1- (i) Find Frent formulas for a differentiable regular curve a = a(s), then show that at every point on a differentiable regular curve a = a(s) with non - zero Curvature k 3 3-orthogonal unit vectors T,N,B.
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: 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: 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: Two tangents AB & BC are intersected by a line having angles 40 o & 35 o respectively. The…
A: Consider the two tangents AB and BC which are intersected by a line that have angles 40o and 35o.…
Q: Find the curvature of the curve y = x^3 at the point (x, y) = (2, 8).
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Q: G1 is a semi-circle Y = √4 - X² starting at (2, zero) ends at (-2, zero) while G2 is a directed line…
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Q: 3. Find the curvature of the curve given by 7(t) = (t, t², t) nt (-1, 1, –1).
A: Given curve is We know that curvature to given curve at point (-1,1,-1) , that is at t=-1 is
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: 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: expresses the curvature k(x) of a twicedi¡erentiable plane curve y = ƒ(x) as a function of x. Find…
A: Given: y=ex ……1 where 1≤x<2 kx is the curvature function of a plane curve y=fx where fx is 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: 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: 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: The curve intersects the plane (in a three dimensional space) y+z = 2 at the point ( (Hint: First,…
A: Topic- intersection point of curve with Plane
Q: at a unit-speed regular parametrised curve 3: (-7, 7)→ R² with T, N) has constant curvature k> 0.…
A: Sol
Q: The curvature of r = 1+ cos 0, 0: 2 2/2 5 2/2 3/3 2 3/3 3.
A: Given: r=1+cosθ,θ=π2 κθ=2f'θ2+fθ2-fθf"θf'θ2+fθ232 r=f(θ)r=1+cosθ f'θ=-sinθf"θ=-cosθ
Q: 2. Find equations of the normal plane and tangent line to the curve of intersection of the surfaces…
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Q: Find the curvature of the function y=ln(3x)y=ln(3x) at x=3x=3. Find the radius of curvature at…
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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: The diagram shows a small block B, of mass 0.2kg, and a particle P, of mass 0.5kg, which are…
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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: 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: 4. Find the radius and center of curvature of the curve y=6x-x³ at the point (1,5).
A: This question is based on radius and centre of curvature formula.
Q: Consider the statements about the surface of equation: n²y z = y² sin x + and the point P (1, 2) in…
A: Directional derivative of a function f(x,y,) in the direction of unit vector u^ Dux,y=∇f.u^…
Q: Find the curvature function κ(x) for y = sin x. Use a computer algebra system to plot κ(x) for 0 ≤ x…
A: Extreme Points of sinx:Suppose that x=c is a critical point of fx then, If f 'x>0 to the left of…
Q: Find Frent formulas for a differentiable regular curve a = a(s), then show that at every point on a…
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Q: What is the radius of curvature at point (1, 2) of the curve 4x – y2 = 0.
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Q: G1 is a semi-circle Y: = √4 - X² starting at (2, zero) ends at (-2, zero) while G2 is a directed…
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Q: (6) Suppose the closed curve C in R? starts at the point (-1, 1), goes along the parabola y = r? to…
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Q: 1 4. Let f(=) =- s · Show that J (z)dz = 0 , where C is a simple closed contour not passing…
A: Moreras theorem
Q: 11) Example: of R' (t) to find the curvature Use the Theorem in 10 to
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Q: Find equations of the normal and osculating planes of the curve of intersection of the parabolic…
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Q: 15. y= 2x +3 at x = 27
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Q: 1 at the point (1,1). Find the curvature of the curve f (x) =
A: We need to find the curvature of the curve fx=1x at the point 1,1.
Q: The two surfaces x2 + y2 + z2 = 6 and 2x² + 3y2 + z2 = 9 intersect at the point(1,1,2). Find the…
A: Given: The two surfaces x 2 + y 2 + z 2 = 6 and 2 x 2 + 3 y 2 + z 2 = 9. The given two surfaces…
Q: 4. (a) Find the mid-point, unit tangent vector and unit normal vector at the mid-point of a planar…
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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: 2. Let a, b, and c be three nonzero constants. Show that the equation of the tangent plane to the…
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Q: 3/Find the equation for the tangent plane and normal line to the surface Z=9-x² + y² at the point…
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Q: 5. Suppose that we have a function f(x, y) and a point P = (a, b) in the domain where ) and = 0.…
A: Given that for the multivariable function f(x,y) , ∂f∂x=0 & ∂f∂y=0 at P=(a,b).
