4. Let r(s) be a unit speed curve with a curvature of zeroc fo. prove that the image r lies in a plane if and only if 3 there exist a point X, E IR such that every osculating plane passes through X, ·

Q: 5. Give an example of a curve C C xy plane so that its curvature as a space curve is different from…

A: The given: A curve  in the -plane defined by parametric equations and .The objective: To…

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

A:

Q: 2. Find the equation of the tangent plane to the surface z = 3++ at point (-4, 3).

A: Topic- application of derivatives  Tangent to surface

Q: The surface z = x² - y² and the surface ryz = -30 intersect along a curve in R³. Find a vector…

A:

Q: TRUE OR FALSE. A plane curve C defined by the parametric equations x = f(t) and y = g(t), where t is…

A: check the given statement is true or not.

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

A: Given: z=x2+y2

Q: Let P, Q be two points on the sphere of radius 1. Assume that P ≠ -Q. Show that there exists a curve…

A: Given P and Q be two points on the sphere of radius 1 P ≠ -Q. We need to show that there exists a…

Q: OShow that every point (b, c) in the plane lies on a tangent line to the graph of y = x³.

A: Consider the given information. y=x3 Find the derivative of the function. dydx=ddxx3y'=3x2

Q: 2.Proof that any tangent plane for the surface F(- point = 0 passses through a fixed

A: Given below clear explanation

Q: Let C be the curve in the x-y plane defined by the equation (x – h) + v – k)* =? where h, k, and r…

A: Equation of the curve is x-h2+y-k2=r2 & a, b is a point on the given curve. a). Slope of the…

Q: 1) Find the distance from the point (1, 2, 3) (a) XZ plane to the (b) Z axis

A: In this question we will use basic of geometry i.e concept of 3D geometry. We will first plot the…

Q: Suppose r(t) = i+t²j+2tk. Find the length of the portion of this curve that goes from the point G,…

A: We will find out the required value of length and expression for curvature.

Q: Let C be the portion of the parabola y = x², oriented from the starting point (0,0) towards the end…

A:

Q: 2. Find an equation of the tangent plane to the parametric surface F(u, v) = (, 2uv, uv?) at the…

A: (2) Consider the provided question,

Q: A Show that the curve F(t) = lies in a plane.

A: