(b) cos(z) -dz, z2 + 2iz – 1 where y is the circle {|z + i| = 1} with counter clockwise orientation.

Q: Зл The parametric curve x = 7 sin 2t and y = 9 cos 2t, where , traces the ellipse 49 = 1 81 a. 3…

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Q: Find the moving trihedral and the curvature at any point of R(t) = (t²,– cos 8t, sin 8t).

A: Moving trihedral are three unit vectors of curve  Unit tangent vector  Unit normal vector  binomial…

Q: Consider the curve C of equation : (x²-2y+y²)² = x² + y². From the following options choose the one…

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Q: Show that the curve y = x? sin 3x touches the parabolas y = x andy =-x² alternately at infinite…

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Q: Find the equation of the plane perpendicular to the curve r(t) = (sin(2t), t, cos(2t)) at the point…

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Q: A curve is defined by the parametric equations a(t) = 2 cos(5t) and y(t) = -7 sin(-10t). Find .

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Q: Consider the plane curve given by the parametric equations x = t² + 3t - 26 y = t² + 3t - 28 - What…

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Q: 3.4.6 Use Green's theorem in a plane to evaluate the line integral 110²-35²) x + ( [(e¹-3y²) dx +…

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Q: • Graph the cardioid r a(1 + cose), a <0? • Graph the cardioid r = a(1+ sin6), a < 0? Graph the…

A: Graph are given below:

Q: 30. Show that the circle with its center at (÷,¿ ) in Figure 20 has polar equation r = sin 0 + cos 0…

A: In the question we have to find the polar equation of the circle and values of A ,B, C, D

Q: Show that 22 dz = 0 1 – cos(2) whenever y is a closed simple smooth curve contained in a very small…

A: Solution: Here the integral is ∮+γz21-cosz We know that    cos0=1cos2π=1cos3π=1... Only z=0 is in…

Q: Find a parametric equation for the path of a particle that moves halfway around the circle…

A: We have to find the parametric equation for the path of a particle.

Q: A Cartesian equation for the parametric curve x=24sin?e, y=21cos20, n ses 31/2 is given by O…

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Q: 27. Show that the curve with parametric equations x = t cos t, y =t sin t, z = t lies on the cone z²…

A: To Show: i. Show that the curve with the parametric equationx=tcost, y=tsint, z=t  lies on the…

Q: 37 <t< y2 = 1 81 The parametric curve x = 9 sin 2t and y = 9 cos 2t, where traces the ellipse 81 O…

A: Both x(t) and y(t) are periodic with period π. Taking the 5 points with an interval π/4.

Q: 32. (sin y, x) • dr, where C is the boundary of the triangle with vertices (0, 0), 0 ), and 2' 2

A:

Q: Sketch the plane curve r(t) = a cos3 ti + a sin3 tj and find its length over the given interval [0,…

A: The curve is r(t)=a cos3ti + a sin3tj   Interval is 0, 2π

Q: 2.2 Prove that e? dz] < 3 +++1 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

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Q: Prove that the curve y:(-,) → R³ is regular with factor H and determine whether the image y (t) = (2…

A: To determine : the      included in a plane in   or not. Here,…

Q: Given the surface of revolution with equation y? = 3.x. 3 3 Find a generating curve of the surface…

A: Solution :-

Q: Find the moving trihedral and the curvature at any point of R(t) = (3t2, sin 2t, cos 2t).

A: Introduction: The Frenet–Serret formulas in differential geometry describe the kinematic properties…

Q: The unit circle x² + y=1 can be wrriten in parametric form as x = cos(0) and y : = sino. dy and…

A: Follow the procedure given below

Q: Show that the arclength of the graph of y=sinx on the interval [0, 2n] is equal to the circumference…

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Q: (e) x = 4 – z²

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Consider the curves:r = 4sin 0 and r = 2 sin(0) + 2cos (0).

