Calculate the curvature of the position vector f(t) = sin tāx + %3D 2cos…

A: Position vector is a Euclidean vector that represents the position of a point P in space.

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 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: Use Theorem 11.24 to prove that the curvature of a linearfunction y = mx + b is zero for every value…

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Q: 1) Calculate the curvature of the position vector r(t) = sintāx + 2costāy + V3 sin t āz is a circle,…

A: Given position vector is  r→t=sintax→+2costay→+3sintaz→ Then unit tangent vector is…

Q: A curve has a vector equation r = ti+ 4 sin(t)j + 4 cos(t)k. Find the curvature, K.

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Q: FInd the curvature of f(t)=ti+t^2j+t^3k at (2,4,8)

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

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: 7.) rCt) = (5+,3cot), 3sind), 7.1-> Find the arc length of the curve. 72-) Find the unit tangent and…

A: as per our company guidelines we are supposed to answer ?️only first 3️⃣ sub-parts. Kindly repost…

Q: 1) Calculate the curvature of the position vector F(t) = sin t ax + 2cost ay + V3 sin t āz is a…

A: Formula of Curvature(K) of the position vector r(t):…

Q: 1) Calculate the curvature of the position vector ŕ(t) = sin t āx+ %3D 2costay + v3 sin t āz is a…

A: To calculate the curvature of the position vector r¯t=sintax¯+2costay¯+3sintaz¯  is a cirlceAlso to…

Q: 7) Find the curvature of y = e* at the point (0, 1). Then find the equation of the osculating circle…

A: Given curve, y=ex

Q: Show that if M" has constant sectional curvature C then Ricci and scalar curvature of M" are Ric =…

A: We know that from given data: If M has a constant sectional curvature C , k. The Uniqueness of this…

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: Question 3: Compute Gaussian & mean curvature on: r(u, v) =u°cosvi+ u'sinvj+uk

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

A: Consider the given equation as x=acost, y=bsint and a>b>0

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> 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: 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.

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.