Q: 4) Find the families of lines of curvature of the surface :=r+y
A: The answer is given in handwritten format --
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: Determine the maximum curvature for the graph of f(x) = 9 In (8x). The maximum curvature is at x=
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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: 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 of the function y=ln(3x) at x=3x=3. Find the radius of curvature at x=3x=3.…
A: Given function is: y = ln(3x) First we will calculate the radius of curvature : Radius of curvature…
Q: Find the curvature K of the plane curve y 3 at x = 3. 2х + K =
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Q: If 7(t) = then the curvature at the point (2,0,0) is
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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: 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)
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- 1. The distance of a point in the 3-D system from the origin a. is defined by the absolute value of the vector from the origin to this point. b. is the square root of the square of the sums of the x-, y- and z-values. c. is the square root of the sum of the squares of x-, y- and z-values. d. can either be negative or positive. e. None of the above. 2. In parametrizing lines connected by two points in 3-D plane, a. there is only one correct parametrization. b. symmetry equations may not exist. c. a, b, and c must not be equal to 0. d. the vector that connects the two points is a scalar multiple of the vector containing the direction numbers. e. None of the above. 3. A plane in 3D-space system a. is generated by at least three points. b. can lie in more than one octant. c. must have a z-dimension. d. must have a point other than the origin. e. None of the above. 4. A quadric surface a. must have either x2, y2, or z2 or a combination of those, on its general expression. b. must have a…Find a vector equation for the curve of intersection between the surfaces y^2-x^2=9 and 2x-3y-z=12.Suppose that z is an implicit function of x and y in a neighborhood of the point P = (0, −3, 1) of the surface S of equation: exz + yz + 2 = 0 An equation for the tangent line to the surface S at the point P, in the direction of the vector w = (3, −2), corresponds to:
- The vector v = <a, 1, -1>, is tangent to the surface x2 + 2y3 - 3z2 = 3 at the point (2, 1, 1). Find a.Dierentiable curves with zero torsion lie in planes That a sufficiently di¡erentiable curve with zero torsion lies in a plane is a special case of the fact that a particle whose velocity remains perpendicular to a fixed vector C moves in a plane perpendicular to C. This, in turn, can be viewed as the following result. Suppose r(t) = ƒ(t)i + g(t)j + h(t)k is twice di¡erentiable for all t in an interval 3a, b4 , that r = 0 when t = a, and that v # k = 0 for all t in 3a, b4 . Show that h(t) = 0 for all t in 3a, b4 . (Hint: Start with a = d2r/dt2 and apply the initial conditions in reverse order.)Jackie and Christine are racing again! We only get some discrete data this time, but we can find the details through calculation! 3 minutes after the race starts, Jackie is 12 miles away from the starting point, and Christine is 27 miles ahead of Jackie. 13 minutes after the race starts, Jackie is 52 miles away from the starting point, but now Christine is only 17 miles ahead from Jackie. Assume they are going at a constant speed. (a) Present their movement on the xy-plane, where the x- and y-axis represent the time since the race starts and their distances from the starting point respectively. Label the four points clearly. (b) Find the speed of Christine and Jackie. You may use the unit “miles per minute” or, if you are comfortable with unit conversion, “miles per hour.” (c) Is anyone given a head start? If so, who is that and how far was that? (d) How far is Jackie from the starting point when he catches up with Christine?
- Steam is rushing from a boiler through a conical pipe, the diameters of the ends of which are D and d; if V and v be the corresponding velocities of the steam and if the motion be supposed to be that of divergence from the vertex of the cone,give the position vectors of particles moving alongvarious curves in the xy-plane. In each case, find the particle’s velocityand acceleration vectors at the stated times, and sketch them asvectors on the curve. Motion on the parabola y = x2 + 1r(t) = ti + (t2 + 1)j; t = -1, 0, and 1i) Show that the curve α is unit speed,ii) Find the Frenet Vector fields of the curve α.iii) Find the curvature and torsion of the curve α.iv) Find the equation of the curve α for the osculator plane at point α (5π/2).
- Where does the parametric line ( 2 + 2 t , t , t ) intersect the plane x+y-3z=4?Find the curvature of the functiony=ln(3x)y=ln(3x)at x=3x=3.Find the radius of curvature at x=3x=3. If the radius is infinite, enter 'oo' for ∞∞. Find the center of curvature at x=3x=3. If the center of curvature does not exist, enter "DNE".Hint if needed: Multiply the normal vector by the radius of curvature, and then add this vector quantity to the position vector. The components of the result will be the coordinates of the center of curvature. :)A particle traveling in a straight line is located at the point (1, -1, 2) and has speed 2 at time t = 0. The particle moves toward the point (3, 0, 3) with constant acceleration 2i + j + k. Find its position vector r(t) at time t.