A:

Q: 3) Sketch a graph of the helix r(t) = (− cos(2ñt), 4 sin(2πt), 3t), listing the rotation direction,…

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Q: Find parametric equations for the circle x2 + y2 = a2, using as parameter the arc length s measured…

A: The given equation of circle is,

Q: 1. Find the moving trihedral and the curvature at any point of the curve R(t) = cos t î + sint ĵ+ k.

A:

Q: Evaluate o y dx – x³ dy, where C includes the two circles of radius 2 and radius 1 centered at the…

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Q: 2 Make a graph of the parametric equations x(t) = t - 3 on the interval 0sts 4 y(t) = ) = t ³ - 5 t

A: Given parametric equations  X(t) = t2-3  Y(t) =t3-5t

Q: The unit circle x² + y = 1 can be wrriten in parametric form as x = cos(0) and y = sin0. dy Find and…

A: The unit circle can be written in parametric form as x=cosθ and y=sinθ We have to find dydx and the…

Q: Scalar line integrals Evaluate the following line integral along the curve C.

A: Given:

Q: (c) (d) Determine the exact length of the curve of the conic section from 0 = 0 to 0 = π/6. Explain…

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Q: Find the exact value of arc length of the semicubical parabola y2 = x³ between the points (1,1) and…

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Q: ( Evalvate $ Cx-4)dx+Cx+y)dy where C is the circle centered at Co,0) and radius 2. a) 4T b) - qT c)…

A:

Q: A particle is moving on the curve defined parametrically by the system of equation x = 1- 2cos?t and…

A: Distance of particle from origin is:

Q: 3. Consider the curves: r 4sin e and r = 2 sin(8) + 2cos (0). =

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Q: (b) Find eiz dz- (z + i)3 Izl=1.5 (the circle is oriented counterclockwise).

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Q: ez 16. Evaluate dz, where: (4z-1) cos(Tz) a) I is the circle |z| = 1/8 traversed once…

A: We need to evaluate the complex integral in the following three regions represented by Γ,

Q: (a) æ = sin t, y = csc t, 0 1. Thus, the curve is the portion of the hyperbola y = 1/x with y > 1.…

A: Given x=sin ty=csc t0&lt;t&lt;π2

Q: Find the asymptotes of the curve defined by the parametric equationsx=φ(t)=1t−2,y=ψ(t)=tt−1.

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Q: Find the moving trihedral and the curvature at any point of R(t) = (t²,– cos 8t, sin 8t). -

A: Given is : R→(t)=t2,-cos(8t),sin(8t) In order to find the curvature use the formula . Formula for…

Q: 6. a. Find the length of the curve using the equation 12xy = 4x4 +3, from x = 1 to x = 3. b.…

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Q: If a projectile is fired with an initial 37. Path of a Projectile speed of t, ft/s at an angle a…

A: To find the path of the projectile motion, write t in terms of x and obtain t=xv0cosα.…

Q: t y = t + 1 II

A: Consider the given parametric equation: x=1tt=1xSubstitute the value of "t" in y=tt+1,y=1x1x+1y=11+x

Q: Prove that the function U(r, 0) = tan- ( ), 0<r<1,0 s0< 2n is harmonic inside the circle Iz| 1.…

A: Given Ur,θ=2πtan-12rsinθ1-r2, 0&lt;r&lt;1, 0≤θ&lt;2π---1. The Laplacian equation in polar…

Q: Sketch the curve r = 3 sin 20. a b Show that its Cartesian equation is (x² + y² )° =36x²y²,«

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Q: The unit circle x² + y=1 can be wrriten in parametric form as x = cos(0) and y = sin0. dy Find and…

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Q: 3. Find the exact length of the parametric curve x = sin-1t, y = In /1 – t2, 0<t<

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Q: Integrate counterclockwise dz by Cauchy's formula 9. - 3 counterclockwise around the circle C : |z –…

A: Given that integrate counterclockwise  ∮Cz2z2-9dz by Cauchy's formula where  C : |z-3-3i|=32π.…